Watson ATreatiseOnTheTheoryOfBesselFunctions Text

G.N.WATSON

G.N.WATSON

PRESS

A TREATISE ON

THE THEORY OF BESSEL FUNCTIONS G. N. Watson The late Professor G. N. Watson wrote his monumental treatise on the theory of Bessel functions with two objects in view. The first was the development of applications of the fundamental processes of the theory of complex variables, and the second the compilation of a collection of results of value to mathematicians and physicists who encounter Bessel functions in the course of their researches.

The completeness of the theoretical account, combined with the wide scope of the of practical examples and the extensive numerical tables, have resulted in a book which is indispensable to pure mathematicians, to applied mathematicians, and to physicists alike. collection

'In Professor Watson's treatise, which is a of erudition and its often too rare accompaniment, clear exposition, we have a rigorous mathematical treatment of all types of Bessel functions, their properties,

monument

representations, asymptotic expansions, integrals containing them, allied functions, series, zeros, tabulation, together integral

with extensive numerical tables.' L. M. Milne-Thomson in Nature

'A veritable mine of information. pensable to all those use Bessel functions.' S.

who have

.

.indis-

occasion to

Chandrasekhar in TheAstrophysical Journal

Also available as a paperback

THEORY OF BESSEL FUNCTIONS

W.

B. F.

A TREATISE ON THE

THEORY OF BESSEL FUNCTIONS BY G. N.

WATSON

SECOND EDITION

CAMBRIDGE AT THE UNIVERSITY PRESS 1966

PUBLISHED BY THE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS Bentley House, 200 Euston Road, London, N.W. 1 American Branch: 32 East 57th Street, New York, N.Y. 10022 West African Office: P.M.B. 5181, Ibadan, Nigeria

First Edition Second Edition Reprinted

1922 1944 1952 1958 1962 1966

First paperback edition

/

1966

~--fcU^6\\*s

First printed in Great Britain at the University Press, Cambridge

Reprinted by lithography in Great Britain by & Viney Ltd, Aylesbury, Bucks

Hazell Watson

PREFACE THIS

book has been designed with two objects in view. The first is the development of applications of the fundamental processes of the theory of

functions of complex variables.

For this purpose Bessel functions are admirably adapted; while they offer at the same time a rather wider scope for the application of parts of the theory of functions of a real variable than

is

provided by

trigonometrical functions in the theory of Fourier series.

The second object is the compilation of a collection of results which would be of value to the increasing number of Mathematicians and Physicists who encounter Bessel functions in the course of their researches. The existence of such a collection seems to be demanded by the greater abstruseness of properties of Bessel functions (especially of functions of large order) which have been

required in recent years in various problems of Mathematical Physics.

While my endeavour has been to give an account of the theory of Bessel functions which a Pure Mathematician would regard as fairly complete, I have consequently also endeavoured to include

all

formulae, whether general or

although without theoretical interest, are likely to be required in practical applications; and such results are given, so. far as possible, in a form appropriate for these purposes. The breadth of these aims, combined

special, which,

with the necessity for keeping the size of the book within bounds, has made it necessary to be as concise as is compatible with intelligibility. Since the book

is, for the most part, a development of the theory of funcexpounded in the Course of Modern Analysis by Professor Whittaker and myself, it has been convenient to regard that treatise as a standard work of reference for general theorems, rather than to refer the reader to original

tions as

sources.

draw attention here to the function which I have regarded namelt the function which was defined by Weber and used subsequently by Schlafli, by Graf and Gubler and by Nielsen. For historical and sentimental reasons it would have been pleasing to have felt justified in using Hankel's function of the second kind; but three It is desirable to

as the canonical function of the second kind,

considerations prevented this.

The

first is

function of the second kind; and, in

my

the necessity for standardizing the

opinion, the authority of the group

of mathematicians who use Weber's function has greater weight than the authority of the mathematicians who use any other one function of the second kind. The second is the parallelism which the use of Weber's function exhibits

between the two kinds of Bessel functions and the two kinds (cosine and

sine)

PREFACE

VI

The third is the existence of the device by which made is possible in Tables I and III at the end of Chapter XX, which seems to make the use of Weber's function inevitable in numerical work.

of trigonometrical functions. interpolation

It has been my policy to give, in connexion with each section, references any memoirs or treatises in which the results of the section have been previously enunciated; but it is not to be inferred that proofs given in this book are necessarily those given in any of the sources cited. The bibliography at the end of the book has been made as complete as possible, though doubtless omissions will be found in it. While I do not profess to have inserted every memoir in which Bessel functions are mentioned, I have not consciously omitted any memoir containing an original contribution, however slight to the theory to

of the functions; with regard to the related topic of Riccati's equation, I have

been eclectic to the extent of inserting only those memoirs which seemed to be relevant to the general scheme. In the case of an analytical treatise such as this, it is probably useless to hope that no mistakes, clerical or other, have remained undetected; but the

number

of such mistakes has been considerably diminished by the criticisms and the vigilance of my colleagues Mr C. T. Preece and Mr T. A. Lumsden, whose labours to remove errors and obscurities have been of the greatest value. To these gentlemen and to the staff of the University Press, who have given every assistance, with unfailing patience, in a work of great typographical

complexity, I offer

my

grateful thanks.

G. N. August

W.

21, 1922.

PREFACE TO THE SECOND EDITION To

incorporate in this work the discoveries of the last twenty years would

necessitate the rewriting of at least Chapters

Bessel functions, however, has

waned

since 1922,

XII

—XIX;

my interest in am consequently not of my other activities.

and I

prepared to undertake such a task to the detriment In the preparation of this new edition I have therefore limited myself to the correction of minor errors and misprints and to the emendation of a few assertions (such as those about the unproven character of Bourget's hypothesis) which, though they may have been true in 1922, would have been

had they been made in 1941. thanks are due to many friends for their kindness in informing me of errors which they had noticed; in particular, I cannot miss this opportunity of expressing my gratitude to Professor J. R. Wilton for the vigilance which definitely false

My

he must have exercised in the compilation of his

list

of corrigenda. G. N.

March

31, 1941.

W.

CONTENTS CHAP. I.

PAGE

BESSEL FUNCTIONS BEFORE

1826

1

THE BESSEL COEFFICIENTS

H

III.

BESSEL FUNCTIONS

38

IV.

DIFFERENTIAL EQUATIONS

85

II.

V.

MISCELLANEOUS PROPERTIES OF BESSEL FUNCTIONS

132

VI.

INTEGRAL REPRESENTATIONS OF BESSEL FUNCTIONS

160

ASYMPTOTIC EXPANSIONS OF BESSEL FUNCTIONS

194

VII.

VIII. IX.

X.

BESSEL FUNCTIONS OF LARGE ORDER

.

....

225

POLYNOMIALS ASSOCIATED WITH BESSEL FUNCTIONS

271

FUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS

308

.

XI.

ADDITION THEOREMS

358

XII.

DEFINITE INTEGRALS

373

XIII.

INFINITE INTEGRALS

383

XIV.

MULTIPLE INTEGRALS

450

XV. XVI.

XVII.

THE ZEROS OF BESSEL FUNCTIONS

NEUMANN

XX.

477

AND LOMMEL'S FUNCTIONS OF TWO

VARIABLES

522

KAPTEYN SERIES

551

XVIII. SERIES

XIX.

SERIES

.

OF FOURIER-BESSEL AND DINI

576

.

SCHLOMILCH SERIES

618

THE TABULATION OF BESSEL FUNCTIONS

.

654

TABLES OF BESSEL FUNCTIONS

665

BIBLIOGRAPHY

753

INDEX OF SYMBOLS

789

LIST OF

AUTHORS QUOTED

GENERAL INDEX

....

791

796

To stand upon every

point,

and go over things at

large,

particulars, belongeth to the first author of the story

and avoid much labouring of the work,

is

:

and to be curious

to be granted to

him that

make an abridgement. 2

in

but to use brevity,

Maccabees

ii.

30, 31.

will

CHAPTER

I

BESSEL FUNCTIONS BEFORE 1*1.

1826

Riccati's differential equation.

The theory of Bessel functions is intimately connected with the theory of a certain type of differential equation of the first order, known as Riccati's equation. In fact a Bessel function is usually defined as a particular solution of a linear differential equation of the second order (known as Bessel's equation)

which

derived from Riccati's equation by an elementary transformation.

is

The

earliest

appearance in Analysis of an equation of Riccati's type occurs

by John Bernoulli in 1694. In paper Bernoulli gives, as an example, an equation of this type and states that he has not solved it"f\

in a paper* on curves which was published this

In various letters! to Leibniz, written between 1697 and 1704, James Bernoulli refers to the equation, which he gives in the form

dy = yydx + xxdx, and

states,

1697): "

Ego

Thus he writes (Jan. 27, dy — yydx + xxdx. meam operam improbum sed Problema performas transmutavi,

more than

once, his inability to solve

Vellem porro ex Te

in mille

scire

num

it.

et hanc tentaveris

Five years later he succeeded in reducing the equation to a linear equation of the second order and wrote§ to Leibniz (Nov. 15, 1702) " Qua

petuo

lusit."

:

occasione recordor aequationes alias memoratae

quam

separare potui indeterminatas a se

simpliciter differentialis

:

differentio-differentialem||

When

sed separavi

illas

dy — yydx + a?dx

qua nuninvicem, sicut aequatio maneret reducendo aequationem ad hanc in

ddy.y — — x& dx*."

had been made, it was a simple step to solve the last and so to obtain the solution of the equation of the first order as the quotient of two power-series. this discovery

equation in

series,

—

* Acta Eruditorum publicata Lipsiae, 1694, pp. 435 437. i f "E8to proposita aequatio differentialis haec x dx + y*dx = a?dy quae an per separationem indeterminatarum construi possit nondum tentavi " (p. 436). t See Leibnizens gesamellte Werki, Dritte Folge (Mathematik), in. (Halle, 1855), pp. 50—87. § Ibid. p. 65. Bernoulli's procedure was, effectively, to take a new variable u defined by the

formula 1 du u dx

in the equation il

dyldx=x'2 + y 2 and then

The connexion between

in §4-3.

,

_ ~*

to replace

this equation

and a

« by

y.

special form of Bessel's equation will be seen

,


2

And, in

[CHAP.

I

form of the solution was communicated to Leibniz by (Oct. 3, 1703) in the following terms*:

fact, this

James Bernoulli within a year

"Reduco autem aequationem dy= yydx+xxdx ad fractionem cujus uterque terminus per seriem exprimitur,

X

3

V ,

X*

XVa 3.

4.7.8.

11

X19 .

12.

15^3.

X 12 3.4.7.8.11.12

X*

T

3.4

quae

X 3.4.7.8.11

X 3.4.7

3

ita

11

7

3.4.7.8

x 3.4.7.8.11.12.15.16 lti

T

quidem actuali divisione in unam non tarn facile patescat, scil.

series

4. 7.8. 11. 12. 15. 16. 19

conflari possunt, sed in

qua

ratio progressionis

_a? y

Of

~

3

+

x1

3.3. 7

+

2x11

373. 3. 7. 11

course, at that time,

+

13-r15

3.3.3.3.5. 7777II

+

mathematicians concentrated their energy, so

far

as differential equations were concerned, on obtaining solutions in finite terms,

and consequently James Bernoulli seems to have received hardly the full credit which his discovery entitled him. Thus, twenty-two years later, the paperf in which Count Riccati first referred to an equation of the type which now bears his name, was followed by a note:,: by Daniel Bernoulli in which it was to

stated that the solution of the equation§

ax dx + uudx = bdu 11

was a hitherto unsolved problem. The note ended with an announcement in an anagram of the solution " Solutio problematis ab 111. Riccato proposito characteribus occultis involuta 24a, 6b, 6c, Sd, 33e, hf, 2g, 4>h, 33i, 61, 21m, :

26n, 16o, Sp, oq, I7r, 16s, 25t, 32m, 5#, By, +, -,

The anagram appears never his solution

||

to

have been solved

4, 2, 1."

but Bernoulli published

;

of the problem about a year after the publication of the anagram.

— 4m/(27/i ±1),

where

m

is

any

namely any one of which the equation is solution will be given in §§ 4*1, 4*11.

integer, for

soluble in finite terms; the details of this

The prominence given

work of Riccati by Daniel Bernoulli, combined equation was of a slightly more general type than

to the

with the fact that Riccati's

See Leibnizens gesamellte Werke, Dritte Folge (Mathematik),

*

t Acta Eruditorum, Suppl. equation was

where

q

= xn

vm.

(1724), pp.

66—73.

m.

The form

(Halle, 1855), p. 75. in

which Riccati took the

xmdq = du + uu dx:q,

.

—

75. Daniel Bernoulli mentioned that solutions had been obtained by three members of his family John, Nicholas and the younger Nicholas. The reader should observe that the substitution

% Ibid. pp. 73

other

—

u=

—bdz— z

dx

an equation which is easily soluble in series. Exercitationes quaedam matheviaticae (Venice, 1724), pp. 77 pp. 465—473. gives rise to |j

±, =,

solution consists of the determination of a set of values of n,

The

§

,

—80;

Acta Eruditorum, 1725,

BESSEL FUNCTIONS BEFORE 1826

1*2]

3

John Bernoulli's equation* has resulted in the name of Riccati being associated not only with the equation which he discussed without solving, but also with

a

still

more general type of equation.

It is now customary to give the namef any equation of the form

where P, Q, It is if

P=0,

R are given

Riccati's generalised equation to

functions of x.

supposed that neither P nor R is identically zero. If U=0, the equation is linear; the equation is reducible to the linear form by taking 1/y as a new variable.

The

equation was studied by Euler J

last

and

linear equation of the second order,

;

it is

reducible to the general

this equation is

by an elementary transformation

to Bessel's equation

sometimes reducible

(cf. §§ 3*1, 4*3, 4'31).

Mention should be made here of two memoirs by Euler. In the first § it is proved that, when a particular integral y x of Riccati's generalised equation is known, the equation is reducible to a linear equation of the first order by replacing y by y x + 1/u, and so the general solution can be effected by two quadratures. It is also shewn (ibid. p. 59) that, if two particular solutions are known, the equation can be integrated completely by a single quadrature; and this result is also to be found in the second of the two papers. A brief discussion of these theorems will be given in ChaDter iv. ||

1*2.

Daniel Bernoulli s mechanical problem.

In 1738 Daniel Bernoulli published a memoir! containing enunciations of a number of theorems on the oscillations of heavy chains. The eighth ** of these

is

as follows

:

"

Defigura catenae uniformiter oscillantis. Sit catena AG flexilis suspensa de puncto A, eaque oscillationes

uniformiter gravis et perfecte

facere uniformes intelligatur: pervenerit catena in

longitudo catenae valorisff ut

= 1:

longitudo cujuscunque partis

si turn

AMF;

fueritque

FM — x, sumatur n

ejus

fit /

n

Jl + 4ww

P_

4 9n» .

+

l^_

I

s '

4.9.16. 25ns

4 9 16n4 .

.

—

See James Bernoulli, Opera Omnia, n. (Geneva, 1744), pp. 1054 1057 it is stated that the is the determination of a solution in finite terms, and a solution which resembles the solution by Daniel Bernoulli is given. t The term Hiccati's equation was used by D'Alembert, Hist, de I' Acad. R. de» Set. de Berlin, *

;

point of Riccati's problem

'

*

xix. (1763), [published 1770], p. 242.

X Iiutitutiones Calculi Integralis, n. (Petersburg, 1769), § 831, pp. 88—89. the reduction, see James Bernoulli's letter to Leibniz already quoted.

In connexion with

Novi Coram. Acad. Petrop. vni. (1760—1761), [published 1763], p. 32. 163—164. If " Theoremata de osoillationibus corporum filo flexili connexorum et catenae verticaliter suspensae," Comm. Acad. Set. Imp. Petrop. vi. (1732 3), [published 1738], pp. 108 122. § ||

Ibid. ix. (1762—1763), [published 1764], pp.

—

** Loc.

cit: p.

116.

ft The length of the simple equivalent pendulum

is n.

—


4

[CHAP.

I

Ponatur porro distantia extremi puncti F ah linea'verticali = 1, dico fore ab eadem linea verticali aequalem distantiam puncti ubicunque assumpti

M

x n

1

He

xx

<*

4>nn

4.9w3

+

i

**

**

4.9.16n 4

4.9.16.25w8

goes on to says "Invenitur brevissimo calculo n

Habet autem

littera

n

f ctc

»

= proxime 0691

I....

infinitos valores alios."

The last series is now described as a Bessel function * of order zero and argument 2 V(#/n); and the last quotation states that this function has an infinite number of zeros. Bernoulli published f proofs of his theorems soon afterwards; in theorem he obtained the equation of motion by considering the forces acting on

viii,

FM

the portion

Euler \

of length

many years

later

The equation

x.

of motion was also obtained

by

from a consideration of the forces acting on an element

of the chain.

The

following

is

the substance of Euler's investigation

:

Let p be the line density of the chain (supposed uniform) and let T be the tension at height x above the lowest point of the chain in its undisturbed position. The motion being transversal, we obtain the equation bT=gpbx by resolving vertically for an element of chain of length bx. The integral of the equation is T—gpx.

The

horizontal component of the tension

zontal) displacement of the element

;

is, effectively, T{dyjdx) where y and so the equation of motion is

is

the (hori-

'*§-»( r 2)we

If

substitute for

T and

proceed to the limit, we find that

dt*

/

If

is

y dx\ dxj

'

the length of the simple equivalent pendulum for any one normal vibration,

we

write

A and

where

f are constants

;

and then

n (x/f) d

A

-\ ( dx\ dx) If

x/f= ii, we obtain the solution

in the

u ul v= l — H 1

*

On

a solution of the equation

+- = f

form of Bernoulli's

series,

namely

+— u* 1.4.9 1.4.9.16 —v?

the Continent, the functions are usually called cylinder functions, or, occasionally, func-

tions of Fourier -Bessel, after Heine,

m.

1.4

is

(1871), pp.

Journal fur Math. lxix. (1868), p. 128; see also Math. Ann.

609—610.

t Comm. Acad. Petrop. vn. (1734—5), [published 1740], pp. 162—179. X Acta Acad. Petrop. v. pars 1 (Mathematics), (1781), [published 1784], pp. 157—177. Euler took the weight of length e of the chain to be E, and he denned g to be the measure of the distance (not twice the distance) fallen by a particle from rest under gravity in a second. Euler's

notation has been followed in the text apart from the significance of g and the introduction of p

and

5 (for d).

~


1*3]

1826

5

—

.

D are constants. If a is the

Since

y

is finite

when #=0, C must be

,

where

C and

zero.

whole length of the chain, y =0 when .» = a, and so the equation to determine/* is a% a3 _ 1

"

—j. + 1

TT4/2 ~

./

1

4. 9/3

.

a

""••'

By an extremely ingenious analysis, which will be given fully in Chapter xv, Euler proceeded to shew that the three smallest roots of the equation in a/f are 1-445795, 7*6658 and 18-63. [More accurate values are 1*4457965, 7*6178156 and 18*7217517.] In

'the

memoir'"' immediately following this investigation Euler obtained the general

solution (in the form of series) of the equation

-j-

(u-r-\+v—0, but

his statement of the

law of formation of successive coefficients is rather incomplete. The law of formation had, however, been stated in his Institutiones Calculi Integrality, n. (Petersburg, 1769), § 977, pp. 233-235.

Euler 's mechanical problem.

1*3.

The

membrane were

vibrations of a stretched

He

1764.

ld?z_d?z e*

where z

is

dt2

and tension of the membrane. (r,

);

To obtain a normal

e is

a,

fi,

B

are constants is

=

?-P

\l

—

2(w+l)e

2

This differential equation

Save

#

for

and

a function of r; and the result of

p\

/a8

is finite

at the origin

+ 2.4(w+l)(n

where n has been written§ in place of

;

is

is

given on

p.

256

it is

{

of order

at the point whose polar

the differential equation

solution of this equation which

w

t

'

+ A) sin (/9<£ + B),

and u

*w,l^w, of Euler's memoir;

3

solution he wrote

substitution of this value of z

The

d*z

1

a constant depending on the density and

z = u sin (at

A,

Idz

~ dr* + rdr* r'cty

the transverse displacement at time

coordinates are

where

investigated by Euler}: in

arrived at the equation

£ may

have

is ||

as Bessel's equation for functions

any of the values

0, 1, 2,

factor the series arje.

,

+ 1.

now known

an omitted constant and argument

coefficient of order /S

2/3

+ 3)e4 ~•••)

The

is

. .

..

now

called a Bessel

periods of vibration, 2irja, of a

* Acta Acad. Petrqp. v. pars 1 (Mathematics), (1781), [published 1784], pp. 178—190. t See also §§ 935, 936 (p. 187 et seq.) for the solution of an associated equation which will be

discussed in § 3*52.

X Novi Comm. Acad. Petrop. x. (1764), [published 1766], pp. 243—260. § The reason why Euler made this change of notation is not obvious. ||

If

/3

were not an integer, the displacement would not be a one-valued function of position,

in view of the factor sin

{p

+ B).


6

[CHAP.

I

membrane of radius a with a fixed boundary* are to be determined from the consideration that u vanishes when r = a.

circular

This investigation by Euler contains the earliest appearance in Analysis of a Bessel coefficient of general integral order. 1*4.

The researches of Lagrange, Carlini and Laplace.

Only a few year* after Euler had arrived at the general Bessel coefficient on vibrating membranes, the functions reappeared, in an astronomical problem. It was shewn by Lagrangef in 1770 that, in the elliptic motion of a planet about the sun at the focus attracting according to the law of the inverse square, the relations between the radius vector r, the mean anomaly and the eccentric anomaly E, which assume the forms in his researches

M

M= E-

e sin

E,

= a(l-e cos E),

r

give rise to the expansions

«

Z A E=M+ n=l in

which a and »

.

n

_9 ~ "m=o

e

00

j,

n

-=l + he + 2 B H cosnM, & 2

sinnM,

n=\

are the semi-major axis and the eccentricity of the orbit,

(-)"» M -n«-i 6 n+»m ti

2«+"*"ro (n !

+

w)'!

n

^ ~~

'

Lagrange gave these expressions

for

=

n

» (_)m H + £ ( ^ l+2 »t

mt

1, 2, 3.

to obtain expressions for the eccentric

is

2'

The

m

)

,

n-t-m-i ro

and

n+2m e

w!(w + m)!

object of the expansions

anomaly and the radius vector

in

terms of the time. In modern notation these formulae are written

A n -2Jn {tu)/n, B n = -2(e[n)Jn '(m). It

was noted by Poisson, Connaissance des j?

Terns, f

1836 [published 1833],

p.

6 that

dA n

n at a

memoir by

Poisson

is

Lefort, Journal de Math. xi. (1846), pp. 142

— 152, in which an error made by

corrected, should also be consulted.

A remarkable investigation of the approximate value of A n when n is large and < e < 1 is due to Carlini J: though the analysis is nob rigorous (and it would be difficult to make it rigorous) it is of sufficient interest for a brief account of

it

to

* Cf. Bourget,

be given here.

Ann. Sci. de VEcole norm. sup.

in. (1866), pp.

55—95, and Chree, Quarterly

Journal, xxi. (1886), p. 298. t Hist, de VAcad. R. des Sci. de Berlin, xxv. (1769), [published 17711, pp. in. (1869), pp.

204—233. [Oeuvres,

113—138.]

X Ricerche sulla convergenza delta serie che serva alia soluzione del problema di Keplero This work was translated into German by Jacobi, Astr. Nach. xxx. (1850),

(Milan, 1817).

197—254 [Werke, yii. (1891), pp. 189—245]. See also two papers by Scheibner dated 1856, reprinted in Math. Ann. xvn. (1880), pp. 531—544, 545—560. col.


14]

shew that

It is easy to

An c

2

^

If u —

(n a

is

/udt e /n

i

!

2

(l-«2 ) = 0.

u or u2 or du/dt must be

large either

equation

and then

(^+«2 ) + «tt-«

«2

differential

+ <^-"-» 2 (l-e 2 )J B ==0.

A n =2nn -

Define n by the formula

Hence when n

a solution of the

is

7

1826

large.

(n a ) respectively and and du/dt to l>e (n 2a ) and on considering the highest powers of n in the various terms of the last differential equation, we find that a= 1. It is consequently assumed that u admits of an expansion in descending powers of n in the form

where

?<

On /-ero

we should expect

)

u lt u 2

,

;

are independent of n.

...

,

u-

substituting this series in the differential equation of the

the coefficients of the various powers of n, 2

«o

where v n'=du

/dt

;

so that

= (l- f 2 )/*2

«

fl

we

«« + 2«o«i)+Mo=0,

,

= +—-

,

«!

=

2,

..

judt-nôgj^—f2)±vA(1-e

2 )

formula

it

now

...

+ l}-ilog(l- f 2 ) + ...,

A n shews that Jude ~ n log ê when be taken and no constant of integration is to be added. Stirling's

order and equating to

and therefore

and, since the value of

From

first

find that

« is

small, the upper sign

must

follows at once that

n exp

f

B v/(^

and

this is the result obtained

much

further

formulae for

which

An

Carlini's

The

by Meissel

he uses

This method of approximation has been carried Cauchy* has also discussed approximate

comets moving in nearly parabolic orbits (see §

in the case of

is

is

much more

arguments employed by Laplace + to establish the for

investigation given

all

8"42), for

obviously inadequate.

investigation of which an account has just been given is

sponding approximation

taking

Carlini.

'

(see § 8*11), while

approximation

plausible than the

The

by

M

{nj{\- t2)J *(i-<2 )*{i+N/(i-«2 )} n

of considerable

Bn

corre-

.

by Laplace

is

method which modified by alternatively, by

quite rigorous and the

importance when

the value of

the coefficients in the series to be positive

—

Bn

or,

is

e is a pure imaginary. But Laplace goes on to argue that an approximation established in the case of purely imaginary variables may be used sans crainte in the case of real variables. To anyone who is acquainted

supposing that '

with the

'

modem

theory of asymptotic series, the fallacious character of such

reasoning will be evident. * Comptes Rendus,

xxxvm.

(1854), pp.

f Mecanique Celeste, supplement, pp. 486—489.

t.

990—993. v.

[first

published 1827].

Oeuvres, v. (Paris, 1882),


8

The

earlier portion of Laplace's investigation is based

that, in the case of a series of positive

crease

up

may

of the series

I

on the principle

terms in which the terms steadily in-

and then steadily decrease, the order of magnitude

to a certain point

sum

of the

[CHAP.

frequently be obtained from a consideration of

the order of magnitude of the greatest term of the

series.

For other and more recent applications of this principle, see Stokes, Proc. Camb. Phil. 362—366 [Math, and Phys. Papers, v. (1905), pp. 221—225], and Hardy, Proc. London Math. Soc. (2) n. (1905), pp. 332—339 Messenger, xxxiv. (1905)., pp. 97—101. A statement of the principle was given by Borel, Acta Mathematica, xx. (1897), pp. 393 Soc. vi. (1889), pp.

;

394.

The

following exposition of the principle applied to the example considered

by Laplace may not be without The

interest

series considered is

iL<" = 2

(n

2 ,„=o

in

which n

is

m—fi, where

large /u

is

and

t.

fi

is

The

has a fixed positive value.

greatest term

(n + 2/i

+ /x)

- 2) ^ (n + 2M ) n2 e2

if

is

that for which

,

approximately equal to

i«W(l + «2 )-l} + *e2/(l+« 2 Now,

'

the greatest integer such that 4/x (n

and so

+ 2m) nn + 2m ~ 2 t n + *>" 2» + 2m m\{n + m)\

u m denotes the general term in BJ~ l \

that, to a first approximation, -H^<**>q&,

U

"

Hence

.BB< 1

it is

).

easy to verify by Stirling's theorem

where

=-2V(l + *2 )/(»e 2 )~M {1 + 2q + 2q* + 2 q9 + log ? f

).

. .

,

.}

~2ttMX/{7r/(l-?)}, since* q

is

nearly equal to

Now, by

1.

theorem,

Stirling's

6"- 1 exp {ws/(l 7T% 2

and

BJ l)r

so

f

\

The

2 v'(l

inference which Laplace

R

jD,l

/ 2V(1

~_ V

+ c2 ))^* )

nn3

e»

-

**) \*

J

'

WU

+ « 2 )} + ^(1 + ,')}»

exp {1

J

drew from

ttw*

+ g2 )|

{l+ x/(l+« 2 )}» {

*

this result is that

6"expW(l-e )} 2

{l

+ V(l-e )} 2

n

'

This approximate formula happens to be valid when e < 1 (though the reason for this restriction is not apparent, apart from the fact that it is obviously necessary), but it is difficult to prove it without using the methods of contour

•The cf.

formula 1 + 2 2
Bromwieh, Theory of

formula in the theory of

2

~ N/W(l-g)} maT

Infinite Series, § 51.

Modern Analysis,

a consequence of Jacobi's transformation

elliptic functions,

*3 see

be inferred from general theorems on series;

It is also

§ 21*51.

(0 \r)

= (-*)-* M0 -TI

1

);

;


1*5] integration

(cf. § 8*31).

1826

9

Laplace seems to have been dubious as to the validity

and

of his inference because, immediately after his statement about real

imaginary variables, he mentioned, by way of confirmation, that he had another proof; but the latter proof does not appear to be extant.

The researches of Fourier. In 1822 appeared the classical treatise by Fourier*, 1"5.

La TMorie analytique

de la Chaleur; in this work Bessel functions of order zero occur in the discussion of the symmetrical motion of heat in a solid circular cylinder.

—

shewn by Fourier (§§ 118 120) that the temperature x from the axis of the cylinder, satisfies the equation

v,

at time

t,

It is

at distance

dv_K_/d?v lcfoA Jt~lW\dtf + xdx)' where K, G, D denote respectively the Thermal Conductivity, Specific Heat and Density of the material of the cylinder; and he obtained the solution v

— e~

gx*

tf<*

22

2 2 .42

f

\

where g

= rnCD/K and

ni

3

x* \ ')

'

has to be so chosen that

+ K(dvldx) =

hv at the

g

2 2 .42 .6 2

boundary of the cylinder, where h

is

the External Conductivity.

—309) by

theorem that and no complex roots. His proof is slightly incomplete because he assumes that certain theorems which have been proved for polynomials are true of integral functions; the defect is not difficult to remedy, and a memoir by HurwitzJ Fourier proceeded to give a proof

(§§

the equation to determine the values of

307

Rolle's

m hasf an infinity of

real roots

has the object of making Fourier's demonstration quite rigorous. It should also be

mentioned that Fourier discovered the continued fraction the quotient of a Bessel function of order zero and its

formula (§ 313) for dervate; generalisations of this formula Another formula given by Fourier, namely 22

2 2 42 .

~

2 2 42 6 2 .

= —1

+

be discussed in

will

§§ 5-6, 9*65.

f* |

cos (a sin x) dx,

.

had been proved some years earlier by Parseval§; are now known as Bessel's and Poisson's integrals

it is

a special case of what

22, 23). memoir deposited in the archives of the French Institute on Sept. 28, 1811, and crowned on Jan. 6, 1812. This memoir is to be found in the Mim. de VAcad. des Sci., iv. (1819), [published 1824], pp. 185—555; v. (1820), *

The

(§§

greater part of Fourier's researches was contained in a

[published 1826], pp. 153—246. t This is a generalisation of Bernoulli's statement quoted in § 1*2. t Math. Ann. xxxin. (1889), pp. 246—266.

—

648. This paper also contains the formal § M4m. des savant itrangers, i. (1805), pp. 639 statement of the theorem on Fourier constants which is sometimes called Parseval's theorem another paper by this little known writer, Mim. des savans Strangers, i. (1805), pp. 379 398, con-

—

tains a general solution of Laplace's equation in a form involving arbitrary functions.


10

[CHAP.

I

The expansion of an arbitrary function into a series of Bessel functions of order zero was also examined by Fourier (§§ 314 320); he gave the formula for the general coefficient in the expansion as a definite integral.

—

The

was examined much more recently by Hankel, 471—494; Schlafli, Math. Ann. x. (1876), pp. 137—142; Diui, Serie di Fourier, i. (Pisa, 1880), pp. 246—269 Hobson, Proc. London Math. Soc. (2) vn. (1909), pp. 359—388; and Young, Proc. London Math. Soc. (2) xvni. (1920), pp. 163—200. validity of Fourier's expansion

vm.

Math. Ann.

(1875), pp.

;

This expansion will be dealt with in Chapter xvni.

The researches of Poisson.

1/6.

The unsymmetrical motions of heat in a solid sphere and also in a solid cylinder were investigated by Poisson* in a lengthy memoir published in 1823. In the problem of the sphere f, he obtained the equation

where r denotes the distance from the centre, p

R is that

integer (zero included), and

is

a constant, n

is

a positive

normal was shewn by Poisson that

factor of the temperature, in a

mode, which is a function of the radius vector. a solution of the equation is cos (rp cos

a>)

It

sin2n+1 coda

Jo

and he discussed the cases

?i

= 0,

(§ 3*3)

that the definite integral

order n

+ ^.

1,

In the problem of the cylinder

V

It will appear subsequently

2 in detail.

is

(save for a factor) a Bessel function of

(ibid. p.

cos (h\ cos

340

co)

et seq.)

the analogous integral

is

sin2n G>cifo>,

.'o

where

w= 0,

In the case n

and is

its

and \

1, 2, ...

now known

integral is

— 0,

is

the distance from the axis of the cylinder.

as Poisson's integral (§

The

2'3).

an important approximate formula for the last integral (ibid., pp. 350 352) when the variable

—

derivate was obtained by Poisson

large; the following is the substance of his investigation:

Let J

J

(k)

=it

Then J

(k) is

I

cos (i cos ») da,

J

'

(k)=

/

ir

J

cos

a>

sin (k cos «) dm.

J

a solution of the equation

*&ffl + 1+ i)„,»_a ( * Journal de I'ficole JR. Polytechnique, xn. (cahier 19), (1823), pp.

249—403. 300 etseq. The equation was also studied by Plana, Mem. della R. Accad. delle Sei. di Torino, xxv. (1821), pp. 532 534, and has since been studied by numerous writers, some of whom are mentioned in § 4*3. See also Poisson, La TMorie Mathimatique de la Chaleur (Paris, + Ibid.

p.

—

1835), pp. 366, 369.

t See also Rohrs, Proc. London Math. Soc. not used by Poisson.

v. (1874), pp.

136

— 187.

The notation J

(k)

was

;


1*6]

When k is

may

large, l/(4£ 2 )

that Jo{k)sJk

be neglected in comparison with unity and so

To determine A and B observe cos k

J

.

Write

(k)

ir

- sin k

- to

A

approximately of the form

is

for

a

cos k

«/

.

'

(£)

=-

J

(k)

{cos 2 ba> cos (2k sin 2 £a>)

I

"J

- sin k

sin £../„(*)

But

by a

VW

lim

—-

(k)

«/„'

.

well

'

cos 2

\

we may expect

B are constants.

+ sin 2 £
2

£«)}

ofo.

and then

= ^^

(£)

Ao cos

^j /"

1

2kJ sin

(2k sin 2 i«)

S

o?
"^'

I

-2-)

°°

fl

-^Y"

nrnx^dx.

V(2 * }

(l-^V.^ \

and

o

+ cos £. J

*-ôo yf o

A

that

= and similarly

cos k + Bsm k where

in the latter half of the integral

.

11

1826

^V.^K/^),

f J „ sin

known formula*.

[Note. It is not easy to prove rigorously that the passage to the limit is permissible the simplest procedure is to appeal to Bromwich's integral form of Tannery's theorem,

Bromwich, Theory of Infinite

Series, § 174.]

It follows that (

cos k

sin

where

**-*-()

and

k -*-0

ri

k

J

(k)

-sink.

J

'

J

(k)

+ cos k J

'

.

.

as k-+-x>

J° ^ =

i It

where

«ô'

(*)

;

.

1 = j—^

(k)

= -ji—p- (1

(

1 4- « t ),

f

rj

k ),

and therefore

J(*k)

K1 +

ffc

C° S ^

*+^+

^

sin

^'

= 77^m t - (!+«*) sin *+(! + ?*) cos

was then assumed by Poisson that

A =.5=1. The

(k)

t/ (£) is

^'3-

expressible in the form

but asymptotic, and the validity

series are, however, not convergent

of this expansion was not established, until nearly forty years later,

—

gated by Lipschitz, Journal fur Math. lvi. (1859), pp. 189

The

J

result of formally operating

on the expansion assumed by Poisson

with the operator -jt2 +

.rz.l.B -jA

* Cf.

-«»*[

&

+ 8m *[_

p

2.

1

+ rn

was

for the function

2B"- (I. 2 + 1)

A'

2.

Ji

+

3B'"-(2. 3 + j) A"

*

+-

£

+-

Watson, Complex Integration and Cauchy's Theorem (Camb. Math. Tracts, no. a proof of these results by using contour integrals.

p. 71, for

investi-

is

& +

it

i

eft

(k) *J(irk)

when

196.

1 J J' 15, 1914),

:


12 and

so,

by equating

to zero the various coefficients, 1

» A'A---H,

AA" -"

—

9 25 2.3.83 '

R

••' '

is

9-25 9 \ /V\*IY 1 - 1 + 2V3."83l3+-" 8I--2T8^ > LV

/""

OM(*ooe.)A,-^J j

o

I

find that

A'"*= A -

A *, 2 g2

and hence the expansion of Poisson's integral

we

[CHAP.

,

C08 *

)

/. 1

V

9

1

9.25

\

,

.

+ 8l"2T8«^~2.3.88ife» + -; 8m ,

,1 *J-

But, since the series on the right are not convergent, the researches of Lipschitz and subsequent writers are a necessary preliminary to the investigation of the significance of

the latter portion of Poisson's investigation.

be mentioned that an explicit formula for the general term in the expansion B. Hamilton, Trans. B. Irish Acad. xix. (1843), p. 313; his result

It should

was first given by W. was expressed thus

-

"["

cos (20 sin a) da =

-^ 1

[0]—

([

- *]») 2 (43)— cos

(20 - $n,r

and he described the expansion as semi-convergent; the expressions to be interpreted as 1/n and (-|)( — f) ... (-«+£)•

[0]

- }n),

-B and [-$]" are

!

A result of some importance, which was obtained by Poisson in a subsequent that the general solution of the equation

memoir*,

is

is

y = Ax%

f

e- hxcoSu> da>

+ Bx$

where

A

and

B are

It follows at

y

m) da>,

constants.

once that the general solution of the equation

dxr is

~ 2 e hxcoaa log (x sin

\

Jo

Jo

= A\

e~ hxcoau da)

.'o

x dx

+B

I

e hxcoa '°\og

(a?

sin2 w)

cZo>.

Jo

This result was quoted by Stokesf as a known theorem in 1850, and it is he derived his knowledge of it from the integral given in Poisson's memoir; but the fact that the integral is substantially due to Poisson has

likely that

been sometimes overlooked J. * Journal de VEcole R. Polytechnique, xn. (cahier 19), (1823), p. 476. The corresponding general integral of an associated partial differential equation was given in an earlier memoir, ibid, p. 227.

t Camb. Phil. Trans, ix. (1856), p. [38], [Math, and Phys. Papers, hi. (1901), J See Encyclopidie des Set. Math. n. 28 (§ 53), p. 213.

p. 42].


1'7]

17.

1826

13

The researches of Bessel.

The memoir* in which Bessel examined in detail the functions which now name was written in 1824, but in an earlier memoir f he had shewn

bear his

that the expansion of the radius vector in planetary motion

-

=

1

Bn =

where

this expression for

+

2

sin

/

Bn should

^£

+ 2

is

Bn cosnM, —

u sin (nu

ne sin u)

du

;

be compared with the series given in

§ 1'4.

In the memoir of 1824 Bessel investigated systematically the function I h k defined by the integral J

If = k— zvrJo

2"

Ikh

He

I

cos (hu

— k sin u)' du.

took h to be an integer and obtained

given in detail in Chapter n.

many

of the results which will be

not adapted for defining the most worth study when h is not an integer (see § 10-1) the function which is of most interest for non-integral values of h is not Ikh but the function defined by Lommel which will be studied in Chapter in.

function which

Bessel's integral

is

is

;

After the time of Bessel investigations on the functions became so numerous it seems convenient at this stage to abandon the chronological account

that

and to develop the theory in a systematic and

An

logical order.

account of researches from the time of Fourier to 1858 has been compiled by Wagner, Bern MittheUungen, 1894, pp. 204—266 a briefer account of the early history was given by Maggi, Atti delta It. Accad. dei Lincei, (Tra/isunti), (3) iv. (1880), pp. 259—263. historical

;

—

* Berliner Abh. 1824 [published 1826], pp. 1 52. The date of this memoir, " Untersuchung des Theils der planetarischen Storungen, welcher aus der Bewegung der Sonne entsteht," is Jan. 29, 1824.

t Berliner Abh. 1816—17 [published 1819], pp. 4'J— 55. J This integral occurs in the expansion of the eccentric anomaly

nA n

= 21\ t

;

with the notation of §1-4,

,

a formula given by Poisson, Connaissance des Terns, 1825 [published 1822],

p.

383.

;

CHAPTEE

II

THE BESSEL COEFFICIENTS of the Bessel

21.

The

The

object of this chapter

definition

of a set of functions

known

is

coefficients.

the discussion of the fundamental properties

as Bessel coefficients.

There are several ways of

the method which will be adopted in this work is to in a certain expansion. This procedure is due coefficients the as them define properties of the functions from his defimany derived who Schlomilch*, to

denning these functions

;

and proved incidentally that the functions thus defined are equal to the definite integrals by which they had previously been defined by Bessel f. It should, however, be mentioned that the converse theorem that Bessel's integrals are equal to the coefficients in the expansion, was discovered by Hansen:}:

nition,

fourteen years before the publication of Schlomilch's memoir. results had been published in 1836 by Jacobi (§ 222).

The generating

function of the Bessel coefficients

Some

similar

is

It will be shewn that this function can be developed into a Laurent series, n qua function of t; the coefficient of t in the expansion is called the Bessel is denoted by the symbol Jn (z), coefficient of argument z and order n, and it

so that

«*('"?)(1)

I

t"Jn

(z).

«=-oo zt can be establish this development, observe that e^

expanded into an and for all values of t, ielt can be expanded into an absolutely converwith the exception of zero, e~ of t. When these series are multiplied powers descending gent series of convergent series, and so it may be absolutely together, their product is an is to say, we have an expansion of the arranged according to powers of t that and t, t = excepted. form (1), which is valid for all values of z

To

absolutely convergent series of

ascending powers of

t

;

;

* ZeiUchriftfiir

namely that of

e

—

Math, und Phys. n. (1857), pp. 137 165. For a somewhat similar expansion, Mem. Soc. Ital. (Modena), xvm. (1820), p. 503. It must be

SCO80 , see Frullani,

pointed out that Schlomilch, following Hansen, denoted by Jftn what we now write as Jn {2z) but the definition given in the text is now universally adopted. Traces of Hansen's notation are to be found elsewhere, e.g. Schlafli, Math. Ann. 22. t Berliner Abh. 1824 [published 1826], p.

m.

(1871), p. 148.

+ Ermittelung der Absoluten Storungen in Ellipsen von beliebiger Excentricitdt und Neigung, der Sternwarte Seeburg : Gotha, 1843], p. 106. See also the French translaab$olues (Paris, 1845), p. 100, and Leipziger tion, Memoire sur la determination des perturbations i.

theil, [Schriften

Abh. n. (1855>, pp. 250—251.

2*1, 2 '11]


If in (1)

we

write

— \jt

for

t,

15

we get n— -x

= 5 i-tyj_ n { Z — x

n—

),

on replacing n by - w. Since the Laurent expansion of a function a comparison of this formula with (1) shews that (2)

J-n{z)

where n

Jn (z)

is

any integer

-

is

unique*,

= (-YJn (*),

a formula derived by Bessel from his definition of

as an integral.

From

(2) it

evident that (1)

is

eiz(t-llt)

(3)

=J

q

{z)

may be

+ J

{t

n

w-1

written in the form

+{_

)n t - n]

Jn {2)

A summary of elementary results concerning Jn (z) has been given by Hall,

The Analyst, of elementary applications of these functions to problems of Mathematical Physics has been compiled by Harris, American Journal of i.

(1874), pp.

81—84, and an account

Math, xxxiv. (1912), pp. 391—420.

The

function of order unity has been encountered by Turriere, Nouv. Ann. de Math. (4) 433—441, in connexion with the steepest curves on the surface z=y (5.r2 -y»).

ix. (1909), pp.

2*11.

The ascending

series

for

Jn (z).

An explicit expression for Jn (z) in of z is obtainable

by considering the

exp (**(«-- 1/0)=

the form of an ascending series of powers

series for

I fl£T v r =o

When n is a

exp

rl

„l=t>

(h zt)

and exp ( - kz/t), thus

L- **>'"'ml

positive integer or zero, the only

term of the first series on the associated with the general term of the second series gives rise to a term involving tn is the term for which r = n + ; and, since n > 0, there is always one term for which r has this value. On associating these right which,

when

m

terms

for all the values of

m, we see that the

(1)

product

is

ml

m=o(n-rm)l

We

coefficient of t n in the

therefore have the result

Jn(z)= S

* For, if not, zero could be

l

^

K

* Z)

expanded into a Laurent

series in t, in which some of the were not zero. If we then multiplied the expansion by t » » and integrated it round a circle with centre at the origin, we should obtain a contradiction". This result was noticed by Cauchy, Comptes Rendus, xm. (1841), p. 911.

coefficients (say, in particular, that of

<"»)

)

.

'


16

where n

is

The

a positive integer or zero.

[CHAP.

II

few terms of the series are

first

given by the formula

Jn («) =

(2)

zn

z*

(

2^1 { 1 ~ 2

2

1

.

.

(n

+

+

2-

1

1

.

.

2

.

* + 1) (n +

(»

1

2)

'"]'

In particular Jo (?)

(3)

—1

22

^ 22 42

22

.

.

"

42 6 2 .

To obtain the Bessel coefficients of negative order, we select the terms involving irn in the product of the series representing exp tyzt) and exp(- \z\t\ where n is still a positive integer. The term of the second series which, when n associated with the general term of the first series gives rise to a term in t~ is the term for which ra = n + r ; and so we have

J -n{Z) ~

rZo

whence we evidently obtain anew the formula

J_»^) = (-)

n

'

+ r)l

(n

r\"

§ 2'1 (2),

namely

^»(*)-

be observed that, in the series (1), the ratio of the (w + l)th term 2 oo for all to the rath term is - £s /{ra (n + ra)}, and this tends to zero as ra — that follows it values of z and n. By D'Alembert's ratio test for convergence, it so and and n, the series representing Jn (z) is convergent for all values of z It is to

,

is

an integral function of z when n = It will

and so

it

appear later (§4"73) that is

0,

+

Jn

(z) is

a transcendental function

;

1,

±

+

2,

3,

. . .

not an algebraic function of z moreover, it is not an elementary

to say it is not expressible as a finite combination of exponential, logarithmic and algebraic functions operated ou by signs of

transcendent, that

is

indefinite integration.

From ance

(cf.

(1)

we can obtain two

useful inequalities, which are of

some import-

Chapter xvi) in the discussion of series whose general term

is

a

multiple of a Bessel coefficient.

Whether

z be real or complex,

we have ao

* and

so,

(4)

854

n!

i

J.

« < &£ exp (|!^) ).

xm.

(1841), pp. 687,

;

.«l< -^«P(l*l , r

lJ

l

l.

was given by Neumann, Theorie der BesseVschen Functionen

(Leipzig, 1867), p. 27.

»

»


2*12]

By

considering

all

the terms of the series for

17

Jn (z)

except the

first, it is

found that

j.(«)-^(i+«),

(«)

where

|»|

For

(*

l

n

' ,'>- 1

+

l

.

1

be observed that the series on the right in § 2*1 (1) converges uniformly in domain of the variables z and t which does not contain the origin in the

It should

any, bounded *-plane.

"P
if 8,

A and

R are positive constants and if d<|t|
|*|

<

|

A.I,

the terms in the expansion of exp (\zt) exp ($z/t) do not exceed in absolute value the corresponding terms of the product exp ($RA) exp QR/8), and the uniformity of the convergence follows from the test of Weierstrass. Similar considerations apply to the series obtained

by term-by -term

differentiations of the expansion 2^*t/n

performed with respect to

z

or

t

or both z and

(z),

whether the differentiations be

t.

The recurrence formulae.

2" 12.

The equations* Jn (z), + Jn+1 (*) = ~ z

(1)

Jn-i (*)

(2)

Jn_ (z)-Jn+1 (z) = 2Jn'(z), 1

which connect three contiguous functions are useful in constructing Tables of Bessel coefficients; they are

To prove the

known

as recurrence formulae.

former, differentiate the fundamental expansion of § 21,

namely

,*'«-™= 5 t»j (z), n with respect to

t

;

we get $z(l

+ l/t*)e

iz(t ~

m=

2 nt^Jniz),

so that

^(1 + 1/e ) 5 2

t

»=—

If the expression on the left

is

n

Jn (z)= 2

»=—

nt^Jniz).

arranged in powers of

t

and

coefficients of n_1

are equated in the two Laurent series, which are identically equal,

tf

it is

evident

that

\z {Jn-i (z) + Jr* which * its

is

the

first

i

(z))

=nJn (z),

of the formulae f.

Throughout the work primes are used to denote the derivate of a function with respect to

argument. f Differentiations are permissible because (§ 2*11) the resulting series are uniformly convergent. of coefficients is permissible because Laurent expansions are unique.

The equating


18

[CHAP.

II

Again, differentiate the fundamental expansion with respect to z and then ;

m=

iz
£

t

n

»= -oo

\(t

so that

- Ijt) 2

t

n

Jn (z) = 5

coefficients of tn

n

t

Jn

'

(z).

w=-oc

»=-oo

By equating

Jn '(z),

on either side of this identity we obtain formula

(2) immediately.

The

and subtracting (1) and (2) are

results of adding

zJn (z) + nJn (z) = zJn__ (z), zJn'(z)-nJn (z) = -zJn+1 (z). '

(3)

x

(4)

These are equivalent

to

^{^« (*)}=*"
(5)

^{z-»Jn (z)} = -z-"Jn+1 (z).

(6)

In the case w=0, (1)

is trivial

while the other formulae reduce to

/,'(#)

(7)

= -/»(*).

(1) and (4) from which the others may be derived were discovered by Abh. 1824, [1826], pp. 31, 35. The method of proof given here is due to Schlomilch, Zeitschrift fur Math, und Phys. ir. (1857), p. 138. Schlomilch proved (1) in this manner, but he obtained (2) by direct differentiation of the series for Jn (2).

The formulae

Bessel, Berliner

A

formula which Schlomilch derived 2r

(g)

where

r

By

Cm

is

^T-)=

a binomial

/

where n

d

\m

\

(is) is

n {*

(6),

we have

Jn(z)}=2 n m Jn-m(z),

m ~ {2 nJn (z)] = (-r*-"^+» (*).

any integer and

in is

any positive

integer.

The formula

to Bessel (ibid. p. 34).

As an example *Ji

of the results of this section observe that

0) = 4t/a 0) - zJs (*)

= 4J"

2

(z)

- 8J, (z) + zJ> (z)

= 4 2 (-Y~ nJi,(z)^(-rzJiN+1 (z) 1

»=i

= 4 1 (-y-*nJm (z), since

zJ^+i (z) -*

is

coefficient.

zTz) (10)

143) from (2)

{-TrCm .Jn -r+»n {z\

2

obvious inductions from (5) and

(d

{ibid. p.

as iV-*- 00

,

by

§

211

(4).

(10)

is

due


2-13, 2'2]

The expansion thus (11)

19

obtained,

*./,(*)

=4 I

{-)n

-

l

nJ2n (z),

n-l is

useful in the developments of

Neumann's theory of Bessel

functions (§ 3'o7).

Jn {?)>

2*13.

The

When

the formulae § 2'12 (5) and (6) are written in the forms

differential equation satisfied by

the result of eliminating J, _, t

d is

and

so

seen to be

f-^Wni*)} = -z*- n Jn (z),

dz that

(z) is

to say

we have

Bessel's differential equation*

The

W)

zi

(1)

analysis

is

+z

dJAz)

by using the operator ^ defined as

simplified

Thus the recurrence formulae (S

+ (z2 _ wl) Jh {z) ^

+ n) Jn (z) = zJn _

x

z (dldz).

are

{z),

(^ - n

+

1)

Jn -i (z) = - zJn (z),

and so n + 1)

(^ that

[z-i

(*

+ n) Jn (z)} = -zJn (z),

is

z-> (^

- n) (* + n) Jn (z) =-zJn (z),

and the equation

C&-n*)Ju {z) = -z*Jn (z) reduces at once to Bessel's equation. Corollary.

The same

differential equation is obtained if

Jn +

l

(z) is

eliminated from the

formulae (S

2*2.

We

+ n + 1) Jn +

l

(z)

= zJn (z), (»-n)Jn (z)=-zJn ^

l

(z).

Bessel's integral for the Bessel coefficients. shall

now prove

(1)

that

Jn (z) =

"'

J-

I

Lit

'o

cos {n$

- z sin B) d6.

This equation was taken by Bessel f as the definition of

Jn (z),

and he

derived the other properties of the functions from this definition. *

Berliner Abh. 1824 [published 1826], p. 34

(1820), p. 504.

t Ibid. pp. 22 and 35.

;

see also Frullani,

Mem.

Soc. ItaL.(Modena),

xvm.

;

THEORY OP BESSEL FUNCTIONS

20

modify

It is frequently convenient to

tegration and writing

2tt —

1

f"

Jn (z)=IT

(2)

by bisecting the range of

(1)

cos (n0

II

in-

This procedure gives

in the latter part.

for

[CHAP.

- z sin 0) dO.

JO

Since the integrand has period

the

27r,

first

may

equation

be transformed

into /*2ir+a

1

J n (z) = 2~

(3)

where a

is

cos(n0-zsm0)d0,

I

any angle.

To prove

multiply the fundamental expansion of

(1),

§

2

-

l

(1)

by t~n

~x

and

integrate* round a contour which encircles the origin once counterclockwise.

We

thus get

»»=-»

2771 J

so

The integrals on the right we obtain the formula

J

m = n;

and

be a circle of unit radius and write t = e~ i0 so that be taken to decrease from 2tt + a to a. It is thus found that

Take the contour

may

Z7rt

vanish except the one for which

all

to

t

rtor+a

I

«*<**-*•«•) <$£

«Jn(*)«^-/

(5)

— 77" J a

a result given by Hansen f in the case a = 0.

= — ir, bisect the range of — 0. This procedure gives

In this equation take a former part, replace

by

Jn (z) = -L and equation

(2),

f

{

e*(«*-z «n »)

may be

from which (1)

integration and, in the

+ e - Une - z sin 9) £Q J

deduced,

is

now

obvious.

Various modifications of Bessel's integral are obtainable by writing 1

Jn (z) = —

IT

f« I

cos

nd cos (z sin 0)d0-i

—If... n0 I

sin

sin (z sin 0) d0.

TT J

Jo

be replaced by 7r — in these two integrals, the former changes sign when n is odd, the latter when n is even, the other being unaffected in each case and therefore

If

s Jn (•&) — J i = -;

(6)

I

T JO

*

Term -by- term

contour. It

is

integration

is

i

n n & sm ( z sm 0) d0 («odd). .

sin

w0 sin (s sin 0) d0 I

uniformly convergent on the to denote integration round a contour encircling

permitted because the expansion

convenient to use the symbol

(a+)

J

the point a once counterclockwise. t Ermiitelung der absoluten Storungen (Gotha, 1843), p. 105.

is

"


2*21]

1

8)

«/n (*)

(9)

The

(z)

«/ii

two

last

[Note.

It

--

dd

cos (z sin &)

d0

f

Jo

in the latter parts of (6)

(-)*<•

1}

cos

I

=-(-)**

results are

6)

("even). cos w

I

ir

(

n$ cos (z sin

,^

2

= 77

cos

I

K) be replaced by £tt -

21

'**

Jn {z)—-

If

.

and

sin (s cos

n?7

(7), it is

found that (w odd),

17)
cos nr) cos (z cos 17) cfy

(n even).

due substantially to Jacobi*.

was shewn by Parseval, Mem. des savans Grangers,

1.

(1805), pp.

639—648,

that .

a

2

2s

and

a

a*

+ 22742 ~

6

2 2.42.62 +

—=

1

[*

J be described as ParsevaVs integral.

the special case in which n = 0, (2) will be seen in § 2-3 that two integral representations of

so, in

will

and Poisson's integral become

identical

8m *) <**»

cos («

,,

when w=0,

Jn (z),

namely

so a special

It

Bessel's integral

name

for

thL

case

is

justified.]

The reader § 2-12 (4)

2 '21.

wijl find it interesting to obtain (after Bessel)

from Bessel's

the formulae § 2-12

(1)

and

integral.

Modifications of Parseval's integral.

Two

formulae involving definite integrals which are closely connected with PaTseval's integral formula are worth notice. The first, namely

Jo{J(*-&)} = \ /"*«*«"• cos (lain*)***,

(1) is

due to Bessel f.

The simplest method of proving

it is

to write the expression on the

right in the form

—

/>

1

J

e v cos 0+vs sin

dfy

expand in powers of y cos d+iz sin 6 and use the formulae J

^

(y cos 6

+ iz sin 6f» +

»

dd = 0,

the formula then follows without

The other

definite integral,

J

(2)

f"

a special case of

(1)

+ iz sin 6f» dd =

2r ( w + *)r(i) (y2

_

^

difficulty.

due to Catalan J, namely

(2»
=IT

is

(y cos 6

/"" e
obtained by substituting

* Journal fUr Math. xv. (1836), pp. 12—13. integrals actually given by Jacobi had limits

See also Anger, Neueste Schriften Comptes Rendus, xxxvm. (1854), pp. t Berliner Abh., 1824 [published Ges. in Danzig, v. (1855), p. 10, and

cos {(l

- z) sin

0} dd,

J 1

[G
-z and

1

+z

Math. Werke,

and w with

for 2

and y

respectively.

vi. (1891), pp.

der Naturf. Ges. in Danzig, v. (1855), p.

910

100—102]

factors I/*- replacing the factors 1,

the 2/*-.

and Cauchy,

—913.

1826], p. 37.

See also Anger, Neueste Schriften der Naturf.

Lommel, ZeiUchrift fUr Math, und Phys. xv_(1870),

t Bulletin de Vdcad. R. de Belgique,

;

(2) xli. (1876), p.

938.

p. 151.

)

THEORY

22 Catalan's integral

may

BESSEL FUNCTIONS

OJF

[CHAP.

II

be established independently by using the formula '(0+) i'(0+)

no that ./

(2i\'*)

=

-i_-J-.

2

= by taking the contour

2 (0

J_.

/"

t-^-Wdt

-, +

p^ + ! )^ = l N

)

ex

to be a unit circle

;

;

[* exp {*»+*-»}

the result then follows by bisecting the range of

integration.

Jacobi' s expansions in series of Bessel coefficients.

2"22.

Two 'series, which

are closely connected with Bessel's integral, were dis-

The simplest method of obtaining them the fundamental expansion § 21 (3). We thus get

covered by Jacobi*. t

= ±

e

ie

in

is

to write

1)

0.

M=l

= J 0) + 2 2 Jm 0)cos 2nd ±

2i

On adding and we

2 Jm+1 (*) sin (2» + w=0

n=\

subtracting the two results which are combined in this formula,

find

cos (z sin 0)

(1)

= J O) + 2 2 Jm (z) cos 2n0, n= \

sin(*sin0)=

(2)

Write \ir

—

r)

for 6,

cos

(3)

2

2 /2n+1 (*)sin(2n + l)0.

and we get

cos

t?)

= Jo (z) + 2 2

(-)»./,»(*) cos 2ni7,

«=i sin (s cos

(4

t/)

=

2

2 (-)M J2n+1 0) cos (2n +

1)

rj.

«=o

The results (3) and (4) were given by Jacobi, while the others were obtained later by Anger t. Jacobi's procedure was to expand cos (s cos 17) and sin (z cos q) into a series of t}, and use Fourier's rule to obtain the which are seen to be associated with Bessel's integrals.

cosines of multiples of integrals

coefficients in the

form of

In view of the fact that the first terms in (1) and (3) are not formed according to the same law as the other terms, it is convenient to introduce Neumann's factor* en> which is defined to be equal to 2 when n is not zero,

and

to be equal to 1 *

when n

is zero.

Journal fUr Math. xv. (1836), p. 12.

The employment [Ges.

Math.

of this factor, which

U'erke, vi. (1891), p. 101.]

t Neueste Schriften der Xaturf. Gen. in Danzig, v. (1855), p. 2. % Neumann, Theorie der BesteVschen Functionen (Leipzig, 1867), p.

7.


2-22]

23

be of frequent occurrence in the sequel, enables us to write (1) and (2) in

will

the compact forms: oo

= X m J2n (z) cos 2nd,

cos (^ sin 0)

(5)

e.

»=o ao

sin (z sin 0)

(6)

= X

Jm+1 (z) sin (2n + 1) 6.

e2w+1

«=o If

—

we pub

in (5),

we

find

1=2

0)

m Jm (z).

e

we differentiate (5) and (6) any number of times before putting 6 = 0, we obtain expressions for various polynomials as series of Bessel coefficients. shall, however, use a slightly different method subsequently (§ 27) to prove

If

We

that zm

of. Bessel coefficients when m is any positive any polynomial is thus expansible. This is a special case of an expansion theorem, due to Neumann, which will be investigated in Chapter xvi. is

expansible into a series

It is then obvious that

integer.

For the present, we will merely notice before 6 is put equal to 0, there results

*= 2

(8)

while, if 6 be put equal to

%-ir

e 2n+1

after

(2n

that, if (6)

be differentiated once

+ l)J2n+1 (z),

two differentiations of

(5)

and

(9)

zsinz

= 2{2*J (z)-4?J (z)+6*J

(z) -...},

(10)

zcosz

=2

(z)-

2

6

i

jl«./;(*)-3 s

J

3

(z)

+

52

J

!i

(6),

then

...}.

These results are due to Lommel*. The expression exp {$z{t-

Note.

in the strict sense.

The generating function t

If this expression be called S,

we

If

1/0} introduced in $ 2 1 is not a generating function

associated with

en

Jn (?)

by using the recurrence formula

solve this differential equation

is

2

*„*"•/„

§ 2-12 (2),

(z).

we have

we get Z

S=ei*«-W+l(t + tyj*V-wJ e-*V-V')j

(11)

A

result equivalent to this

was given by Brenke,

Bull.

(t

)di:

American Math.

Soc. xvi. (1910),

pp. 225—230. * Studien ilber die Bessel'schen

t

It will

Functionen (Leipzig, 1868), p. 41. be seen in Chapter xvi. that this is a form of " Lommel's function of two variables."


24 2

Poissons integral for the Bessel

3.

[CHAP.

II

coefficients.

Shortly before the appearance of Bessel's memoir on planetary perturbations, Poisson had published an important work on the Conduction of Heat*, in the course of which he investigated integrals of the types f

^ cos (z cos 6) sin

2"-1" 1

Odd,

where n

is

(' cos (z cos 6) sin*1 0d0,

Jo

Jo

He


proved that these integrals are

and gave the investigation, which has already been reproduced in § 1*6, to determine an approximation to the latter integral when z is large and positive, in the special case n = 0. solutions of certain differential equations^

We

shall

now prove

that

and, in view of the importance of Poisson 's researches,

it

seems appropriate to J (z). In the

describe the expressions on the right § as Poissons integrals for n case n = 0, Poisson's integral reduces to Parseval's integral (§ 2'2). It is easy to prove that the expressions

Jn (z);

we expand the integrand term-by-term we have for, if

under consideration are equal to and then integrate

in powers of z

||,

I

I

fir

-

cos (* cos

7tJ

/

oo

0)8^0(20 = - t 7T m =o

= j!

~(>0r

\m »sm

V(2W)!„

'

is

cos2'»0sin=»0
2.4.6... (2n

~ -1.3.5...(2»-1) 2 and the result

fir

Jo

+ 2m)

'

H+m m!(n oli 2

+ m)r

obvious.

* Journal de VEcole R. Polyteckuique, xn. (cahier 19), (1823), pp. 249—403. previously been t Ibid. p. 293, et seq. ; p. 340, et seq. Integrals equivalent to them had examined by Euler, hut. Calc. Int. u. (Petersburg, 1769), Ch. x. § 1036, but Poisson's forms are more elegant, and his study of them is more systematic. See also § 3'3.

J E.g. on p. 300, be proved that,

if

R=rn+1

™" cos (rp cos «) sin 2

" 1 1

I

udu,

Jo then

R

satisfies the differential

equation

dr1 § Nielsen,

Handbuch der

r-

Tfieorie der Cylinderfunktionen (Leipzig, 1904), p.

51, calls

but the above nomenclature seems preferable. The series to be integrated is obviously uniformly convergent ; the procedure adopted

them

Bessel's second integral, ||

to Poisson, ibid. pp. 314, 340.

is

due


2-3, 2*31]

25

Poisson also observed* that (' e^«»* sin8'* Odd Jo this is evident left

when we

=

* f

cos (z cos 0) sin™ 0d0\

Jo

consider the arithmetic

and the integral derived from

it

mean

by replacing

by

of the integral on the it

— 0.

We thus get A

namely

slight modification of this formula,

/„(*)

(3)

=

r(n

4|»^V,

has suggested important developments

(1

_ PY - iM>

§ 6*1) in the theory of Bessel

(cf.

functions. It should also be noticed that !

(4)

' cos (z cos 0) sin2"

0d0 = 2

' \

cos {z cos 0) sin1"

0d0

cosCssin^cos*1

^,

Jo

Jo

=2

f*»

Jo

and each of these expressions gives rise to a modified form of Poisson's integral.

An interesting application of Bessel's and

Poisson's integrals

was obtained

by Lommelf who multiplied the formula *

, / ,m cos2n0= 2 (-)

«-o

»

by cos (z cos 0) and

2'31.

The Bessel J,

.

(zm)\

It thus follows that

r

(

J

s

integrated.

J (z)-(-Y I

(5)

4w8 {4n8 -28 }...{4n2 -(2ro-2)2} „. -sin*"^ /a v.

4 » 8 f4n»-2r... (W-(2iii -*Y)

Jm (s)

Bessel's investigation of Poisson's integral.

Jn (z)

proof, that is

is

equal to Poisson's integral, which was given by

somewhat elaborate;

substantially as follows

on differentiation that

It is seen

sin2"

-^ cos I

-1

cos (z cos 0)

(2w - 1) sin8"-8

=

it is

—9

,-

- 2n sin " + 8

sin2*1 **

" " 1

1

sin (z cos 0) sin2"*8

cos (* cos 0),

|

* Poisson actually sinSH+1 6

;

but, as

made

(p. 293) concerning the integral which contains odd powers may be replaced by even powers throughout

the statement

he points out on

p. 340,

his aualysis.

f Studien Uber die BesseVscken Functional (Leipzig, 1868), p. 30. + Berliner Abh. 1824 [published 1826], pp. 36—37. Jacobi, Journal filr Math. xv. (1836), p. 13, [Get. Math. Werke, vi. (1891), p. 102], when giving his proof (§ 2*32) of Poisson's integral formula, objected to the artificial character of Bessel's demonstration. w. B. jr. 2


26

and hence, on integration, when n > (2ra

cos (z cos 0) sin8"" 2

-1)1

6d0 - %n ^2

now we

H

1,

cos (z cos 0) sin3"

j

0d0

r*

+ 2n + i J If

[CHAP.

cos (* cos 0) sin2"-*" 8

0d0 = 0.

write 008 <* cos 0) sin2n

rin + ^ra) /^

^

s * <»>

the last formula shews that z

so that

$

(w)

Jn (z)

and

(n

- 1) - 2w0 (w) +

satisfy the

it is

=

cos (* cos 0) sin2

^J l r»

=and

so,

sm (z cos

(z),

0d0 = -

Odd

0) cos

+ 1) = 0,

evident that

=J (0) (1)

(»

same recurrence formula.

But, by using Bessel's integral,

*<-6

^

-J

——J

'

(z)

isin (s cos 0)1 sin

—J

x

0d0

(z),

by induction from the recurrence formula, we have

when w

= 0,

1, 2, 3, ....

Jacobus investigation of Poisson's integral.

2*32.

The problem integral

of the direct transformation of Poisson's integral into Bessel's

was successfully attacked by Jacobi*;

method

this

necessitates the use

of Jacobi's transformation formula

dM -i sin2M -i0

si^— = ^

where

fi

=

cos

0.

We

1.3.5. ..(2/i-l) . sinn *> » .

}

assume this formula for the moment, and, since no seems to have been previously published, we shall

shall

simple direct proof of

it

give an account of various proofs in §§ 2*321

we observe that the vanish when fi = ± 1, it If

fi,

zn j

Odd

cos (z cos 0) sin-'1

= (-) n *

is

n—

— 2323.

— /i2 ) M-i with respect to evident that, by n partial integrations, we have

first

J_

1 derivates of (1

= zn

I

COS foa i

cos (zfi) (1 .

- \ mr)

,

-

i.

(1836), pp.

195—196.

2

)"-* dfi

-^ £ — d». di

Journal fiir Math. xv. (1830), pp. 12—13. [Ges. Math. Werke,

also Journal de Math.

/a

}

vi. (1801),

pp. 101—102.] See


2*32, 2*321] If

we now use

becomes

Jacobi's formula, this

1.3.5...(2n-l)f If

.

3 5 .

. .

.

.dsinnd dsinnd

1

(2n

,

.

cos

=1

27

{Zfi

- 1)

— \n-jr)

—

cos (z cos

,

dfi

- ^nir) cos n0d0

Jo

= 1.3.5...(2n-l)wJ n (5), by

Jacobi's modification of § 2'2 (8)

to

cos (z cos 6) or (-)^ n-1)

(— )***

and

(9), since

cos^cos 6

sin(^cos 9) according as n

is

— \nir)

is

equal

even or odd; and

this establishes the transformation. 2*321.

Proofs of Jacobus transformation.

Jacobi's proof of the transformation formula used in § 2-32 consisted in deriving

it

as a special case of a formula due to Lacroix*; but the proof which Lacroix gave of his formula is open to objection in that it involves the use of infinite series to obtain

a result of an elementary character. A proof, based on the theory of linear differential equations, was discovered by Liouville, Journal de Math. vi. (1841), pp. 69 73; this proof will be given in § 2*322. Two years after Liouville, an interesting symbolic proof was published by Boole, Camb. Math. Journal, in. (1843), pp. 216—224. An elementary proof by induction was given by Grunert, Archio der Math, und Phys. iv. (1844), pp. 104 109. This proof consists in shewing that, if

—

2 n+1 = (l- M )-

then

and that ( -

n~ )

1.3.5.;. (2» —

'

1) (sin nff)jn satisfies

B rfM ,

the same recurrence formula.

Other proofs of this character have been given by Todhunter, Differential Calculus (London 1871), Ch. xxviil, and Crawford t, Proc. Edinburgh Math. Soc. xx. (1902), 15, but all these proofs involve complicated algebra. pp. 11

—

A proof

depending on the use of contour integration is due to Schlafli, Ann. di Mat. (2) The contour integrals are of the type used in establishing 202. Lagrange's expansion; and in § 2*323 we shall give the modification of Schlafli's proof, in which the use of contour integrals is replaced by a use of Lagrange's expansion. v.

(1873), pp. 201

To prove

—

by Leibniz' theorem, thus

Jacobi's formula, differentiate

i.l!5^-l) g^« 1 -^'

t

«»=0

•"

<

1+ '->- , »

a

•

a"

••• \

m ~t"2J

M

=

2 (-)"«.C'!m + i(ainitf)*» + »(oo8itf)«"-*»- 1

m=0

=sin(2wx£0),

and *

this is the transformation required

TraiU du Calc.

J.

—

183. See also a note written by edition), pp. 182 Catalan in 1868, Mim. de la Soc. R. des Sci. de Liege, (2) in. (1885), pp. 312—316. t Crawford attributes the formula to Rodrigttes, possibly in consequence of an incorrect statement by Frenet, Recueil d'Exercices (Paris, 1866), p. 93, that it is given in Rodrigues' dissertation, Diff.

i.

(Paris, 1810,

2nd

Corresp. sur VEcole R. Poly technique, in. (1814—1816), pp. X I owe this proof to Mr C. T. Preece.

361—385.


28 2*322.

[CHAP.

II

LiouvMds proof of Jacob?* transformation.

The proof given by Let y=(l —fi2 )

n ~k

Liouville of Jacobi's formula is as follows

and

D be written for d/dp

let

then obviously

;

(l-V)Z)y+(2n-l) My=0. n times ; and then

Differentiate this equation

- fi2 )

(1

Dtt+l

i/

- (iD^+^D^-^Ô

:

(l"^-^-^^^-"^.

but

(^ + »«

so that

D*~ 1 y= A

Hence

J

Z)»- »y = 0.

sin nd +

BcoB n6,

n~ l

A and B are constants since D y is obviously an odd function of 6, B is zero. To determine A compare the coefficients of 6 in the expansions of D*- 1 y and A sinnd in ascending powers of 6. The term involving 6 in D*~ l y is easily seen to be where

;

n"1

^" 1=( "

(-^)

)

"~ 1(2w " 1)(2w ~ 3)

-3

-

1

-

<9'

%J = (-)»- 1 1.3.5...(2n-l),

so that

and thence we have the

result,

namely

d»-»sin2»-i0

2*323.

.1.3.5...(an-l) v

dp*~ l

.

a

n

'

SchlaJWs proof of Jacobi's transformation. '

We first recall

Lagrange's expansion, which

so that

*•

(,)

| = £*

is

that, if z=fi+hf(z), then

*l{ fWjr if 0)],

subject to the usual conditions of convergence*.

Now it

/(2) 3 -|(l-2*),

take

),

when A-*-0. (2) reduces to ^(1 j«a function of h are at A=>e± *' ; and so, when 6 is real, the exin powers of h is convergent when both h and 2 are less than unity. -/*2

being supposed that 0'

The

8

v/(l-*

<£'(*) ), i.e.

to sin 6

singularities of 2

pansion of

,J{\

— 22 )

|

Now de

*

-

Jfl

,n

it

follows that

„J(l

- 22 )

^*

z

.

rr-r \ ; 2n_1 .(n-l)!

-

(62/8/*)

in

J)

1

powers of

,

is

. -1.

the coefficient of hn ~ l in the ex-

d/t"

But

h.

(l-^y-i-q-^-*)-* -

and a consideration of the

(l^-*)*-(l^*)»

d« -1 sin 2B_1 d

/_\»-i

pansion of

1

J

and so

Hence

\

2={l- v'(l-2/iA+# )}/A,

coefficient of A*

it is

j?

evident that

1.3.5...(2n-l) e»* - «"" ...,

-1 in the last expression establishes the truth of

Jacobi's formula. * Cf.

Modem

Analyst*, § 7-32.


2-322-2*33]

233.

An

29

application of Jacobi s transformation.

The formal expansion /'(cos x)

cos

nxdx =

Jo

2 Jo »t-0

(-)m am fi n+2m >(cosx)dx,

which am is the coefficient of tn+2m in the expansion of Jn (t)/J (t) in ascending powers of t, has been studied by Jacobi*. To establish it, integrate the expression on the left n times by parts it transforms (§ 2*32) into

in

;

1.3.5...(2n-l) Sl and,

when

sin2n #

w

1

We

now

is

f

f(n) (C °S

J.

XM ,

/

(n,

of the type stated:

2n(n-l)

2n

.(cosa?)cos2#, /<"> (cos x) cos

continual repetitions of this process,

When

we

/(cos x)

[

4a?,

...

1

A

by

parts,

,

and by

evidently arrive at a formal expansion

is

a polynomial in cos

obviously terminates and the transformation

J

>

replaced by a series of cosines of multiples of x, this becomes

integrate

To determine

*} 8inm * dx

is

a?,

the process

certainly valid.

the values of the coefficients a^ in the expansion

5 (-)m am f in+sm) (cosx) dx f (cos x) cos nxdx=jJOw-0

thus obtained, write

/(cos x) according as n

is

= (-)*» cos (t cos x),

(-)*<«-»> sin (t cos x),

even or odd, and we deduce from

Jn (t) = 2

(_)-«,„«»+«

§ 2*2 (8)

{(_),» J%

and

(9) that

^

so that am has the value stated. It has been stated that the expansion is valid when /(cos x) is a polynomial in cos a;; it can, however, be established when /(cos x) is merely restricted to be an integral function of cos x, say

* 6w cos n a?

provided that lim %\ bn Jo (t)

—

;

j

is less

than the smallest positive root of the equation

the investigation of this will not be given since

it

seems

to be of

no practical importance. *

Journal filr Math. xv. (1836), pp.

also Jacobi, Astr. Nach.

xxvm.

25—26

(1849), col.

[Qes. Math. Werke, vi. (1891), pp. 117—118]. 94 [Ges. Math. Werke, vn. (1891), p. 174].

See


30

The addition formula for the Bessel

2*4.

The Bessel

coefficients possess

[CHAP.

II

coefficients.

an addition formula by which

may be

expressed in terms of Bessel coefficients of y and which was first given by Neumann* and Lommelf, is

z.

Jn (y + z)

This formula,

00

J»(y

(1)

+ z) = 2 Jm (y)Jn-r,i(z). m-

The simplest way

-oo

of proving this result

from the formula § 2*2 (4), which

is

gives 0+)

1

Jn {y

f<

+ *) = 5^-.

'~

~

po+)

I

=

M_1 «* {y+* oo

X

t

X

~ vt '

dt

m -n- Jm (y)e*z
1

1/t

>dt

no+)

oo

i

= _i-.

{t

Jm (y)\

p»-»-i eft-Milt

00

-

Jm (y)Jn-m{z),

2 wi— -oo

on changing the order of summation and integration in the third and this is the result to be established.

analysis

Numerous

2'5.

line of the

;

generalisations of this expansion will be given in Chapter xi.

Hansen's series of squares and products of Bessel

Special cases of early as 1843.

The

coefficients.

Neumann's addition formula were given by HansenJ as first system of formulae is obtainable by squaring the

fundamental expansion

§ 2*1 (1), so that

«.«-!/»,

.j I- rJr (z))\ I-oo tm Jm (z)\. [

r=

oo

)

Km-

J

By expressing

the product on the right as a Laurent series in t, and equating n the coefficient of tn in the result to the coefficient of t in the Laurent expansion of the expression on the

J„(2s) In particular, taking (1)

Jo (2s)

?i

left,

we

find that

= £ Jr {z)Jn-*(z).

= 0, we

= Jo* (:) + 2S

have§

(- )' Jr2 (*)

- £

(-)' e r

Jr

*

(z).

r=0

r=l

* Theorie der BesseVschen Functionen (Leipzig, 1867), p. 40. f Studienilber die BesseVschen Functionen (Leipzig, 1868), pp. 26

—27

;

see also Schlafli,

Math.

Ann. in. (1871), pp. 135—137. + Ermittelung der absoluten Stiirungen (Gotha, 1843), p. 107 et seq. Hansen did not give (4), and be gave only the special case of (2) in which n = l. The more general formulae are dne to Loramel, Studien Uber die BesseVschen Functionen (Leipzig, 1868), p. 33. §

For

brevity,

JH2 (z)

is

written in place of

{<•/„

(z)) 2 .


2'4-2'6]

From

we

the general formula

31

find that

Jn (2z) -S/f (#) Jn-r (*) + 2 £ (-Y Jt (*) Jn+r (z), r=0 r=l

(2)

when the

Bessel coefficients of negative order are removed

by using

§ 2*1 (2).

Similarly, since

rjr (s)\\ 5 {-rv*J

I

j

ni

{z)\

= exp [±z (t =1

1/t)}

exp [\z (-

t

+

1/t)}

>

it

follows that

J >(z) + 2% Jr*(z) = l,

(3)

2X Jr (z) Jm+r (z) = 0.

2 (-y Jr (z) Jm_r (z) +

(4)

r-0

r-1

Equation (4) is derived by considering the coefficient of tm in the Laurent expansion the result of considering the coefficient of tm+1 is nugatory. ;

A

very important consequence of

\J,{*)\*1,

(5)

where r 2"6.

(3),

= 1,

It is evident

that,

when x

is real,

|^,(*)j
was noticed by Hansen.

2, 3, ...,

Neumann's

namely

integral for

Jn

2

(z).

from § 2*2 (5) that

Jn (z) = l-

I" eUn0-z amo) d0)

and so

Jn2 (z)

To reduce this double by the equations

1 = |-5

f*

f*

«»*«>+*> e -«(«nfl+sin +,

I

dOdj).

j

integral to a single integral take

8-

= 2X

,

0+

=

new

variables defined

2yJr,

so that d(0,)

=

2.

9 (x» Vr ) It follows that

J„ 2 (*) =

where the

field

*"""*

2^*11

of integration

is

""** 8in * cos *

d*;cty,

the square for which

— tt^x— k^'7r — "" < X + ^ ^ 7r y

>

Since the integrand is

increased by

gration

may

is

unaffected

while

both

if

% and

yfr

-

are increased by

simultaneously decreased by ir, the evidently be taken to be the rectangle for which ir

yfr

is

0
l/r

^

7T.

tt,

field

or

if

^

of inte-

)

THEOEY OF BESSEL FUNCTIONS

32

[CHAP,

n

Hence Jn

- 2^-

(*)

a

f f*

2"**- 2** sin

e

1 C"

=If

we

replace

% by |tt T

J** (2* cos x)

\

Jo

if

according as

0,

^

* «» * cty dy

<%

is

acute or obtuse,

we

obtain the

result

Jn*

(1

t

W = -f

Jm (2z sin 0) dd.

If J

This formula

may

obviously be written in the form

Jn

*

(2)

(z)

= - f J*n (2z sin 0) d0, IT

JO

which is the result actually given by Neumann*. It was derived by him by some elaborate transformations from the addition-theorem which will be given in §H'2. The proof which has just been given is suggested by the proof of that addition- theorem which was published by Graf and Gublerf.

We obtain a different form of the integral with respect to x instead of with respect to yjr. Jr? (s)

=

^f

Jo (2* sin

we perform the integration This procedure gives

if

yjr)

j***

so that r*

i

(3)

^""^"Srl a»

1 -

f*

h

if

«

J

/o(2*sin^)cos2n^(fy

(2z sin

i/r)

cos 2nyjr

dijr,

a result which SchlafliJ attributed to Neumann.

2'61.

Neumann's

series

a for J„

(z).

taking the formula §26(1), expanding the Bessel coefficient on the right in powers of z and then integrating term-by-term, Neumann§ shewed

By

that

(-)* zm+wt

lf»

_ ~

• TO =

(

-)» (2n

-f

sin 2n+^B,

JQ

2m) (^zYn+im !

m\{tn+m)\{(n + m)l\ %

'

• Theorie der BesseVtchen Funetionen (Leipzig, 1867), p. 70. t Einleitung in die Theorie der BesteVtchen Funktionen, n. (Bern, 1900), pp. 81 85. of Neumann's treatise. J The formula is an immediate consequence of equation 16 on p. 69 was given, was first pub§ Math. Ann. in. (1871), p. 603. The memoir, in which 4his result lished in the Leipziger Berichte, mi. (1869), pp. 221—256.

—

— THE BESSEL COEFFICIENTS

2*61, 2'7]

Neumann

This result was written by

W

,

Jn{Z)

~

l(2n

(n\f I

+

l)

+

33

in the form

1.2.(2n

+

l)(2n

+ 2)

J'

where 1\

4

(2)

l = 2w + 2' 2m-

(2n+l)(2n + ~(2n + 2)(2w + (2,.

3) 4)'

+ l)(2tt + 3)(2n + 5)

"(2n + 2)(2n + 4)(2n +

This expansion

,

6)

a special case of a more general expansion (due to any two Bessel functions as a series of powers with

is

Schlafli) for the product of

comparatively simple coefficients

(§ 5*41).

Schlomilch's expansion of z m in

2*7.

a

series

of Bessel

coefficients.

We shall now obtain the result which was foreshadowed in §2*22 conis any cerning the expansibility of zm in a series of Bessel coefficients, where = has already been given in § 2*22 (7). positive integer. The result for

m

m

In the results §2*22(1) and (2) substitute for cos 2nd and sin(2n+ 1)0 These expansions are*

their expansions in powers of sins 6.

n.(n + s -1)!

cos2n#= 2

(-)'

,=o

The (

|

sin (x sin 9)

1

= 1 Jm+

-/i\

•

,

(X)

£(-) ij&±"î (2 sin *)»} !

J„ (x)

2 ^1

j

{

I

(-)•

£±gj£$| (* - •)""(

on the right as power we have

series

permissible to do

, n\ sin (z sin 6)= •

- J, (x) +

we rearrange the

it is

/

(

(2 sin ey»,

results of substitution are

cos (x sin 6)

If

that

wo„\t (n-s)!(2s)! ,

series in sin 6

•

(assuming

so),

irMLPvrnL? (-)"(2sin0)»f 3 2n.(n + s-l)\ (~)*(2sin0)» +1 5 X L -L

ro— TT\i

* Cf.

Hobson,

f

\

" (2n

2

+

—

l).(n +

r^-^r

s)!

r

,

J

«J»+i(*)

PZajie Trigonometry (1918), §§80, 82.

•

T

\


34 If

we

we expand

[CHAP.

II

the left-hand sides in powers of sin $ and equate coefficients,

find that

5 J2n (4

l=Jo(s) + 2

n=l

( I

*

"»-.

'

Jm+1

Jn^7)\

The first of these is the result already obtained bined into the single formula (i .

(1)

I C + *.)•(» + r = w=0

. -Dl

Jm>

(z >-

;

(

*

* °»

the others

w

*' 2>

• •

•>

may be com-

(m _ 1(

2|

3...)

m = 1, 2, 3, were given by Schlomilch* shewed how to obtain the general formula which was given explicitly some years later by Neumann f and LommelJ. The

He

particular cases of (1) for which

also

The rearrangement of the double series now needs justification the rearrangement if we can establish the absolute convergence of the double series. If we make use of the inequalities

is

;

permissible

IA» +1

^'<1,

(*)|<||êxr(iM*),

in connexion with the series for sin

« I2sin4j*+i » (2w

{z

+ l) .(n+«)!,

we

sin 6) T

,

see that

-

Nl

|2sin^p +

= sinh and so the series of moduli a similar manner.

is

convergent.

<»>.),

The

(|

2 sin

1)

»

-

|i*|

exp (£

series for cos (z sin $)

fc+ »

U

,. ,

l2N

2 |

),

may

be treated in

The somewhat elaborate analysis which has just been given is avoided Lommel's proof by induction, but this proof suffers from the fact that it supposed that the form of the expansion tion.

If,

following

is

known and merely needs

is

verifica-

Lommel, we assume that

i tor- n=0 * ZeitschriftfUr Math,

<» + * t

und Phys.

+

-

1 )'

>-iT n. (1857), pp.

jM ,(A

140—141.

f Theorie der BesseVschen Functionen (Leipzig, 1867), p. 38. t Studien liber die BesseF?chen Functionen (Leipzig, 1868), pp. 35 in

in

given later in this section.

—

36.

Lommel's investigation


2*71]

[which has been proved in

§ 2'22 (8) in .

n

n=0

35

m = 1],

the special case

S (m + 2n) (m + n — 1)

\«,+,

/i

v

,

r

,

!

i

2

(wi -Mi)!

»=0

n\

f

r

,

(m+l + 2n).Q + n)

»=0

w!

+ m)' «î»+sn (z)/n\-*-

as

line of the analysis is permissible.

duction holds for

Let t

m = 2,

w

,

n

i

/

,

x

,,

the rearrangement in the third

It is obvious

from this result that the

let

is

u be denned by the equation u= -

describes a small circuit round the origin (inside the circle

in-

1 1

= 1),

\

as follows

—

2

:

so that

,

when

u does the same.

have

wi!

^/ —

/"

lw

expi-ia^-l/O}---^*

r

-p<-^'-'/'»,

• (m +2n).(m + n-l)! . / m+2n — * »!

+2n)

,

-'

i!

-ir'

'--"

*

•

'

n=0

when we

,

3, 4,

be a complex variable and

We then

r

I

extremely elegant proof of the expansion, due to A. C. Dixon*, t

.

£* «J~+» (*)

.

,

r

-*• oo

y

r

,

,

"

_ Since (m

^-

x.

\

.

we have

-

(ra + w — 1)!) ((wi + n)! / = \z m Jrm (z) + r _l + 5 2^ r—

An

—

-

•

calculate the

sum

\

zh

of the residues at the origin for the last integral

;

the inter-

change of the order of summation and integration is permitted because the series converges uniformly on the contour and the required result is obtained. ;

Note.

2*71.

When m

is zero,

-y-

has to be replaced by

Schlomilch's expansions of the type

2n r JH

—~—

.

(z).

The formulae 1 (?nypJin {z)= 2

(1)

/^"S

2 (2n+lJ*+i J2n+1 (*)- 2 m=0

(2)

i^?* -* 9

1 ,

n=0 in

which p

is

any positive integer

[zero included in (2) but not in (1)]

and 1*£

cal coefficient, are evidently very closely connected with the results of § 27. * Messenger, xxxii. (1903), p. 8; a proof

viously given by Kapteyn,

on the same lines for the case Nieuw Archief voor Wiskunde, xx. (1893), p. 120.

»i

=l

is

a numeri-

The formulae had been

pre-

•

:


36

were obtained by Schlomilch,

Zeitschrift

gave, as the value of P^ p \ m

™ {)

fur Math, und Phys.

II.

[CHAP. (1857), p. 141,

II

and he

<* i-r-c*(»-**r

w»>

2m

m ~A=o

.m!

»

where m Ck is a binomial coefficient and the last term of the summation is that for which k hm - 1 or \{rti — 1). To prove the first formula, take the equation § 2*22 (.1), differentiate

is

2p times with respect «/<

2

xn

2

/o

x*.

to

0,

zero.

It is thus found that

^—

d *p

(-yfaiiPT]

cos

/x R**" w _[__

r

- ).£<*.). *.

The terms

and then make 6 equal to (* sin *)"! ;

of the series for which

contain no term in 0*, and so

(2^Ti

[_d6*>JLo

m>p,

it is sufficient

JL

J»-o"

j,. o =

V Z [^ Mi;

when expanded

(2m)

j,,,,

,

in ascending powers of

6,

to evaluate

(20""*

(a*»)l".iJoL«M*l

iJ»-o

fAz^ 2m 2m

P

m=0 v * wl7

!

Jfc-0

p

= 2(-)»lS

jP-Zff,

»=o

since terms equidistant from the beginning I-

The truth of equation manner from § 2*22 (2).

are equal.

similar

The reader

(1) is

and the end of the summation with respect to evident, and equation (2) is proved in a

now

will easily establish the following special cases,

which were stated by

Schlomilch

P J (*) + 33 Ja (i + 53 Jb (*) + =£ («+**), '2V2 ?)+4V4 W+eV6 («)+...=^,

4)

...

)

x

(

(

2.3.4«73 (*) + 4.5.6./6 (*)+6.7.8.f7 (*)+...=$*3.

2*72.

Neumanns expansion of z™ as a seizes of squares of Bessel coefficients.

From

Schlomilch's expansion

of even order,

it is

Bessel coefficients,

Thus,

if

by using Neumann's

z™

1

as a series of Bessel coefficients

integral given in § 2*6.

we take the expansion

a»m (2sin0)»»=

v 2

(2m v

and integrate with respect to

n-l)! + 2n).(2m '-A- +

0,

I?.? &>»*- I ir

(I) (1)

of

easy to derive an expansion of z*™ as a series of squares of

,

so that

(§ 2'7)

(when

Jo

»=o

we

/a J,Wi+m (2zsm0), .

find that

(^ + 2«)-(2m + n-l),

m

m > 0)

v (2m + 2n).(2m + r»-l) a *.„S2£ "(2m)!*, .*! M (**>

!

«/-*•<*>

-.


2-72]

This result was given by Neumann*.

this is true

As

when

special cases,

is

m = 0,

~ tJ.

*

4»=2

«»

*6= Il-f-'-J I

4 5 b »=3 .

we

tA\

4n* (4»»

€,

.

uh»- 1

.»»!(«»- 1)1

- 22) J„* (*),

4n* (4n2

- 22 )

(4n* - 4*)

./„* (*),

.

differentiate (1), use § 2*12 (2) /i

then reduces to Hansen's formula of

for it

we have

**= (3)

If

alternative form

<»^-S.L-ri£^<*

(2)

and

An

37

and then rearrange,

it is

Z (2m + 2n-l).(2m+»-2)!

,

readily found that

ST

an expansion whose existence was indicated by Neumann. * Leipziger Berichte, xxi. (1869), p. 226.

[Math. Ann. in. (1871), p. 585.]

§

25

CHAPTER

III

BESSEL FUNCTIONS 3*1.

The

The generalisation of

Bessel's differential equation.

Bessel coefficients, which were discussed in Chapter

of two variables, z and

n,

of which z is

required to be an integer.

We

now

shall

II,

are functions

unrestricted but n has hitherto been generalise these functions so as to

have functions of two unrestricted (complex) variables. This generalisation was effected by Lommel*, whose definition of a Bessel function was effected by a generalisation of Poisson's integral

;

in the course

shewed that the function, so defined, is a solution of the linear differential equation which is to be discussed in this section. Lommel's definition of the Bessel function Jv {z) of argument z and order v wasf

of his analysis he

J

v

(*)

= rfr +ffirft)

C

cos {z cos 6) sin2 "

0d0

'

and the integral on the right is convergent for general complex values .of v for which R(v) exceeds — |. Lommel apparently contemplated only real values of v, the extension to complex values being effected by HankelJ; functions of order less than - \ were defined by Lommel by means of an extension of the recurrence formulae of § 212.

The reader

on comparing § 3*3 with § 1*6 that Plana and Poisson had investigated Bessel functions whose order is half of an odd integer nearly half a century before the publication of Lommel's treatise. will observe,

We shall now replace the integer n which occurs in Bessel's differential equation by an unrestricted (real or complex) number§ v, and then define a Bessel function of order v to be a certain solution of this equation it is of ;

course desirable to select such a solution as reduces to

the integral value

We

v assumes

n.

shall therefore discuss solutions of the differential equation

z>*l

(1)

which

Jn (z) when

will

+z

^+

(z>- v>)

y = 0,

be called Bessel's equation for functions of order

v.

* Studieji iiber die BesseVschen

Functionen (Leipzig, 1868), p. 1. + Integrals resembling this (with v not necessarily an integer) were studied by Duhamel, Cours d Analyse, n. (Paris, 1840), pp. 118—121. %

Math. Ann.

§

Following Lommel, we use the symbols

m

i.

(1869), p. 469. v,

ft.

to denote unrestricted numbers, the symbols

is customary on the Continent, though it has not yet come into general use in this country. It has the obvious advantage of shewing at a glance whether a result is true for unrestricted functions or for functions of integral order only. «,

being reserved for integers.

This distinction

BESSEL FUNCTIONS

3-1]

39

Let us now construct a solution of (1) which is valid near the oiigin; the form assumed for such a solution is a series of ascending powers of z, say

y= 2 m-0

where the index o and the

+m cm z* , be determined, with the pro-

coefficients c m are to

viso that c is not zero.

For brevity the

which occurs in (1)

differential operator

be called V„,

will

so that

w£+»s+*-*

< 2>

It

easy to see that *

is

V, 2

cm z* +m

= 2

+ m)

c w {(a

2

-i/2 }s" +m

+ 2 cm z a+m+a

.

The expression on the right reduces to the first term of the first series, c (a8 — Vs) za if we choose the coefficients cm so that the coefficients of

namely

,

corresponding powers of z in the two series on the right cancel.

This choice gives the system of equations

+ l)*-* ca {(a + 2) -*, j+c c {(a + 3) -i/ + Cl 2

/(^{(a

}

2

2

2

2

8

}

=0 =0 =0

(3) c,„{(a

If,

+ ra) -v }+cM_2 = 2

2

we have

then, these equations are satisfied,

2

V„

(4)

cTO ^+'tt

=c

(aa -i/2 )2

a .

m-0

From

this result, it is evident that the postulated series can

of (1) only

values of

if

=+

a

v; for c is not zero,

and za vanishes only

z.

Now

consider the mth. equation in the system (3) written in the form cm (o

and

a-i>ora + i/isa

integer (when

We 3*5), *

- v + m) (a + v + m) +

so it determines cm in terms of

unless

a= — v)

Cm^

is

and then

(o

+ m) — p* 2

w > 1.

It can

be

= 0,

m

for all values of

negative integer, that

or unless 2v

Cm-*

when

greater than 1

— 2v

is a negative a negative integer (when a — v). is,

unless

disregard these exceptional values of v for the

When

be a solution

for exceptional

does not vanish

when

moment

m—

1,

(see §§ 3" 11,

2, 3, ....

It

now

the constants a and c m have been determined by the following analysis, the series

obtained by formal processes

is easily seen to be convergent and differentiable, so that the formal procedure actually produces a solution of the differential equation.

>

;


40

follows from the equations (3) that

pressible in terms of c c"*

=

d = c3 = c5 =

...

= 0,

and that

Ill

c 2Ml is ex-

by the equation

(-r*

.

(a-v + 2)(a-v + 4,)...(a-v + 2m)(a+v +

The system

[CHAP.

of equations (3)

is

now

satisfied;

+ v + 4>)...(a + v + 2my and, if we take a = v, we see

2)(a

from (4) that

—

1+ 2

c zv

(5)

m

«-i is

soli a formal solution of equation

!

(v

(-)" (**)""

+ 1) {v + 2)

(1).

we take

If

+ m) — a= v, we (v

. . .

obtain a second

formal solution c

(6)

(-)m (^y

'z-"\i+ s

m\(- v + 1) (- v + 2)

,-i

...

(-v + m)]'

In the latter, c has been written in place of c because the procedure of obtaining (6) can evidently be carried out without reference to the existence of (5), so that the constants c and c ' are independent. '

,

Any

may be

values independent of z

assigned to the constants

but, in view of the desirability of obtaining solutions reducible to v -*• n, we define them by the formulae * co

(')

The

= oi»rv..

i \ 2T(v+l)' and (6) may now be

series (5)

c°

„,=o

m\r(v + m +

*

l)'

and

c

'

2-T{-p + l)'

written

(-)m (bz) v+2m

|

=

°

i

c

Jn (z) when

„.

(

(-) w (i-g)~" +m

m!r(-i/ + m + l)'

In the circumstances considered, namely when 2v is not an integer, these series of powers converge for all values of z, (z = excepted) and so term-by-term

The operations involved in the analysis f by which they were obtained are consequently legitimate, and so we have obtained two solutions of equation (1).

differentiations are permissible.

The

first

of the two series defines a function called a Bessel function of

order v and argument

the symbol

J

¥ (z).

the present, 2v

is

z,

of the

Since v

is

first kindX',

and the function

not an integer), the second series

Accordingly, the function

is

denoted by

unrestricted (apart from the condition that, for

J

v

(z) is defined

is

evidently J_„

(z).

by the equation

MZ) = ? m!I> + m+l)m

(8) It is evident

from

when v is a a Bessel function of integral order being

§ 2*11 that this definition continues to hold

positive integer (zero included),

identical with a Bessel coefficient. * For properties of the Gamma-ftmction, see Modern Analysis, ch. xn. t Which, up to the present, has been purely formal. X Functions of the second and third kinds are denned in §§ 3*5, 3 54, 3-57,

3-6.

'

'

BBSSBL FUNCTIONS

3*11]

An interesting symbolic

41

solution of Bessel's equation has been given

by Cotter*

in the

form

[l+zv D- 1 z- 2 where Dsd/dz while

A

and

"- 1

D- 1 zv+1 ]- 1 (Azv +Bz- v

B are constants.

may be

This

),

derived by writing successively

[D(zD~2v)+z]zv y=0,

[zD-2p+D~ 1 z]z"y= -2vB, zD (z~ vy) + z~ 2v D~ V+V= - 1vBz- % \ x tv z- vy+D- z-^- x D- x z v+l y=A +Bz~ which gives Cotter's

Functions whose order

3'11.

,

result.

is

half of an odd integer.

two cases of Bessel's generalised equation were temporarily omitted from consideration, namely (i) when v is half of an odd integer, (ii) when v is an integerf It will now be shewn that case (i) may be included in the general In

§ 3*1,

.

theory for unrestricted values of

When

v is half of

an odd

v.

integer, let

S = (r + $r, where r If

is

we


take a = r

+£

we find =0, 1 / ,v r! (cTO .m(m + 2r+l) + cm _ 2 = 0,

in the analysis of § 3*1, (c 1

(1)' v

that

.l(2r + 2) .

,,

.

(m>l)' v

ci

and so Ctm

(2)

which If

we

is

=

(— y» q 2.4...(2m).(2r + 3)(2r + 5)...(2r +

c^ given by

the value of

when a and

§ 3'1

27/i

+ l")

v are replaced

by r

+ \.

take 1

c

»-2~r"+4r(r + f)'

we

obtain the solution

§

31

is

by the symbol

naturally denoted

however,

we take a = — r —

As

§,

so that the definition of

the equations which determine cm become

=0,

foi.l(-2r)

(S) (3)

\cm before, c lt c3 ,

...

}

cv -i are all zero,

this equation is satisfied

defined by the equation . 27,1+1

= (2r

Cgr+i

but the equation to determine c^+x

+ Czr— — l

-r c

(-)w

2r

is

0,

by an arbitrary value of

+ 3) (2r + 5) ... (2m +

* Proc. R. Irish.

(m>1)

.m(m-l -2r) + Crn-* = 0. •

is

Jr+ ^(z),

(8) is still valid.

If,

and

m + f)'

m!r(r +

,„=

which

Car+r>

when

+i

1)

.

2

.

4

...

(2m- 2r)

Acad. xxvn. (A), (1909), pp. 157—161. f The cases combine to form the case in which 2v is an integer.

ra

> r,

C2m+i


42 If

Jv (z)

be defined by

§ 3*1 (8)

when v» — r —

\,

[CHAP.

the solution

now

Ill

con-

structed is* Co

It follows that

v=±(r + £);

2-"-*

T(\-r) /_,_! (z) + Czr+1 2*+* r (r + f) Jr+i (z).

no modification in the definition of

Jv (z)

is

necessary

the real peculiarity of the solution in this case

is

when

that the

negative root of the indicial equation gives rise to a series containing two arbitrary constants, c

and

c^+i,

i.o.

to the general solution of the differential

equation.

A fundamental system

3*12.

of solutions of Bessel's equation.

It is well known that, if yx and ya are two solutions of a linear differential equation of the second order, and if y( and y2 denote their derivates with respect to the independent variable, then the solutions are linearly inde'

pendent

if

the Wronskian determinant^ y*

y*

yl

y*

if the Wronskian does vanish identically, two solutions vanishes identically, or else the ratio of

does not vanish identically; and then, either one of the

the two solutions If the

is

a constant.

Wronskian does not vanish

identically,

differential equation is expressible in the

form

cx

yx

then any solution of the

+ d y2

where

cx

and

ca are

constants depending on the particular solution under consideration; the solutions

y and y2 are then x

said to form a fundamental system.

For brevity the Wronskian of yx and y% W,{y,,y»}, the former being used

We now If

we

when

it is

will

8K{yi,y.},

necessary to specify the independent variable.

proceed to evaluate

"multiply the equations

V v J_ v {z) = 0,

V v J,(z) = Q

by J„ (z), J_ v (z) respectively and subtract the which

may be

v {z),

we obtain an equation

j-^yi-q,

connexion with series representing this solution, see Plana, Mem. della R. Aeead.

Sci. di Torino, xxvi. (1821), pp.

f

results,

written in the form

jz [zm{j * In

be written in the forms

For references to theorems concerning Wronskians, see Encyr.lqpidie des

(§ 23), p.

109.

delle

519—538.

Sci. Math. n. lfr Proofs of the theorems quoted in the text are given by Forsyth, Treatise on

—

Differential Equations (1914), §§ 72

74.

BESSEL FUNCTIONS

3*12]

43

and hence, on integration,

m[j

a)

G is

where

J-Az))=-,

v {z),

z

a determinate constant.

To evaluate we have

J

"

we observe

C,

=

{z)

{l

l>TT)

with similar expressions for

when

that,

v is not

{z% J"' {z) =

+

J-V (z) and

an

integer,

m>5

+

1

^

and

and hence

«/'_„ (z);

{

-.

2sini/7r

If

we compare

right which

is

m

.

is

= _ *!E^r.

[jv {z\ j_ v (z)}

TTZ

not zero (because v

is

not an integer), the functions

form a fundamental system of solutions of equation

«/_„ {z)

When

v is

and when v

an

integer, n,

made equal

is

we have seen

to

nK) Since the

first

+ °«

this result with (1), it is evident that the expression on the 0(z) must vanish, and so*

(2)

Since sini^r

1 )}

'

V

TTZ

small,

W>

W^W- J-W: W-i r(^T(^)-r W r(-, +

J.

\z\ is

—n

in § 3*1 (8),

mZ

n terms of the

that,

§

31

(1).

with the definition of §

we

Jv (z),

21

(2),

find that

m\r(-n + m + Ty

last series vanish,

the series

is

easily reduced to

(— )n Jn(z)y so that the two definitions of J-n(z) are equivalent, and the functions Jn (z), J_n (z) do not form a fundamental system of solutions of Bessel's equation for functions of order n. The determination of a fundamental system jn this case will be investigated in

§

363.

Jv (z) is defined, for all values of v, by the and J¥ (z), so defined, is always a solution of the equation V„t/ = 0. When v is not an integer, a fundamental system of solutions of this equation is formed by the functions Jy (z) and «/_„ (z). To sum

up, the function

expansion of §

A

31

(8);

generalisation of the Bessel function has been effected

researches on "basic numbers."

Briefly,

Gamma

function

the base, and the basic

Tp

(u

+ l) = [u].rp (v).

The

*

rp

is

H. Jackson in his vn — 1 defined as * , where pis

(v) is defined to satisfy

F.

the recurrence formula

by replacing the numbers which occur in the by basic numbers. It has been shewn that very many theorems

basic Bessel function is then defined

series for the Bessel function

This result

making

a basic number [»]

by

z

-*•

is due to Lommel, Math. Ann. iv. (1871), and using the approximate formulae which

p. 104.

He

derived the value of

will be investigated in

C

Chapter vn.

by


44

[CHAP.

Ill

concerning Bessel functions have their analogues in the theory of basic Bessel functions, but the discussion of these analogues is outside the scope of this work. Jackson's main

be found in a series of papers, Proc. Edinburgh Math. Soc. xxi. (1903), pp. xxn. (1904), pp. 80—85; Proc. Royal Soc. Edinburgh, xxv. (1904), pp. 273—276; Trans. Royal Soc. Edinburgh, xli. (1905), pp. 1—28, 105—118, 399-408; Proc. London Math. Soc. (2) I. (1904), pp. 361—366; (2) II. (1905), pp. 192—220; (2) ill. (1905), pp. 1—23.

results are to

65—72

;

The more obvious generalisation of the Bessel function, obtained by increasing the number of sets of factors in the denominators of the terms of the series, will be dealt with in § 4 4. In connexion with this generalisation see Cailler, Mem. de la Soc. de Phys. de #

Geneve, xxxiv. (1905), p. 354; another generalisation, in the shape of Bessel functions of two variables, has been dealt with by Whittaker, Math. Ann. lvil (1903), p. 351, and Peres,

Comptes Rendus, clxi. (1915), pp. 168

General properties of Jv (z).

3'13.

The closed

series

domain

which defines

when

Jv (z) converges absolutely and uniformly* in

of values of z [the origin not being a point of the

R (y) < 0], and in For,

— 170.

\v\

any bounded domain of values of

%

N and

-I*

is

m (m — A ) r

m(v + m) whenever

v.

% A, the test ratio for this series

\z\

any domain when

m is taken to be greater than the positive root of the equation m'-miV-|A = 0. 8

This choice of to being independent of v and

the result stated follows from

z,

the test of Weierstrass.

an analytic function of z for all values ofz(z = possibly an analytic function of v for all values of v. An important consequence of this theorem is that term-by-term differentiations and integrations (with respect to z or v) of the series for Jv (z) are Hence f

Jv (z)

being excepted)

is

and

it is

permissible.

An

inequality due to Nielsen J should be noticed here, namely

W=tv£t)( 1+ »>

(i)

where

and

j

v

|

+l

\

is

the smallest of the

6\

1'

{j—Ji]}v+2 numbers v + 1 |

1,

|

|,

1

1»+3|,

This result may be proved in exactly the same way as § 2'11 pared with the inequalities which will be given in § 3*3. Finally, the function

z",

which

is

a factor of

* Bromwich, Theory of Infinite Series, § 82. + Modern Analysis, % 5-3. % Math. Ann. lii. (1899), p. 230; Nyt Tidsskrift, (1902), p. 494.

ix.

B

Jv (z),

....

(5)

;

it

should be com-

needs precise specifica-

(1898), p. 73; see also

Math. Ann.

lv.

l

BESSEL FUNCTIONS

3-13, 3-2] tion.

We

given

its principal

define

be exp (v log z) where the phase (or argument) of z

it to

When

it is

3*2.

is

value so that

— ir < arg z values of arg

45

%

ir.

Jv {z) outside this range of be made of the process to be carried out.

necessary to "continue" the function

z,

explicit

mention

will

The recurrence formulae for

J

v

(z).

Lommel's generalisations* of the recurrence formulae

for

the Bessel co-

efficients (§ 2'12) are as follows:

o„

(1)

^(fJ + ^W-v/rW z

(2)

J _ (z)-Jv+1 (z)=2J,'(z), v

l

(3)

zJv'(z) + vJv (z) = *J*-^)>

(4)

zJ: (z)

-vJ

v

= - zJv+1 (z).

{z)

These are of precisely the same fonn as the results of being the substitution of the unrestricted number v

To prove them, we observe

±

i

,

first

« **

for

the only difference

the integer

n.

that

(-)»j*h«»

it V - A v

~

§ 2*12,

/ 7\.

Zo2"-

_ \m ^2 v—

1+im

i

+ai»

.m\r{v + m)

- z" Jv^ (z). When we

differentiate out the product

on the

left,

we

at once obtain (3).

In like manner,

£ dz

i \

z-krJT

(-) m z*m

\\-A 5

'WJ-rfj mt 2"™.»i!r(v + » =1

whence

(4)

tracting (3) *

when

is

.

/

+m+1 w = 2'

obvious; and (2) and (1)

and

+

l)

/_\r»^2wi—

2 K+2m-i

oc

=

»»

.

(

W _ 1)! T

(l/

+W+

1)

\m+i £»n+i

ml r

may be

(v

+

w + 2)

obtained by adding and sub-

(4).

StUdien Uber die BesseVschen Functionen (Leipzig, 1868), pp. 2, 6, 7. Formula (3) was given v is half of an odd integer by Plana, Mem. delta R. Accad. delie Sci. di Torino, xxvi. (1821),

p. 533.

—

1

THEORY OF BESS EL FUNCTIONS

46

We

now obtain the

can

Ill

generalised formulae

(AJ [zvj

(5)

[CHAP.

>

{z)]

= *-~ j-« <*>'

W (6)

ir-J.{*)} =(-)'» z—>"Jv+m (z)

(^)

by repeated

Lommel

differentiations,

obtained

all

m is any positive integer.

when

these results from his generalisation of Poisson's

which has been described in

integral

The formula

(1) has

§ 3*1.

been extensively used* in the construction of Tables

of Bessel functions.

By and

expressing

(4),

we can

«/„_!

J_ x

v

(z) in

terms of J±„ (z) and J' ±v

by

(z)

(3)

/vtv + JL„ T-/\r/\2 = -— J

T

J. (z) J^ v

(7)

from formula (2) of

An

0) and

derive Lominel's formula f

sin vtr

(z)

(z)

v .,

(z)

7TZ

312.

§

interesting consequence of (1)

aud

(2) is that, if

Q y («) = J"„2 (2),

then

(8)

this formula

was discovered by Lommel, who derived various consequences of it, Studien Functionen (Leipzig, 1868), pp. 48 et seg. See also Neumann, Math.

iiber die Bessel'schen

Ann.

111.

(1871), p. 600.

321. Bessel functions of complex order. r

and imaginary parts of the function J „ +£M (x), where v, /* and x been discussed in some detail by Lommel J, and his results were have real, are subsequently extended by B6cher§.

The

real

In particular, after denning the real functions

K

VtH.{x)

and

S

v>ll

(x)

by the

equation 1

Lommel (1)

obtained the results

~ {K

dx*

Vi

*(a)

± iS„(x)}

+ {K Vth (x) ± iS„(w)}

+ 2(»±fr)

+ l*

x

(2)

K

(3)

S, +I>M (x)

w+ltM (x) .

dx

[Kv ^x) ± iS„{*))

- 0,

= K {x) + K" (x), = 8VtH (x) + S"„(x), VttL

Vtll

.

Lommel, Milnchener Abh. xv. (1884—1886), pp. 644—647. t Math. Ann. iv. (1871), p. 105. Some associated formulae are given * See, e.g.

X Math. Ann. in. (1871), pp. 481—486. § Annals of Math. vi. (1892), pp. 137—160. The reason for inserting the factor on the right I!

established in § 3-3.

is

in § 3*63.

apparent from formulae which

will

be

BESSEL FUNCTIONS

3-21, 3-3]

47

with numerous other formulae of like character. These results seem to be of no great importance, and consequently we merely refer the reader to the memoirs in which they were published.

In the special case in which v = 0, Bessel's equation becomes

solutions of this equation in the form of series were given

by Boole* many

years ago.

33. Lommel's expression of Jv (z) by an integral o/Poisson's

We

shall

now shew

when R(v) > -

that,

type.

£, then

Jw (z) = _^g.'_ JJcos (* cos 6) sm*>0d0.

(1)

was proved by

when 2v is a positive integer (zero inon the right is a solution of Bessel's equation and expression was adopted by Lommelj as the definition of Jv (z) for positive

It

Poisson-f- that,

cluded), the expression this

;

+ \.

values of v

Lommel subsequently proved generalised equation

defined

Jv (z)

and that

that the function, so denned,

a solution of Bessel's 32 and he then ;

(-£, -#),(_#, _£), (J| -J),

for values of v in the intervals

t

cessive applications of § 3 - 2

To deduce

is

the recurrence formulae of §

it satisfies

Jv {z) adopted in this work, we Jv (z) in the following manner:

(1) from the, definition of

form the general term of the series (-)rw (i*),+swt

...

by suc-

(1).

for

(-y*Q*Y

=

** r(y+j)r(m + i)

m!I> + m + l) I> + £)r(*)-(2ro)f

R (v) > — |. Now when R (v) > £, the

trans-

I> + m + l)

provided that

series

„l= i

(2wz)!

converges uniformly with respect to t throughout the interval (0, ), and so it 1 may be integrated term-by-term on adding to the result the term for which ;

* Phil.

Tram, of

the

Royal Soc. 1844,

p. 239.

See also a question set in the Mathematical

Tripos, 1894. t Journal de VEcole R. Poly technique, xn. (cahier 19), (1823), pp. 300 et teq., 340 et $eq. Strictly speaking, Poiason shewed that, when 2v is an odd integer, the expression on the right

multiplied by Jz is a solution of the equation derived from Bessel's equation by the appropriate change of dependent variable.

t Studien

liber die

BetseVschen Functionen (Leipzig, 1868), pp. 1

et seq.


48

m = 0,

namely

— t^dt,

<"~*(1

/

which

is

convergent,

[CHAP.

we

find

that,

Iir

when

whence the result stated follows by making the substitution t = sin2 6 and using the fact that the integrand is unaffected by writing ir — 6 in place of 6-

When -•&< R(v)<$,

the analysis necessary to establish the last equation is a little to be to take the series with the first two

The simplest procedure seems terms omitted and integrate by parts, thus more

elaborate.

m _2

(2m)!

Jo

m=2>'

_

+i

(2m)!

Jov+iU

The interchange

(1

(2mj!

2

»+**ui

/

on integrating by parts a second time.

./

(2m)i

(1

'>

°

of the order of

J* j*

summation and

integration in the second line of analysis is permissible on account of the uniformity of

convergence of the

series.

On

adding the integrals corresponding to the terms

m»0, m = 1

(which are convergent), we obtain the desired result. It follows that,

J- <" "

when R{v) > — \, then

r(5(i) />-* V ~ ** cos

Obvious transformations of this <2 >

J

-

<*>

-

<3>

* <*> -

<5)

^

<6 >

J "« =

(z)

=

nMm fj 2

+i)r(» C'

is

v

(*>

*

- ,r * cos (zt)

*

cos (z cos 9) sin" " d "'

"^

partial integration of (5),

^

= ^TlVffe

sometimes usefuiy^t

cos <*>

>0,,sin8

1

j

1

<'>"""'

i>^f)Ta)J.

The formula obtained by a (7)

G - '« dt

result, in addition to (1), are the following:

(1 nffir-w ,C -

r(-

I*

^s

valid,

only

f

^

namely

""^ cos (« cos ^) sin2

when R{v)>

\.

^'

BESSEL FUNCTIONS

3*31]

49

An expansion involving Bernoullian polynomials has been obtained from with the help of the expansion rfl

f-*-(i-.-)i

'1

n (£)

[Note.

+ 1)!

denotes the nth Bernoullian polynomial and a—izt.

Integrals of the type (3) were studied before Poissou by Plana, Mem. delta R. Sci. di Torino, xxvi. (1821), pp. 519 538, and subsequently by Kummer,

—

Accad. delle

Journal

by Nielsen*

*" +,tf+1) (n

'„=„ in which

(4)

—

Math, xn. (1834), pp. 14-4 147; Lobatto, Journal fiir Math. xvn. (1837), pp. 363—371; and Duhamel, Cours d' Analyse, II. (Paris, 1840), pp. 118 121. filr

—

A function, substantially equivalent to Jv J{fjL,x)=\

(1

(2),

denned by the equation

- v2 )* cos vx

.

dv,

J o

was investigated by Lommel, Archiv der Math, und Phys. xxxvu. (1861), The converse problem of obtaining the differential equation satisfied by

£

e

f*

zv

pp.

349—360.

1

{v-af- (v-P)"- 1 dv

was also discussed by Lommel, Archiv der Math, und Phys. XL. (1863), pp. 101—126. In connexion with this integral see also Euler, Inst. Cole. Int. II. (Petersburg, 1769), § 1036,

and

Petzval, Integration der linearen Differentialgleichungen (Vienna, 1851), p. 48.]

3*31.

Inequalities derived from Poissoris integral.

From

§

33 (6)

(i)

By

it

follows that, if v be real

!-/.<*)!«

rj+ffrft-j

using the recurrence formulae §

and greater than

/„' exp

3-*2

(1)

l

'

w

and

'

sin "

(4),

— \,

then

ede

we deduce

in a similar

manner that (2)

(3)

By

|JpW

|

i^jl + jFi lggL

}exp|/W|

i^wK^lJi + j-^yî/wi using the expressionf

(1) is valid

when

v

{2/(7rz)}* cos

<-|<„«-i>,

J]

z for

J^ (z)

(„>-*). it

may be shewn

that

= — \.

These inequalities should be compared with the less stringent inequalities obtained in § 313. When v is complex, inequalities of a more complicated character can be obtained in the same manner, but they are of no great importance. *

Math. Ann.

lix. (1904), p. 108.

The notation used

in the text is that given in

Modern

Analysis, § 7*2; Nielsen uses a different notation.

t The reader should have no

difficulty in verifying this result.

general theorem will be given in § 3*4.

A

formal proof of a more

* THEORY OF BESSEL FUNCTIONS

50

Gegenbauer's generalisation of Poisson's integral.

3*32.

The

integral formula

~« - fRPp|^ />--- 9.CJ

J

(1 >

in

[CHAP. HI

which

Gn v (t)

is

the coefficient of a" in the expansion of (1

and n

is

rfft

— 2at + o

-"

2

)

in

the formula is valid when a, is due to Gegenbauer* any of the integers 0, 1, 2, .... When n = 0, it obviously

ascending powers of

R (v) > — \

(cos *)

;

reduces to Poisson's integral.

In the special case in which v =

Jn +

(2)

this equation has

h

(z)

= (_i)«

,

,

the integral assumes the fonn z cos 9

(£ff/

Pn (cos 0) sin dd

;

been the subject of detailed study by Whittakerf.

To prove Gegenbauer's

'»*

'

\

formula,

we

take Poisson's integral in the form

W - r(, + i'5*)r( )i>

- (=) ' + ""'

(

(1

t

and integrate

^ Now

» times

,,()

it is

by parts

(-2i)nr(,

the result

;

is

e

+ ,,4-^ra)J_ 1

p-

<**»

1

known thatj

dn (l- t*)*+»-l

(- 2)» n T ( v + n + !

T (2i/)

$)

(1

-«•)-*

Cn "(0,

whence we have

w

^

J

(3)

t/

n

- (-O -r(2,).7 ,!(^ p ^-r( v + i)r^)r(2 + n)J_ ?

I;

and Gegenbauer's result

A

this

e

(2) (

is

°

-

c »w<«.

evident.

symbolic form of Gegenbauer's equation

was given by Rayleigh§

The reader

(1

1

is

in the special case v

will find it instructive to establish (3)

=h

by induction with the

aid of the

recurrence formula

* Wiener Sitzunasberichte, lxvii. See also Bauer, (2), (1873), p. 203; txx. (2), (1875), p. 15. Milnchener Sitzungsberichte, v. (1875), p. 262, and 0. A. Smith, Giornale di Mat. (2) xn. (1905), The function Cn " (t) has been extensively studied by Gegenbauer in a series of pp. 365 373.

—

memoirs

in the

Wiener Sitzungsberichte

;

some

of the

more important

given in Modern Analysis, § 15'8. + Proc. London Math. Soc. xxxv. (1903), pp. 198-206.

t Cf. Modern Analysis, § 15*8. § Proc. London Math. Soc. iv. (1873), pp. 100, 263.

results obtained by

See §§ 6-17, 10

5.

him

are

"

BESSEL FUNCTIONS

3*32, 3*33]

A formula which is a kind of converse of (4), ***

(5)

in

which

1

i*"* denotes

(1903), p. 198

namely*

(7(^)) - r(î^+V^ (P »)

a generalised Legendre function,

the proof of this formula

;

51

is left to

is

due to Filon,

Gegenbauer's double integral of Poisson's type.

It has

been shewn by Gegenbauerf that, when

Jv («r ) =

v

* I

Mag.

(6) vi.

R (v) > 0,

— tV (cos cos0 + sin <£ sin

exp [iZ cos

I

Phil.

the reader.

3*33.

(1 )

*"'

sin

2

""1

^ sin

cos ^r)] 3-

0d^d0,

«r = Z* + z* - 2 Zz cos <£ and Z, z, are unrestricted (complex) variables. This result was originally obtained by Gegenbauer by applying elaborate integral transformations to certain addition formulae which will be discussed in Chapter xi. It is possible, however, to obtain the formula in a quite natural

where

3

manner by means of transformations of a type used

in the

geometry of the

sphere J.

when

After noticing that,

z

= 0,

the formula reduces to a result which

is

an obvious consequence of Poisson's integral, namely

Jv (Z) =

%-r-r

TT 1

we proceed unit sphere this point

to regard

yfr

(* eiz «»• sin8"

and

f sin *" 8

.

(V) Jo

1

yjr dyjr,

.'

and colatitude of a point on a

as longitude

we denote

;

by

(I,

the direction-cosines of the vector from the centre to m, n) and the element of surface at the point by dto.

We

then transform Poisson's integral by making a cyclical interchange of the coordinate axes in the following manner§ :

J* (•)

=

e ^FJ-\ I"!*

ttJ

_

iv *

*9

sin2 '

sin8-- 1 fd0dyfr

{v))} m> o

(W

= -Â

[(

/

e%v sm 9 cos * cos2" _1 e sin

Ofy d0-

* It is supposed tbat

9*y _ Y{y + l)z v ->x ~ r("-M+i) dz*

—

t Wiener Sitzungsberichte, ucxiv. (2), (1877), pp. 128 129. t This method is effective in proving numerous formulae of which analytical proofs were given by Gegenbauer ; and it seems not unlikely that he discovered these formulae by the method in question

;

cf.

§§ 12*12, 12-14.

(1880), p. 328, for

§

The symbol

which

The

device is used by Beltrami, Lombardo Reiidiconti, (2)

xiii.

a father different purpose.

m ^ means

jj «i is positive.

that the integration extends over the surface of the hemisphere on


52

Now

[CHAP.

Ill

an integral periodic function of yjr. and so the limits of yfr may be taken to be a and a + Itt, where a is an arbitrary (complex) number. This follows from Cauchy's theorem. the integrand

is

integration with respect to

We

thus get

J

v

(w)

/'i"'/" a

(ktffY

=

+2w e iv sin

/.

-,

e cos

* cos2 " -1

sin Bdslr d$

cos2 ""1

sin ddylrdd.

(v)Jo J*

TTl

W

= -te K

Wio

irl

We

now

define a

'

e

I *l

iw8i,l * C08 (* + °>

.'o

by the pair of equations

«r cos

= Z — z cos

o

«r sin

,

a

— z sin

,

so that

J

v

(a)

=

/*" f 2 "

(Aw)" L,

-

exp

I

1

[i

(Z — z cos

cos

sin

)

yfr

— iz sin 2

con

The only

Jv {m) = is in

between

difference

/

exp [iw

sin

t/t

sin 0]

6 sin Bdyfrde.

and the formula

this formula

v

l, 7Tl (v) Jo

'- 1

cos -f] cos 2

sin

"- 1

sin 0ety> d0

.'

the form of the exponential factor

and we now retrace the steps of the

;

When

analysis with the modified form of the exponential factor.

the steps are

retraced the successive exponents are

i(Z — z cos )l — iz sin i(Z — z cos

%{Z—z cos The

)n )

m,

.

(f>

— iz sin — iz sin <£ cos

.

I,

cos

yfr

sin

0.

cos

\jr),

last expression is

iZ cos

— iz (cos

cos

+ sin

sin

so that the result of retracing the steps is

ff ex P [*^ cos

(IwY y,,

I

— ** ( C(>s

<}>

+ sin

cos

sin

sin

and consequently Gegenbauer's formula The

[Note.

1789, p. 372,

3*4.

We

cos

yfr)]

^ sin

It will

0d\frd0,

seems to be due to Legendre, M4m. de

now deduce from

appear later

not so expressible

;

manner of I'

Acad, des

Sci. in. (1818), p. 126.]

{z) infinite terms.

Poisson's integral the important theorem that,

when v is half of an odd integer, the function Jv (z) is expressible by means of algebraic and trigonometrical functions of z. is

2"

established.

and Poisson, Mem. de FA cad. des

The expression of
"" 1

device of using transformations of polar coordinates, after the

this section, to evaluate definite integrals Sci.,

is

2

when

in finite teivns

such a value, then ,TV (z) but of course this converse theorem is of a much more (§

474)

that,

v has not

recondite character than the theorem which

is

now about

to be proved.

.

BESSEL FUNCTIONS

3*4] [Note. Solutions in

finite

terms of differential equations associated with

tained by various early writers

;

J

)t

2*

Jn+i

Jn+

i

was observed by Euler, Misc. Taurinensia,

it

1765), p. 76 that a solution of the equation for e i2

the equation satisfied by

53

was solved

(.s)

+

,

were ob-

(1762—

expressible in finite terms

;

while

terms by Laplace, Conn, des

Terns.

(z) is

in finite

(z)

in.

1823 [1820], pp. 245—257 and Mecanique Celeste, v. (Paris, 1825), pp. 82—84 della R. Accad. delle Sci. di Torino, xxvi. (1821), pp. 533—534; by Paoli,

;

by Plana, Mem. Mem. di Mat. e

di

Fis. (Modena), xx. (1828), pp. 183—188; and also by Stokes in 1850, Trans. Camh. Phil. Soc. ix. (1856), p. 187 [Math, and Phys. Papers, n. (1883), p. 356]. The investigation

which will now be given is based on the work of Lommel, Studien Fimctionen (Leipzig, 1868), pp. 51 56.]

—

It

is

by

then,

BesseVsehen

convenient to restrict n to be a positive integer (zero included), and § 3-3 (4),

^(^ST.^d-^ n

!

dt

\Zv '_!

e

nly/ir

when we

liber die

r+x r=0 z

\_

+ times degree 2n, the process then terminates. integrate by parts 2n

1

;

dtr

since (1

J_i'

-

t*)

n is a polynomial of

To simplify the last expression we observe that if dr (l -t2 n /dtr be cal) culated from Leibniz' theorem by writing (1 -t2 n = (1 - t)n (l + t) n the only ) term which does not vanish at the upper limit arises from differentiating n ,

times the factor (1 - t)n and therefore from differentiating the other factor r— n times; so that wo need consider only the terms for which r^n. ,

[tOfff]

Hence

[*&=!*j

and similarly

» -5^

=(-)», 0.

_(_>-., Q, .„

J^

.

It follows that

Um

* KZ)

Jir

(

}

L

r

rn

^+1 .(r-n)!(2n-r)! 2«

+ (_yi+i e-i2 v V '

/"_ v

,.-«s' +1

«V+l Oan-r V L

-, -

1

~\

t •

.(r-w)!(2n-r)!

'

and hence (1)

This result (2)

=

-J— [>

may be

*~"
i

+ r)

;-|

WL

J^(.)-fiy[«n(,+1W i „)'S' -,<-£<»±*2L_ 5 (2r)I(n-2r)i(2*)» r-c,

2

P

A

(- t-r-n-x (n

written in the form*

cos^-i^r' _o *

•

compact method of obtaining

this

Soc. Sci. de Bruxelles, xxix. (1905), pp. 140

formula

— 143.

(-M* +»+!)! (2?is

+ l)!(n-2r-l)!(2^'+'

1 '

given by de la Vallee Poussin, Ann. de la

J THEORY OF BESSEL FUNCTIONS

54

[CHAP.

Ill

In particular we have

= (—)

Jj (*)

(3)

the former of these results

sin

J

m,

=

(z)

%

S

(-)

(

-^ - cos z)

also obvious from the

is

power

;

series for J± (z).

Again, from the recurrence formula we have

= (")n * n+i (^)" {*"* J* (*)i.

•/»* <*)

and hence, from 6

i

v

iz

r

(1),

-n

-(n

+ r)l

»

(~i)r n .(n+r)\

~

rl(n-r)l(2zy

_.

2

,~ rl(n-r)\(2zy

*

we can express

But, obviously, by induction

\zdz) as a polynomial in \jz multiplied o±iz

•

'

,_"o

e

(±»y-(~ + r)

- r)

r (n !

S

,

l

(2*)

r

~

V

z

and

so

we must have

y.W<*

V****.

WW

;

*

'

the preceding identity would lead to a result of the form

for, if not,

where fa

by

±iz

(z)

and fa (z) are polynomials in

and such an identity is obviously

1/z;

impossible*.

Hence

it

follows thatf "

r -o

r!(n-r)!(2^"

1

e r!

,=

(w-r)! (2-) r

n

= (-r (2tt>

*»+>

{*-*

i)

(

= (-)»(27r^./_n_

J

j_j (*>i

(^).

Consequently (4)

* Cf.

+

J

-J— [> W _ v(te) M;a)= ^

| 2

**».(» + + r)n _____ + Î (-»Y+».(n _ ^ r)i

e

rS(n

Hobson, Squaring the Circle (Cambridge, 1913),

From

m-_r(j)

it is

p. 51.

the series *

j-* u obvious that

J,

,

(e)

(-r <**)»»

I

„,to»»4-# ...(m-i)'

=

(

—\

cos

z.

r)!(2

j

BBSSEL FUNCTIONS

3*41]

55

and hence

j^ w -(i)*r

00 .(,

(5)

+ j«,? *

77

yr^L!^_

v

(2r+l)!(>i-2r-l)!(2«)2

'

r-o

•

+1 _

In particular, we have

•mo- £)'«.* ^w-(i,y(-sf'-™.)

(6)

We

have now expressed in

finite terras any Bessel function, whose order is by means of algebraic and trigonometrical functions.

half of an odd integer,

The explicit expression of a number of these functions can be written down from numerical results contained in a letter from Hermite to Gordan, Journal fur Math. lxvi. (1873), pp.

3*41.

303—311.

Notations for functions whose order


is

Functions of the types «/±(n+j)(s) occur with such frequency in various branches of Mathematical Physics that various writers have found it desirable to denote them by a special functional symbol. Unfortunately no common notation has been agreed upon and none of the many existing notations can

be said to predominate over the others. Consequently, apart from the summary which will now be given, the notations in question will not be used in this work. In his researches on vibrating spheres surrounded by a gas, Stokes, Phil. Trans, of the clviii. (1868), p. 451 {Math, and Phys. Papers, iv. (1904), p. 306], made use of

Royal Soc. the series

"

r

which

is

n(» + l ) 2.imr

(»-!)»(» + !)

(» + 2 ) " 2. 4. fair) 2

K "'

annihilated by the operator

d* dr2

.

d

n(n + l)

dr

r2

This series Stokes denoted by the symbol /„(»') and he wrote >•*»

where

and

Sa

'

Sm e ~ *»<•/„ (r) + Sn

are zonal surface harmonics

«'*•/.

'

(

so that

;

" *'), fa

is'

annihilated by the total

operator rf2

dr1

and by the

+ 2 d 4-,»a r

M (" + 1 )

dr

'

r2

partial operator 2

—

'•' l 2 cr*

2 d

1

6

f

.

55 c6

\ {

+ -*-+., - r- sin 6. " or r ,

,

.

.

v)

sin 6 52}

+ m2

.

dff)

In this notation Stokes was followed by Rayleigh, Proc. London Math. Soc. iv. (,1873), 103, 253—283, and again Proc. Royal Soc. i.xxii. (1903), pp. 40—41 [Scientific v. (1912), pp. 112—114], apart from the comparatively trivial change that Rayleigh would have written/,, fair) where Stokes wrote/, (/•).

—

pp. 93 Papers,

—

\

-

'


56

In order to obtain a solution

Sn = '

(

n+ )

1

Sn

finite at

the origin, Rayleigh found

from

§ 3-4 that °

i (ir\ Q?-' =

-

r \ n e~

ir

/

rn+1

( %

Jn+i {r) = -j^

and that

n [e- ir i +

it

Ill

necessary to take

and then

in the course of his analysis,

t>

It follows

[CHAP.

-4-

ir

)

,

\ rcrj

r

fn (ir) + e»-i-»-ifn {-ir)].

i

)

ir In order to have a simple notation for the combinations of the types e* /„(±tr) which write convenient to are required for solutions finite at the origin, Lamb found it

** ®~ l ~ 2(2n + 3)

+2

4.

.

(jSh

+ 3)

(2/i+5)~

"*'

—

London Math. Soc. xm. (1882), pp. 51—66; 189 212; xv. 139—149 xvi. (1885), pp. 27—43 Phil. Trans, of the Royal Soc. clxxiv. (1883), 519—549; and he was followed by Rayleigh, Proc. Royal Soc. lxxvii. A, (1906), 486—499 [Scientific Papers, v. (1912), pp. 300—312], and by Love* Proc. London

in his earlier papers, Proc.

(1884), pp.

pp. pp.

;

;

Math. Soc. xxx. (1899), pp. 308—321. With this notation it is evident that

« .=p^,«»=(-)-»...(

!.

!

Subsequently, however, ingly in his treatise on pp. 11

Lamb

found

it

convenient to modify this notation, and accord-

Hydrodynamics and

e~"

-j-\

* n («) =

so that

>t+ ^ pr+i

"

Soc.

xxxn.

(1901),

"" 4(2» + 3) (S» + 5) ~ J

1

l)

n

London Math.

also Proc.

— 20, 120— 150 he used the notation f *n W = 1.3.5... (2n + L ~ 2^+3) + 2

(d —

r

+ .).( i)'^.

.

=*n(*) -l'V'i»(*)»

to Y~

= vnw
, '

while Rayleigh, Phil. Trans, of the Royal Soc. cciii. A, (1904), pp. 87—110 [Scientific Papers, Love, v. (t912) pp. 149—161] found it convenient to replace the symbol /„(*) by x*(*)Phil. Trans, of the Royal Soc. ccxv. A, (1915), p. 112 omitted the factor (-)» and wrote

„

,

.

(

dye-**

.

,

/rf\"sin2

x

while yet another notation has been used by Sommerfeld, Ann. der Physik

xxvm.

Chemie, (4) (1913), p. 191

;

Chemie, (4)

this notation is

^(.HW^«-^ C»(*)«(MH'»+i (*) and

und

665—736, and two of his pupils, namely March, Ann. der Physik und xxxvu. (1912), p. 29 and Rybczyriski, Ann. der Physik und Chemie, (4) xli.

(1909), pp.

it is

l

(-ai)"T'

+ (-)* tf-»-*(*>l

certainly the best adapted for the investigation

on

electric

waves which was the

subject of their researches. *

In this paper Love defined the function

E n (z)

as

(

- )» . 1

.

3

...

(2n - 1)

(

^ — J

,

but, as

stated, he modified the definition in his later work.

t This is nearer the notation used p.

82

;

except that Heine defined

by Heine, Handbuch der Kugelfiinctionen, i. (Berlin, 1878), to be twice the expression on the right in his treatise, but

^ n (z)

—141.

not in his memoir, Journal filr Math. lxix. (1869), pp. 128

BESSEL FUNCTIONS

3*5]

57

Sommerfeld's notation is a slightly modified form of the notation used by L. Lorenz, who used vn and vn +( — ) n iwn in place of >frH and fn see his memoir on reflexion and refraction ;

Danske Videnskabernes Selskabs pp. 405—502.]

of light, K. (1898),

A

3*5.

It has

Skrifter, (6) vi. (1890), [Oeuvres scientifiqucs,

second solution of BesseVs equation for functions of integral order.

been seen

(§ 3'1 2) that,

whenever v

system of solutions of Bessel's equation the pair of functions

Jv {z)

and

./_„(z).

formed by an integer (—n), this is no

for functions of order v is

v

is

J_ n (z) = (—) n Jn (z)-

therefore necessary to obtain a solution of Bessel's equation which

is

linearly independent of

The

Jn (z);

and the combination of this solution with

is

JH (z)

a fundamental system of solutions.

will give

the

not an integer, a fundamental

is

When

longer the case, on account of the relation It

I.

solution which will

now be

constructed was obtained by Hankel*;

of the analysis involved in the construction were first published

full details

by BOcherf.

An

method of constructing HankeFs solution was discovered by Forsyth method of Frobenius, Journal fiir Math, lxxvi. (1874), pp. 214—235, for dealing with any linear differential equation. Forsyth's solution was contained in his lectures on differential equations delivered in Cambridge in 1894, and it alternative

his procedure is based on the general

has since been published in his Theory of Differential Equations, iv. (Cambridge, 1902), 102, and in his Treatise on Differential Equations (London, 1903 and 1914), pp. 101

—

Chapter

vi.

note

1.

It is evident that, if v be unrestricted,

and

if

n be any integer

(positive,

negative or zero), the function

j {z)-(-yj_ v {z) v

is

a solution of Bessel's equation for functions of order v

vanishes

when

v

=

Consequently, so long as v

± n, v

v

v

(z )

—n

also a solution of Bessel's equation for functions of order v

assumes an undetermined form \ when v

We

shall

shall

and this function

shew that

it is

^-(-y^-fr)

a solution of Bessel's equation for functions of

* Math. Ann. i. (1869), pp 469—472. t Annals of Math. vi. (1892), pp. 85 90.

—

xxv. (1880), pp.

;

= n.

now evaluate lim

and we

this function

the function

J {z)-(-TJ_ is

and

;

n.

See also Niemoller, Zeitsehri/t fiir Math, und Phys.

65-71

X The essence of Hankel's investigation is the construction of an expression which satisfies the equation when v is not an integer, which assumes an undetermined form when v is equal to

the integer

w.

B. F.

n and which has a

limit

when

v-*~n.

3


58 order n and that

it is linearly

independent of

Jn (z)

;

[CHAP.

Ill

may be taken

so that it

to be the second solution required*. It is evident that

J, (*)

- (- V J- v (*) _ Jv («) - Jn (M) v — n v — n l

dv as v

)n (

;

J- v (Z) - J-n (z) v — n

du

since both of the differential coefficients existf

•-» n,

.

Hence v

„^» w

a Bessel function of the second kind of order

exists; it is called

To

distinguish

the second kind

it

we

it

shall

denote

described as HankeVs function. by the symbol J TTn (z) so that

YH (s) =

j v

lim v-*-

Following Hankel,

{z)-(-yj_ v ( Z ) v

n

—n

also

*.(*)-

(2)

It has

now

to be

dJA z ) dJA

shewn that Y„

Since the two functions

( K

=dv

J± v (z)

(z) is

Vt dJ"-,(fY >

dv

a matter of indifference §.

_

a solution of Bessel's equation.

are analytic functions of both z and

order of performing partial differentiations on is

n.

from other functions which are also called functions of

it

may be

(1)

and

—n

Hence the

J

v,

the

with respect to z and v

±v (z) result of differentiating the pair of

equations

V v J±v {z) = with respect to v

may be

d*

dJ±v (z)

dz%

dv

-

When we

written

d bJ±v (z) dz

combine the

*

The reader

+ See §

3-1.

dJ_ v (z) K

}

will realise that, given

a limiting form of this solution It is

„

is

dv

,

T

,

_n

dv

results contained in this formula,

dJv {z) dv

^ dJ ±v (z)

dv

=

*2v{Jv

we

find that

(z)-(-rJ-Az)}>

a solution of a differential equation,

it is

not obvious that

a solution of the corresponding limiting form of the equation.

conventional to write differentiations with respect to

z

as total differential

coefficients while differentiations with respect to v are written as partial differential coefficient-.

Of course,

in

many parts of the theory, variations in v are not contemplated. Yn (z), which was actually used by Hankel, is used in this work

% The symbol function equal to § See, e.g.

ljir

times Hankel's function

(§ 3*54).

Hobson, Function* of a Real Variable

(1921), §§ 312, 313.

to denote a

BESSEL FUNCTIONS

3-51]

59

so that

+ 2v{Jv (z)-(-yj_v (z)}.

Now make functions of

where v

v,

to

is

All the expressions in the last equation are continuous

v --n.

and so we have

be made equal to n immediately after the differentiations with

respect to v have been performed. (3)

so that

We

have therefore proved that

V n Y n (*) = 0, Yn (s) is a solution of BesseVs equation for functions of order n. •

It is to be noticed that

r* J J T-.(,)-lim 'W-<-> V + H „^_ n

=

l

im

M ^»

whence

-W

J-A*)-(-YJAz) — fi + n

follows a result substantially

due

to

'

Lommel *,

Y_ n (*)=(-)»Yn (s).

(4)

Again,

while, because

Jv (z) is a

monogenic function of v at v

—

0,

we have

dJ. v (zy dv

and hence

it

A result equivalent to 3*51.

J„-0

^(-^Jv-O

L

dv

J"""'

follows that

this

The expansion of

was given by Duhamelf as early as 1840.

Y

(z) in

an ascending

series.

Before considering the expansion of the general function

Y„ (z),

venient to examine the function of order zero because the analysis

and the resulting expansion

We

*

is

it is is

con-

simpler

more compact.

use the formula just obtained,

SUidien

this result for

t Court

iiber die

what

is

d Analyse, 1

BesseVschen Functionen (Leipzig, 1868), p. 87. Lommel actually proved sometimes called Neumann's function of the second kind. See § 3-58 (8). n. (Paris, 1840), pp. 122

— 124.

:


60

and the result of term-by-term differentiation

Y

(*)

^w-^'Tt

= 2["I

=2 2 where

yfr

x ]

lo g

[CHAP. HI

is

fa) ~ J- log r (v + m +

1) 11

rra^ (log^-^(m + l)},

denotes, as

is

customary, the logarithmic derivate of the

Gamma-

function*. Since

may be

0< ^ (»i + 1) < m

when w=l,

replaced by

in.

The convergence

The

is

Y.(*)-2Sl

Y. («) -2

The reader

fry

{log(^)-^(m + l)},

'J**' .

J

(s)

- X

yy * -2 JS^ yy ^

+ log ft*)} ./.(*)

(m

+

,

1)J

|

1 + raj !

I+ g +

will observe that

iY.(*) is

is

also

following forms of the expansion are to be noticed

log (£*)

(3)

^(m+1)

an immediate consequence of a general theorem See Modern Analysis, § 5*3.

concerning analytic functions.

(1)

the convergence of the series for YoG?)

2, 3, ...

established by using D'Alembert's ratio-test for the series in which

+ (log2- 7 )J.(*)

a solution of Bessel's equation for functions of order zero.

The expansion

of

this function is

This function was adopted as the canonical function of the second kind of order zero by Theorie der BesseVschen Functional (Leipzig, 1867), pp. 42 44; see § 3 57. But the series was obtained as a solution of Bessel's equation, long before, by Euler \.

—

Neumann,

own notatiou

Eider's result in his

is

-

that the general solution of the equation

xx ?dy + xdxcy +gx*y Cx* —

V~

+A +a *

Modern

\JW X + 1.8.27»'^

X n*

1 ( \

- 3- r « + JL- X2« _

nn

_£L_

.

nn

1

.

Atialy-iis,

Ch. xn.

1.4. 9»

4m'

X+ 1.4n* X

1

.

It is to be

*U)=-7,

4

11.

9» f

tT 3» ('

+1

C

-

.

remembered

V'('»

+

where 7 denotes Euler 's constant, 0-5772157 t Inst. Calc. Int.

1.8.27.64«^

1 )=

J

1.

.

4

.

that,

9

.

.*•»

Ion8

16»«

when

- etc

)

Ix

/

*'

m is a positive

integer,

then

+ 2+ - +-->»

....

(Petersburg, 1709), §977, pp.

(1781) [published 1784], pars

g + 1.4.9.

+et °-

233—235.

Mathematica, pp. 186—190.

See also Acta Acad. Petrop.

v.

:

BESSEL FUNCTIONS

3*52]

where A and a are arbitrary constants. numerators in the first line 6 = 3.2-1.0,

He

gave the following law to determine successive 100 = 7.22-9.6,

22=5.6-4.2,

548 = 9.100-16.22,

w

61

3528 = 11.548-25. 100

2(1 \1

etc.

= + *1+...+!) mj ^,

m

!

this law is evidently expressed by the formula

m+ i = (2m+l)
ii

3*52.

»<*) We

shall

Y

where n

fl

(^),

Yn (z)

The expansion of

in

m.1

(r

an ascending

now obtain Hankel's* expansion any positive

is

.

integer.

[Cf.

series

of the

and

the definition

of

more general function

equation (4) of

§ 3-5.]

It is clear that

?^.<*>dv

3

|

m =()dv

(- )m (i*) H * w | \m\ T(i> + l)j (

m+

w=0 m!r(i/

OT =o

when

a)

i/-*-w,

|^£>] o"

J^=n

L

The

where

»i

is

ra!(w

+ w+l)

+ ra)!

l

6V2

l

6V2

7

n

7

/;

rv

a positive integer. That

is

/J

'

to say

- {7+ k g(W) j;(.)- 5 (-ra^- |i -om!(n + m)! (1 m

i

2

_li ^ + n+wj

[3«/_„(.s)/3i/],,_ n is a little more tedious because of the pole (- v + m + 1 ) at v = w in the terms for which m = 0, 1. 2, ., n — 1. We break the series for «/_„ (z) into two parts, thus

of

evaluation of

yjr

.

" V

l m!

'

TO

r(-i/ +

m + l)

*„»»»! r(-i/

+m+

.

l)'

and in the former part we replace 1

,

V(-v + m+l) Now, when

^

ra

<

by

r (v — in) sin (v — m) it *

'

n,

(^)~'" m r (y - wt) sin (v - m) tt "

[~—

t

(

)

1

= [(J*)~ r+m r(i»-m) {tt-i ^r — m) sin (»>— ra)7r + cos (v— m) tr —tt' - (£*)-»+»« r (n - m) cos (n - m) tt. (i»

*

Math. Ann.

i.

(1869), p. 471.

1

log ($z) sin (i/— »»)«}]»_»


62

[CHAP. HI

Hence

p«M*)1

.

v +

that

is

1

(-) n r(»-TO)(|^)-" 4 * ,t

J(-,+.) it- h «<*') + »<- + '» +i »'

,;.

to say

()

_U-

dp

L

(

}

+(

( **>

m!

w =o

x

when we

replace

w

(1)

and

we have Hankel's

(2)

a

»*!

m=o

x {

(-)'"(^)^ m =o m:(n + m)l first

(m+1)},

term (m =s 0) of the

|l

last

w!(n + m)!

(n

1}

l

~r,"

(h^

,111

1

m

1

,

2

1

} {

is

1

n

Lommel*)

It is frequently convenient (following

d

)

w + m)

summation, the expression in

12

%(*) =

1

s

(12

11,

(4)

namely

formula,

m=0

-(^r

7 + log g *)}/,(*)

j

In the

lo g(i*)-"f

{2 log ($z) - yfr (m + 1) - i/r (n + m + 1)} w

=2

{

m by n + m in the second series.

On combining

w

.t m!(n + m)!

>

Z)

-^

to write

-Mz)logz,

so that <5

^^-S J<.nXLi) ll082+n ^ m+1)];

>

when

v is a negative integer,

^„ (s)

is

defined by the limit of the expression

on the right.

We

thus have

Yn(z) = 2JJz))ogz + %(z) + (-)»$_

(6)

The complete

solution of

.v

7l

(z).

-v^ + ay=0 was given in the form of a series (part of which

contained a logarithmic factor) by Euler, Inst. Calc. Int. n. (Petersburg, 1769), §§ 935, 936 solutions of this equation are ;

*•*

Euler also gave

(ibid.

J (2aM), x

#*

Yi (2aM).

§§ 937, 938) the complete solution of x% -r^ + ay =0

this equation are *•*

J2 (4a* **),

x±

Y

2

(4a* .v*).

* Studicn iiber die Besscl'schen Fvnctionen (Leipzig, 1868), p. 77.

;

solutions of

BESSEL FUNCTIONS

3-53, 3-54] 3-53.

The

o/Y v

definition

63

(z).

Hitherto the function of the second kind has been defined only when its order is an integer. The definition which was adopted by Hankel* for unrestricted values of v (integral values of

Y

(1)

v

(*)

= 2ne»«

2v excepted)

JAz)C0S

is

™- J

-' (z) .

sin2i/7r

This definition fails both when v is an integer and when v is half of an odd integer, because of the vanishing of sin 2vtt. The failure is complete in the latter case

but, in the former case, the function

;

of the expression on the right and it

the definition of §

To prove

is

3'5.

Y

„

(z)

=

ire"**

lim

v

-n J

v

J z COSv7r (-)* lim \ "( ) v-n v+n L

= YM (z) + lim >— r+n L = Y n (s),

~n

J-AzY

\

J

Jv (z) J

we have proved that limY„(*) =

(2)

now

It is

(i)

integer

Yn

(*).

Y„ (z), defined either by (1) or by the limiting form a solution of Bessel's equation for functions of order v both v has any value for which 2v is not an integer, and when (ii) v is an evident that

of that equation,

when

v

— «/_„ (z)~\ — n v

(z) cos vtt

cos vrr sin vir

=

so

defined by the limit

this statement, observe that

lim

and

is

easy to reconcile this definition with

is

the latter result follows from equation (2) combined with § 3*5 (3). function Y„ (z), defined in this way, is called a Bessel function of the second kind (of Hankel's type) of order v ; and the definition fails only when :

The

v

+1

is

an integer. The reader should be

Note.

careful to observe that, in spite of the change of form, the qua function of v, is continuous at v = n, except when z is zero; and, in fact, Jv {z) and Y„(z) approach their limits J (z) and Y„ (z), as v-*~n, uniformly with n respect to z, except in the neighbourhood of z = 0, where n is any integer, positive or negative.

function

Y„ (z),

3*54.

The

The Weber-Schlafli function of the second kind. definition of the function of the second kind

which was given by

Hankel

(§ 3*53) was modified slightly by Weberf and Schlafli} in order to avoid the inconveniences produced by the failure of the definition when the

order of the function *

Math. Ann.

i.

is


(1869), p. 472.

f Journal fur Math, lxxvi. (1873), p. 9 ; Math. Ann. vi. (1873), p. 148. These papers are dated Sept. 1872 and Oct. 1872 respectively. In a paper written a few months before these, Journal filr Math. lxxv. (1873), pp. 75—105, dated May 1872, Weber had used Neumann's

function of the second kind (see §§ 3-57, 3-58). + Ann. di Mat. (2) vn. (1875), p. 17 ; this paper

is

dated Oct. 4, 1872.


64

The

function which was adopted by

second kind

is

Weber

[CHAP.

Ill

as the canonical function of the

expressible in terms of functions of the

first

kind by the formula*

Jv (z) cos vir — J., (z ) sin vtt (or the limit of this, Schlafli,

when

an

v is

integer).

however, inserted a factor \nr\ and he denoted his function by

the symbol K, so that, with his definition,

V

<*\

- J 0) COS I/7T -
1

Subsequent writers, however, have usually omitted this factor \tr, e.g. Graf and Gubler in their treatise f, and also Nielsen, so that these writers work with Weber's function.

K

The symbol

is,

however, used largely in this country, especially by

Physicists, to denote a completely different type of Bessel function (§ 3'7),

and so

produce least confusion

to

manner

after the

The procedure which seems

advisable to use a different notation.

it is

is to

of Nielsen J,

use the symbol

and to adopt

Yv {z) to denote

Weber s function,

this as the canonical function of

when the use of the number tr in

the second kind, save in rare instances

of Hankel's function of

integral order saves the insertion

certain formulae.

We

thus have

(1)

Yv (z) =

(2)

Yn (z) =

J

"

(jOcgg

lim

'" --*--(*>

= cos^;

JM^^JLrl-A^ = I Sin VTT

,-*. n

7T

Yn (0

.

Schlafli's fuuction has been used by B6cher, Annals of Math. vi. (1892), 85—90, and by McMahon, Annals of Math. vm. (1894), pp. 57—61; ix. (1895), pp. 23 30. Schafheitlin and Heaviside use Weber's function with the sign changed, so that the function which we (with Nielsen) denote by Yv (z) is written as — Yv (z) by Schafheitlin § and (when v = n) as — GM (z) by Heaviside||.

[Note.

pp.

—

Gray and Mathews sometiinesIT use Weber's function, and they denote symbol Yn

it

by the

.

*

Weber's definition was by an integral

(see § 6*1)

which

is

equal to this expression

;

the

expression (with the factor \ir inserted) was actually given by Schlafii. + Einleitung in die Theorie der Bexsel'xchen Funktioiien,

index, thus

Y v (z), Handbuch

and we

(§ 3*58).

§ See, e.g.

number

indicating the order as an

der Theorie der Cylinderfuiilctionen (Leipzig,

are obvious objections to such a notation,

Neumann

(Bern, 1898), p. 34 et seq.

i.

% Nielsen, as in the case of other functions, writes the

Journal fUr Math. cxiv.

(181)5),

pp. 31

reserve

it

190-1), p. 11.

for the obsolete function

— 44, and other papers;

There used by

also Die Theorie der

BesseVschen Funktioiien (Leipzig, HK)8). I!

Proc. lloyul Sue. liv. (1893), p. 138, and Electromagnetic Theory, n. (Loudon, 1899), p. 255;

made from his Electrical Papers, n. (London, 1892), TreatUe on Bessel Functions (London, 1895), pp. G5 66.

a change in sign has been 11

A

—

p. 445.

;

BESSEL FUNCTIONS

3-55]

Lommel,

Neumann's function

in his later work, used

65

of the second kind (see § 3'57), but

—

in his Studien iiber die BesseVschen Functionen (Leipzig, 1868), pp. 85

86,

he used the

function

£7rFM where

Yn (z)

(*)

+ {,/,(»+4) + log2} Jn (z), One disadvantage

the function of Weber.

is

presence of the term

of this function is that the

(»+£) makes the recurrence formulae

for the function much more complicated; see Julius, Archives Nterlandaises, xxviii. (1895), pp. 221 225, in this connexion.] yjr

—

3'55.

Heine's definition of the function of the second kind.

The definition given by Heine* of the function of the second kind possesses some advantages from the aspect of the theory of Legendre functions it enables certain generalisations of Mehler's formula (§ 5*7 1), namely ;

lim to be expressed in a

the symbol equal to

Kn

(z), is

Pn (cos $/n) = J

compact form.

The

'

(0),

function,

which Heine denoted by and it is

expressible in terms of the canonical functions,

-%irYn (z) and

to

-\Yn {z);

the function consequently differs only

in sign from the function originally used

by

Schlafli.

The use of Heine's function seems to have died out on the Continent many years ago the function was occasionally used by Gray and Mathews in their treatise t, and they term it On (z). In this form the function has been extensively tabulated first by AldisJ: and Airey§, and subsequently in British Association Reports, 1913, 1914 and 1916.

This revival of the use of Heine's function seems distinctly unfortunate, both on account of the existing multiplicity of functions of the second kind and also on account of the fact (which will become more apparent in Chapters vi and vn) that the relations between the functions cosine

JH (z)

and

sine

YH (z) present many points of resemblance to the relations between the

and ;

so that the adoption

parable to the use of cos« and

out that the symbol

On (z) has

||

of

Jn (z)

— %ir sins as

On (z)

as canonical functions It

must

is

com-

also be pointed

been used in senses other than that just explained by at least liv. (1893), p. 138 (as was stated in § 3*54),

two writers, namely Heaviside, Proc. Royal Soc. and Dougall, Proc. Edinburgh Math. Soc. xvm. Note.

and

canonical functions.

(1900), p. 36.

An

error in sign on p. 245 of Heine's treatise has been pointed out by Morton, Nature, lxiii. (1901), p. 29 ; the error is equivalent to a change in the sign of y in formula § 3*51 (3) supra. It was also stated by Morton that this error had apparently been copied by various other writers, including (as had been previously noticed by GraylT) J. J. Thomson, Recent Researches in Electricity and Magnetism (Oxford, 1893), p. 263. A further error *

Handbuch der Kugel/unctionen,

t

A

—

i. (Berlin, 1878), pp. 185 248. Treatise on Bessel Functions (London, 1895), pp. 91, 147, 242. t Proc. Royal Soc. lxvi. (1900), pp. 32 43. § Phil. Mag. (6) xxn. (1911), -pp. 658—663. ||

—

From

and also

the historical point of view there is something to be said for using Hankel's function, Neumann's function ; but Heine's function, being more modern than either,

for using

has not even this in

1 Nature,

its

favour.

xli?. (1894), p. 359.

:


66

[CHAP.

;

Ill

noticed by Morton in Thomson's work seems to be due to a most confusing notation employed by Heine for on p. 245 of his treatise Heine uses the symbol to denote the function called - \n J'q in this work, while on p. 248 the same symbol denotes - \n (Y -iJ ).

K

;

K

3*56.

Recurrence formulae for

Y

v

(z)

and"Y v {z).

The recurrence formulae which are satisfied by Yv (z) are of the same form by Jv (z) they are consequently as follows

as those which are satisfied

;

(1)

Y ^(z)+Yv+1 (z)^Yv (z),

(2)

Y ^(z)-Yv+1 (z) = 2Yv'(z),

v

z

v

(3)

zYv'(z) + v Yv (z) = zYv_

(4)

zYv'(z)-vYv (z)

and in these formulae the function function Y.

To prove them we take

jz if

{*"

we multiply

whence

J

v

(*)}

§ 3*2 (3)

= *" J*-i (*)

,

Y

and

(3) follows at once.

may be

replaced throughout by the

(4) in the

vrr,

Equation (4)

(z),

zYv+1 (z),

jz {? J-

these by cot vir and cosec

1

is

v

forms

(*)}

= - *' J-v+

>

(*)

;

and then subtract, we have

derived in a similar manner from

the formulae

£ {*- J

v

By

(z)}

= - jr- Jv+1 (z),

{*- J. v

(z)}

= r- «/_„_> (zy

J*-

addition and subtraction of (3) and (4)

we

obtain (2) and

(1).

The formulae are, so far, proved on the hypothesis that v is not an integer but since Yv (z) and its derivatives are continuous functions of v, the result of proceeding to the limit when v tends to an integral value w, is simply to replace v

is

by

n.

Again, the effect of multiplying the four equations by equal to 7re " ±1,,ri sec {y ± l)7r, is to replace the functions (

ire ¥rri

sec

Y by the

vir,

which

functions

Y throughout. In the case of functions of integral order, these formulae were given by Lommel, Studien


instructive to establish

Neumann's

them

Functionen (Leipzig, 1868), for such functions directly

p. 87.

from the

The reader

will find it

series of § 3-52.

investigation connected with the formula (4) will be discussed iu § 3*58.

;

BESSEL FUNCTIONS

3*56, 3*57]

67

Neumanns function of the second kind. The function which Neumann* adopted as the

3'57.

second kind possesses the advantage that

it is

canonical function of the

represented more simply by

integrals of Poisson's type than the functions of the second kind which have

been hitherto discussed

We

first

;

but this

is its

only merit.

define the function of order zerof, which will be called

The second solution of Bessel's equation for functions of order known to contain logarithms, Neumann assumed as a solution the

J where

w

(z) log z

(z).

zero being

expression

a function of z to be determined.

is

by V we must V w = -V {J (*)log*} = -2zJ '(z). ,

have

§212 (11),

But, by

so,

<0)

+ w,

If this expression is to be annihilated

and

F

since

2z

^

=

(z)

2z J,

= 8 2 {-T~'nJ2n (*)

(s)

V Jm (z) — 4>n2 JM (z), we

have

VoW = 2 2(-)'«V 4(z)/n n=l

= 2V 2(-y-iJm (z)/n; »=i

the change of the order of the operations

Hence a

possible value for

2

w

2 and V

is

easily justified.

is

2(-y->Jm (z)/n,

n-l

and therefore Neumann's function F<« \z)

(1) is

=J

F

(0)

(z),

(z) log z

+

defined by the equation

2

2

(-)""1

^-^

,

a solution of Bessel's equation for functions of order zero. Since

«/-*Oas«+0, (the

the origin), solutions,

it is

evident that

and hence

Y

(z) is

series for w being an analytic function of z near J (z) and F (z) form a fundamental system of expressible as a linear combination of J (z) and ,0)

Yw (z); a comparison of the behaviours of the three functions near the origin shews that the relation connecting them is F<°> (*)

(2) *

(log 2

- 7 ) Jo (*).

(§ 9-1).

But, because.

—

Neumann calls

this function

and he describes another function, On (z), as the function

of the second

Theorie der Bessel'schen Functionen (Leipzig, 1867), pp. 42

Bessel's associated function,

kind

= J Y, (z) +

H

(z)

is

44.

not a solution of Bessel's equation, this description

is

un-

and it has not survived. t Neumann's function is distinguished from the Weber- Schlafii function by the position of the suffix which indicates the order. desirable

)


68 3*571. It

The integral of Poissons type for

Y

{0)

[CHAP.

Ill

(z).

was shewn by Poisson* that

h

eixcoa

a

log (x sin 2

<

dm

a solution of Bessel's equation for functions of order zero and argument x and subsequently Stokes obtained an expression of the integral in the form of an ascending series (see § 3*572). is

;

The

associated integral

2

-

f*»

cos (z sin 0) log .

(4iz

cos2 0)

dd

was identified by Neumann f with the function F (z) and the analysis by which he obtained this result is of sufficient interest to be given here, with some slight modifications in matters of detail. ,0 >

;

From

§ 2-2 (9)

we have

(-)M «/*»(*)

/•*

2

cos (z cos 0) cos 2n0d0,

nir J

and

o

we assume that the order we deduce that

so, if

changed,

I

2,

n=i

n

—=-

summation and integration can be

cos (z cos 0)7

7rJ

z

n

W=1

2

=

of

do

f*» I

cos (z cos 0) log (4 sin3 0) .

from this result combined with Parseval's integral F (z), we at once obtain the formula

(§

d0

;

2 2) and the definition of

,0

»

(1

F<°> (*)

= !/"

from which Neumann's result

"cos (z cos 0) log (4* sin3 0) .

is

<*0,

obvious.

The change of the order of summation and integration has now to be examined, because 1n~ l cos 2n0 is non-uniformly convergent near 6=0. To overcome this difficulty we observe that, since 2 ( - ) n J.in {z)jn is convergent, it follows from Abel's theorem that %

S(-)»^.W/»= lim I (-)««»Ji»<«)/n= lim * j /*% os (s C os0)?Wcos2 ^ca?. »=l o^-i-o n=l M «-*M5Tn-U0 * Journal de. VEcole 2?. Poly technique, hi. (oahier 19), (1823), p. 476. The solution of an associated partial differential equation had been given earlier (ibid. See also Duhamel, p. 227). Cowr« d? Analyse, n. (Paris, 1840), pp. 122—124, and Spitzer, Zeitsehri/t filr Math, und Phys. n. (1857), pp. 165—170. t Theorie der Bessel'sr.hen Functionen (Leipzig, 1867), pp. 45 49. See also Niemoller, Zeitschriftfllr Math, und Phys. xxv. (1880), pp. 65—71.

—

I Cf. Bromwich, Theory of Infinite Series, § 51.

+

a

1

+

.

BESSEL FUNCTIONS

3-571, 3*572]

69

Now, since a is less than 1, 2 (on cos 2n6)/n does converge uniformly throughout the range of integration (by comparison with 2a"), and so the interchange is permissible that is to say ;

2 » - 2

« »=1

n 2 / a .a cos2n8 . A eos(«cos0) d6**>-

/"*"" /

n

Jo

/"i*

« a" cos 2nd

.

,

cos(«cos#) 2

I

n

n

n-1

J

lh"

1 = --

cos

/

n Jo

(«

cos d) log (1

dOn ,

- 2a cos 26 + a2

)

flW.

Hence we have (")* »(*'

£

=_

M

n-l

lim o-*.l-0

i "

*""

cos

|

(a

cos d) log (1

- 2a cos 20

a2 )

y o

We now proceed to shew that* lim

cos(*cos0){log(l-2acos20-M2 )-log(4asin 2 0)}c?0 = O.

/

a-*-l-0 J

It is evident that

- 2a cos 26 +

1

and so if

A

a2 )

be the upper bound t of cos

(z

j

'It

cos

- 4a sin2 6 « ( 1 - a) 2 ^ 0,

- 2a cos 26

log (1

Hence,

2

(«

cos 6) {log

(1

^ log (4a sin 2 6).

cos 6)

|

when

0^6^ £«,

we have

- 2a cos 26 + a2 ) - log (4a sin2 6)} dd \

I/,o

**

^

{log (1

f

- 2a cos 26+ a8 ) - log (4a sin2

6)}

d6

Jo

«~ ( |-22

/**"

-A i

\

io

a"cos2«0

„

.

.

.

„.

+log (l/a>- 2

,„

= J*iUog(l/a), term-by-term integration being permissible since

.

log (2 sin

"

«*i

I

J

„

6)\d6 J

a< 1.

Hence, when

a< 1,

/"*» I

cos

/

as a-»-l

- 2a cos 26 + a2 ) - log (4a sin2 0)} dd < Jtt4 I

(z

cos 0) {log (1

— 0; and

log (l/a)-*-0,

this is the result to be proved.

Consequently

2 n=j

(-)w,7*»(*) n

= _

=

I ( ** cos ii a-*i-o ,r y o

-1

m

/"*"

cos

(z

The

tf)

log (4a sin2 6)

.

dd

cos 0) log (4 sin 2 8) dd, .

and the interchange is finally justified. The reader will find it interesting to deduce combined with § 3*5 (5). 3*572.

CoS

(*

this result

from Poisson's integral for Jv (z)

Stoked series for the Poisson- Neumann integral.

differential equation considered

by Stokes \

in 1850

was -rf

+-

-r-

- m2 y =0, where

m is a constant. This is Bessel's equation for functions of order zero and argument imz. Stokes stated (presumably with reference to Poisson) that it was known that the general solution was /itr

{C+ D log (z sin 2 0)} cosh (mz cos 6)

d&.

o

*

The value

t If 2

of this limit

is real,

A — l\

if

was assumed by Neumann,

not,

J Trans. Camb. Phil. Soc. p. 42.]

A < exp

{

1

(z)

} j

ix. (1856), p. [38].

[Mathematical and Physical Papers, in. (1901),

+

'


70

[CHAP.

Ill

with Neumann's notation, the value of the expression on the right

It is easy to see that, is

\n {C- D log (4m)} J (imz)+\nD F(°) (imz). The expression was expanded bn (C + D log z)

J

into a series

*

(imz)

+ 2D 2

by Stokes

m 2" z2n ,_

f in I

.,

»=o v* n )

it is

;

equal to

cos2" 6 log sin 6 dd,

J o

•

and, by integrating by parts, Stokes obtained a recurrence formula from which

it

may

be

deduced that

-j*%o82^logsin^^=^^|^log2 + i

3*58.

Neumann's

The Bessel

Neumann*

definition of

F

(n)

ff

(I

+ ^ + ... + i)}.

(z).

function of the second kind, of integral order n, was denned

F

in terms of

<0)

(z)

^

dYi z

(1)

- nY^

(z)

= - zYW (*),

which is a recurrence formula of the same type as §2*12(4). It from this equation that

is

evident

FW(jr)-(^)»(^)*FW(«).

(2)

Now Y m (z)

and, if

by

by induction from the formula

the equation

satisfies

we apply the operator f

theorem, <3>

-j- to this equation n times,

and use Leibniz'

ZCLZ

we get

'

&T

F

'°'

{l)

+ (2M + 2) (a)"" 7 "

w

(a)"

7 " (J) " °*

and so *"

{zlzf

This equation

^^

is

{Z)]

+ (2n + 2)

so

F

(s) is

a solution of Bessel's equation

may be l

*

^^

{Z)

= °-

for functions of order n.

written in the form

zj{- z~n~ F(n+1

>

(*)}

- (2w + 2) z-*-

1

F
>

Theorie der BetteVschen Functionen (Leipzig, 1867), p. 51.

order

+

V,Fw(5)-0, (w)

Again, (3)

its

{Z)]

at once reducible to

(4)

and

~ nYW

{z

{zlz)

is

not an integer.

t The analysis

is simplified

by taking \z* = f, so that d _ d zdz~ df

(*)

+ *~ n F(w

The function

>

(2:)

is

= 0,

undefined

when

;

BESSEL FUNCTIONS

3*58, 3*581]

71

so that

dF(n+I >(*)

n+ 1 + -^t

Tz

F<»+-> (z)

z

- F«

= 0,

(*)

whence we obtain another recurrence formula z

(5)

When we combine

d7

(Z)

2

+ nYm (*) = zY«-»

(1) with (5)

we

(z).

at once deduce the other recurrence

formulae (6)

F'"- 1 (z)

+

F<» +1 (z)

=

—

(7)

F<"--> (*)

-

F<»+ -> (*)

=

2

)

Consequently

Yn

and

F

(w »

F<"> («),

dF< "'^ as

.

the same recurrence formulae as

(*) satisfies

It follows from §

(s).

>

357

(2) that

Jn (z),

F» (s)

<

F« (,) - *7rFn (,) + (log 2 - y) Jn (z)

(8)

= ^Y n (z) + (\og2-y)Jn (z). A

solution of the equation V H (y) = in the form of a definite integral, which reduces to the integral of § 3*571 when »=0, has been constructed by Spitzer, Zeitschrift fiir Math, und Phys. ill. (1858), pp. 244-246; cf. § 3-583.

3*581.

The

Neumann's expansion of

F

(n)

(z).

357 (1) has been given by Neumann*;

generalisation of the formula §

it is

m-1

7«(,)-/. W{kw .-^-jE

(1)

i

v. where

sn

To

111 Z

On— ro-i

«,

o

1

...+-,

w

establish this result,

we

s

r /«\

•

i

m (n + m)

,«=i

= T + B + 5+ 1

|_ -^ ^

= 0.

first

define the functions

Ln (z)

and

Un (z)

by

the equations

,_££_ -ig. ^()-^(-)+j "ff++y-/"^^ Z.

(2)

W.J

9n-m-i

n-l

*>

r

j

/_\

.(.) log

5

(

(3)

1

- U (z). We shall prove that Ln (z) and Un (z) L^ (z) = - Zn (z) + (n/z) Ln (z), (4)

so that

F<°> (*)

=Z

(z)

'

and then

(1) will

satisfy the recurrence formulae

U^ (z) = - Un (z) + (n/z) Un '

be evident by induction from

§

358

* Theorie der Bessel'schen Funetianen (Leipzig, 1867), p. 52.

—84

die Bessel'schen Functionen (Leipzig, 1868), pp. 82

and Haentzschel,

Zeitschrift fiir Math,

;

und Phys. xxxi.

Otti,

(z),

(2).

See also Lommel. Studien liber

—35

Bern Mittheilungen, 1898, pp. 34

(1886), pp. 25

—

33.

— THEORY OF BESSEL FUNCTIONS

72

[CHAP.

Ill

It is evident that d_

[

L n (z)

\

zn

)

dz dz\

j

Jn (z)

...

_

1

Mz) _

d (JJs)) S dz\ zn

.

n z} 2

z n+1 n - r"- 1

•£» 2 -»->.nl _d

m. .«!

(

Jm (z)

)

(n-m).w!^Vsn - M (.. .Jm (z) J +i(z)\l j

vl

z1

-74 _,7n+1 ^ )log ^ +

—

i

+

1+ ^^ri| ^HriJ 2

io~^r-|

L n+1 Q)

_

zn

and the

'

first

part of (4)

d [Un(z)]_ dz\ zn

n

)

J

is

d [Jn (z)) dz\ zn

)

1

(z)

To prove the second

proved.

°°

= - *» -=•£- + 7n *

=i

part,

we have

| (-y»(n + 2m) d J^fr) m(n + m) dz\ z n f

TO =i

(— )m

~ -^ [mJr*^ (n +

(z)

)

j

- (n + m) J1l+M (z)}

*7n + l(*)

n

»

and the second part of

It follows from §

(4) is proved.

Y<™ (z) - L n+X (z) +

358 (2)

that

d[ Y m (,) - L n (z) + Un (z)

U^ {z) _

zn

zn

dz\

)

j'

and since the expression on the right vanishes when n = 0, it induction that it vanishes for all integral values of n. Hence

is

evident by

Y™(z) = L n (z)-Un (z), and the truth of equation (1)

is

The power series for

3*582.

The function coefficients, coefficients,

U

n (z),

therefore established.

Un (z)>

which was defined in

has been expressed by namely

«

§ 3*581 (3) as

Schlafli* as a

a series of Bessel power series with simple

g-<'>-. s v.i(;v»)- »when n =

by

and by straightforward differentiation, the expression on the right satisfies the same recurrence formula as that of § 3*581 (4) for Un (z) equation (1) is then evident by induction.

To

establish this result, observe that

§ 3*57 (1);

and

it is

true

§ 3*51 (3)

that,

;

Note.

It will be found interesting to establish this result

of (£«) n+2m in the expansion *

on the right of § 3*581

by evaluating the

(3).

Math. Ann. in. (1871), pp. 146—147.

coefficient

BBSSBL FUNCTIONS

3*582-3*6]

The reader

now

will

easily prove the following formulae

(2)

3fn (*)

(3)

F««> (m)

(4)

i*Yn{»)

3 '583.

The

iivbegral

73

= {7 - log 2} Jn (z) - Un (z), = Ln (z) + 3» (s) + {log 2 - 7} Jn (z), = Ln(*) + %*<*).

of PoUson's type for F<»)

(«).

The Poisson-Neumann formula

of § 3-571 for F(°) (z) was generalised by Lommel, Sttidien ilber die BesseVschen Functionen (Leipzig, 1868), p. 86, with a notation rather different from Neumann's; to obtain Lommel's result in Neumann's notation, we first

observe that, by differentiation of Poisson's integral for

^^-Jy(z)logz= and

so,

from § 3*582

rW (>) "

in

which

3*6.

8

{log

«*«*-* (*+*)-?} d$+Ln (*),

- 2 log 2 = - y - 2 log 2, we have the

(

r(w + ^r(i) j

formula

** cos sin 6) cos2w * log (4 cos8 6) (*

to be remembered that and powers of z.

it is

coefficients

coa(z Sme)coH^B{log(ico^0)-ylr(u+i)}dB,

J**«»(*sin 0)co*** 6

•£ (£) «• \/r (1)

jw (») =

we have

(3),

r(n+l?r(i)

and hence, since (l)

^^^ f^

F

Jv (*),

Ln (z)

is

de

expressible as a finite combination of Bessel

Functions of the third kind.

In numerous developments of the theory of Bessel functions, especially

vn) on integral and asymptotic expansions of Jv (z) and Yv (z), two combinaBessel functions, namely J, (z) ± iY, (z), are of frequent occurrence.

those which are based on Hankel's researches (Chapters vi and representations tions of

The combinations

also present themselves in the theory of "Bessel functions

of purely imaginary argument" (§

3*7).

seemed desirable to Nielsen* to regard the pair of as standard solutions of Bessel's equation, and he describes them as functions of the third kind; and, in honour of Hankel, Nielsen denotes them by the symbol H. The two functions of the third kind are defined by the equations f It has consequently

Jv (z)±iYv (z)

functions

H*\z) = Jv {z) + iYv (z\

(1)

From '

„(!),.,

t9\

When

these definitions,

v

is

an

Hf{z) = Jv {z)-iYv {z).

combined with

J- v {z)-e-™Jv (z)

§

354

(1),

.»

we have

J. v (z)-e^Mz)

integer, the right-hand sides are to be replaced

by

their limits.

Since Jp {z) and F„ (z) satisfy the same recurrence formulae (§§ 3*2, 3'56), the functions enter linearly, and since the functions of the third kind which in * Ofversigt over det K. bitch

Danske Videnskabemes Sehkabs Forhandlinger,

der Theorie der Cylinderfunktionen (Leipzig, 1904), p. 16.

t Nielsen uses the symbols

Hf(z)

t

Hv 2

(z).

1902,. p. 125.

Hand-


74

are linear functions (with constant coefficients) of

[CHAP.

Jv (z) and Y

(z), it

v

Ill

follows that

these same recurrence formulae are satisfied by functions of the third kind.

Hence we can at once write down the following formulae

:

(3)

*.•>)+*&<.)-**»«.

*«?,«+*»«-**»(.),

(6)

.f^-^w.-^w

.f^'-^w-.dr*,.,,

-&-- -<

-Jr- --<<•>.

(7)

(2) >

Vv ff™(*)=0,

Vv ff™(z)=0,

(8)

™ (^-{£^.(^^. £)-{^_ -^. (

Note. Bayleigh on several occasions, e.#. PAt7. J/ogr. 350—359 [Scientific Papers, iv. (1904), p. 290; v.

(1907), pp.

symbol

Dn (z) to denote the function

which Nielsen

calls

\ triH

;

(2)

(s).

Relations connecting the three kinds of Bessel functions.

3*61.

It is easy to obtain the following set of formulae,

some of the formulae are simply the

which express each

The reader

function in terms of functions of the other two kinds. that

266 (6) xiv. 410—418], has used the

(5) xliii. (1897), p.

(1912), pp.

will observe

definitions of the functions

left.

/„(*)

(1)

=

H«\z) + Hf(z)

Y_ v {z)-Y v {z) COS VTT

W^

g

e™ JB™ (z) + e-" #?> CO '

J-' (jr)l

r

(3)

(5)

(6)

r

H

{x)

(z)

.

«r- J_ r <*)

sm

#J

w

= J-M-c-iJriz)

"

t

"

(5)

= Y.

sin vtt

<*)

" #f (*)

v

(z)-e-"- i

Y

(z)

_

'

r.,(*)-e-'r,,(*) sini/7r

(6) it is obvious that

H™ (z) = e- J5T™

v

sin vir

^sini/7r

and

1}

2t

JjWm /^'^" ^^)

From (7)

,/„<*) cob

= F_„ (#) cos „tt Yv (#)

nnw

2

(jr)«

'

(s),

if

« (,) . «--

i

iff

(*).

'

on the

BESSEL FUNCTIONS

3-61-3-63]

Bessel functions with argument

3*62.

Since Bessel's equation the functions

J± v (— z)

—z and

unaltered if z

is

is

75 ze mvi

.

-

replaced by

we must expect

z,

to be solutions of the equation satisfied

by

J±v (z).

To avoid the slight difficulty produced by supposing that the phases of both of the complex variables z and — z have their principal values*, we argument

shall construct Bessel functions of

arg z has

its principal value,

and

it is

Jv (z)/zv

is

defined by the

when

the phase of z

same convention

is

any

integer,

+ arg z.

definable as a one-valued function,

venient to assume that,

m

where

,

supposed that

arg (zemni ) — mnr Since

ze mni

as that

by which

it is

obviously con-

Jv (z) is to be and accordingly

unrestricted,

is

z" is defined;

we have the equations

Jy (zemni - emviri J

(1)

)

J_ v (ze

(2)

The

miri

)

v

=

e- m

(z),

™ J-

v (z).

functions of the second and third kinds will

now be

defined for

all

values of the argument by means of the equations § 3"54 (1), § 36 (1); and then the construction of the following set of formulae is an easy matter:

Y

(3)

v

(zemvi ) ,Hiri

= e~ m ™

= e~ m

Y

(z)

v

™ F_„ (z) + 2i sin mvir cosec

(4)

r_„ (ze

(5)

m"i = e- m ™ir w H?\ze ) v "

)

+ 2i sin mvrr cot vir Jv (z),

v

"

sini/7r

Hf (zem«) - a—* H v

v

7

(2)

'

_ e_

(z),

v

.

yit

'

sinmi/Tr

Rm

sinv7r

+ 2e™ J^T 8

(z) v '

"

J

(z)-2e-^~^^ JJz) sini>7r

= sin(l-w)i/7r ff(1) (6) v '

vir

"

7r

sini/7r

JyvV(z)'

= sin(l+«t)»^r H m (z) + evni sinmvTr ff "

sinv7r

sinv7r

(l)

''

"

Of these results, (3) was given by Hankel, Math. Ann. vin. (1875), p. 454, in the special when m=\ and v is an integer. Formulae equivalent to (5) and (6) were obtained by Weber, Math. Ann. xxxvu. (1890), pp. 411, 412, when m=l see § 6-11. And a memoir by Graf, Zeitschrift fur Math, und Phys. xxxvm. (1893), pp. 115—120, contains the general case

;

formulae.

3*63.

Fundamental systems of solutions of Bessel's

It has

been seen

-

(§

3 12) that

J

v

solutions of Bessel's equation when,

now examine the Wronskians

(z) and J_„ (z) form a fundamental system of and only when, v is not an integer. We shall

of other pairs of solutions with a view to deter-

mining fundamental systems in the *

equation.

For Arg ( - z)

critical case

= Arg z =f t,

when

v is

according as I (z)

^

an integer.

0.

,

:

,

,

.


76

[CHAP,

III

It is clear from § 3"54(1) that

m

Y

[Jv (z),

(z)}

v

= - cosec vir WL [Jv (z), J.

v

(z)}

_2 TTz'

This result

established on the hypothesis that v is not

is

an integer; but con-

shew that

siderations of continuity

M

(1)

whether v be an integer or mental system of solutions.

{J, (*), *,(#)}

J

Hence

not.

v

-2/(™», and

(z)

Y

v

(z)

always form a funda-

deduce that

It is easy to

aa {j, (*),T

(2)

r

(#)}-

z cos vrr

and, in particular*,

m{Jn (z),Yn

(3)

When we

Y

v (z), it is

(4)

(z)}

= 2Jz.

express the functions of the third kind in terms of J„ (z) and

found that

5129

{H™ (z),

hT

(z)}

=-2im{J

v

(z),

Yv

= - 4i/(w),

(jr)}

so that the functions of the third kind also form a fundamental system of solutions for all values of

v.

Various formulae connected with (1) and (3) have been given by Basset, Proc. London Math. Soc. xxi. (1889), p. 55 they are readily obtainable by expressing successive differ' ' ential coefficients of Jv (z) and v (z) in terms of Jv (z), Jv (z) f and v (z), v (?) by re;

Y

Y

peated differentiations of Bessel's equation.

Y

which the

Basset's results (of

earlier ones

are frequently required in physical problems) are expressed in the notation used in this work by the following formulae

--

(5)

Jv

(z)

IV

(z)

-

Yv

(z)

Jv"

(z)

(6)

JJ

(«)

JV

(«)-

17

(,)

Jv"

(«)-

(7)

Jv

(z)

YJ" (z)-Yv {z)JJ" (.)-!

(8)

j;

(z)

iv"

w - JV

(9)

JJ'iz)

JV"

(«)

(10)

Jv

(z)

YvW{z)-Yv

(z)JvW(z) =

(ii)

j;

(,)

jy>(*)- r;

(,)

-

- sin vw

;

jv>"

Yv"{z) Jv'"

Throughout these formulae are multiplied by

(z)

(•)

Lommel, Math. Ann.

A (l -^ (^-l),

A (^ _ y x

(.)--£ (i

./,<">(*)=

J^

-

-^- + --^r-)

(i-^±3),

A (î— - —

J-

+i)

Yv may be replaced by J_ v if the expressions on the right Yv may be replaced by W^ ET throughout if the

and J„

,

expressions on the right are multiplied by * Cf.

=

J.

iv. (1871), p.

— 2i.

106,

and Hankel, Math. Ann. vnr.

(1875). p. 457.

BESSEL FUNCTIONS

3*V]

An

associated formula, due to

Math. Ann. (

vm.

Lommel* Math. Ann.

77 iv. (1871), p. 106,

and Hankel,

(1875), p. 458, is

JAz)Yv ^{z)-Jv ^{z)Yv {z)=-~.

12 )

irz

This

is

3'7.

The

proved in the same way as §

32

(7).

Bessel functions of purely imaginary argument. differential equation

which differs from Bessel's equation only in the coefficient of y, is of frequent occurrence in problems of Mathematical Physics; in such problems, it is usually desirable to present the solution in a real form, and the fundamental systems

J

v

(iz)

and J_„ (iz) or

Jv (iz)

Y

and

v

(iz)

are unsuited for this purpose.

However the function e~ iviri J„ (iz) of the equation.

It is

is a real function of z which is a solution customary to denote it by the symbol /„ (z) so that

y+2m <**)

oo

Iw(s)wg

(2)

When z

J: 9 m\r( v +

m + iy

regarded as a complex variable, it is usually convenient to define its phase, not with reference to the principal value of argiz, as the consideration of the function Jv (iz) would suggest, but with reference to the principal value of arg z, so that is

The introduction argument "

is

due

/„ (*)

= e-W Jv (ze^%

(- nr < arg z < £tt),

/„ (*)

= e»™ J„ (ze~^),

(J

of the symbol

to Basset f

and

w < arg z < w).

Iv (z) to denote "the function of imaginary now in common use. It should be men-

it is

tioned that four years before the publication of Basset's work, Nicolas} had suggested the use of the symbol v (z), but this notation has not

F

been used by

other writers.

The relative positions of Pure and Applied Mathematics on the Continent as compared with this country are remarkably illustrated by the fact that, in Nielsen's standard treatise §, neither the function /„(*), nor the second solution v {z\ which will be defined immediately, is even mentioned, in spite of their importance in

K

physical applications.

The function I_ v (z) §312) that (3)

is

also a solution, of (1),

«{/.(*X

/-„(*)}

=-

and

28in

it is

easy to r prove

Ccf.

VTT

irz

#

Lommel gave the corresponding formula for Neumann's function of the second kind. f Proc. Camb. Phil. Soc. vi. (1989), p. 11. [This paper was first published in 1886.] Basset in this paper, defined the function of integral order to be i+» Jn (iz), but he subsequently changed it, in his Hydrodynamics, u. (Cambridge, 1888), p. 17, to that given in the text. The more recent definition is now universally used. + Ann. Sci. de Vticole norm. sup. §

(2) xi. (1882),

Handbuch der Theorie der Cylinderfunktionen

supplement,

p. 17.

(Leipzig, 1904).

THEORY OF BBSSEL FUNCTIONS

78 It follows that,

when

[CHAP.

Ill

an integer, the functions Iv (z) and /_„(*) form

v is not

a fundamental system of solutions of equation

(1).

In the case of functions of integral order, a second solution has to be con3*54. structed by the methods of §§ 3'5

—

The

K

function

second solution,

n (z), which will be adopted throughout defined by the equation

is

It

equivalent definition

may be

verified,

the order v

The

work as the

*.«-£¥ ['-^]-

<«>

An

this

is

3*5) is

by the methods of §

equal to

K

function

(cf. §

v

3*5,

that

Kn (z)

is

a solution of (1)

when

n.

has been defined, for unrestricted values of

(z)

v,

by

Macdonald*, by the equation

K

(6)

and, with this definition,

it

v

(Z)

= A-7T

may be

:

,

verified that

#„(*)=

(7) It is easy to

v

(z\

deduce from (6) that iT,(z)

(8)

K

lim

= i«^'ir v

(iz)

= $>me-i"n H_ v (™)-

K

fact that it is a v (z) lies in the physical importance of the function as z— through zero to exponentially tends which equation (1) solution of

The

positive values.

This fundamental property of the function will be established

in § 723.

The definition of n (z) is due to Basset, Proc. Camb. Phil. Soc. vi. (1889), p. 11, and the infinite integrals by his definition is equivalent to that given by equations (4) and (5) ; 61 5. Basset subsein 614, discussed be will §$ which he actually defined the function

K

(Cambridge, 1888), quently modified his definition of the function in his Hydrodynamics, n. rdl-y(z) 37, (*)1 1 , Ai equivalent to c£ pp. 18—19, and his final definition is .

^Ti^—

In order to obtain a function which

Gray and Mathews omit the factor

in their work,

1/2", so

A

satisfies

'

cV~J v -n

the same recurrence formulae as 7„

(z),

Treatise on Bessel Functions (London, 1895), p. 67,

that their definition

is

equivalent to

'

2|_

The only simple extension

ov

Cv

Jv= n

of this definition to functions of unrestricted order is

formula

K* (2) = \ir cot vrr{I- v (z) - 7„ («)}, • Proc.

London Math.

Soc. xxx. (1899), p. 167.

by the

3*71]

BESSEL FUNCTIONS

79

Modern Analysis, § 1771) but this function suffers from the serious disadvantage that vanishes whenever 2» is an odd integer. Consequently in this work, Macdonald's function will be used although it has the disadvantage of not satisfying the same recurrence formulae as Iv (z). (cf.

it

An inspection of formula (8) shews that it would have been advantageous if a factor %n had been omitted from the definition of v (*) but in view of the existence of extensive tables of Macdonald's function it is now inadvisable to make the change, and the presence of the

K

;

factor is not so undesirable as the presence of the corresponding factor in Schlafli's function (§ 354) because linear combinations of /„(«) and v (z) are not of common occurrence.

K

3'71.

We

Formulae connected with Iv {z) and

shall

K

(z).

v

K

now

give various formulae for Iv (z) and v (z) analogous to §§3'2—3*6 for the ordinary Bessel functions. The proofs

those constructed in

of the formulae are left to the reader.

(1)

/„_, (,)

- J„+1 (,) = ^ I, (z),

K.

(2)

/„_, (z)

+ Iv+l (z) = 2/; (*),

K^

(3)

zi; (z) + vlv (z)

(4)

zlj (z)

^L)

K

}

\zdzj

z"

(7)

/,'(,)«/,(*),

(8)

I_ n

The (9)

(z)

(z)

'

j

+ Kv+1 (z) = - 2KJ (z), + vKv (z) = - z Kv . (z), x

(-*-)

[z

\zdz)

\

K '(z)

= In (z),

- Kv+l (z) = - ^ Kv (z),

zK: (z) - vK v (z) = - zKv+1 (z),

= zÎv_ m (z), *'+»

(z)

x

zKJ (z)

(z),

x

- vlv (z) = zlv+1 (z),

{z"Iv (.)}

\

= zlv_

v

K_

v

{z)

*Kv (z)} =

z"

r

y

——

=K

v

when R(v +

sm^BdO

^

- i>?|rr (i)

<**)'

=

>

L

(1

£-

" t2y

,(1

'

h

cosh

ôs» s i n2 ,

r^r|^)/rcosh

-t7^fc)/.

~^~'

(z).

cosh (z cos 0)

I

K

K,{z),

following integral formulae are valid only

Iv (z) =

]~

{-r*—K„

"

{zt) dt

dQ

{z cos d) sin2v0d0

\)

>

:

(z),

'


80

[CHAP.

;

Ill

We also have L_L*I (-)»>+- r)i

These results are due to Basset.

n0

x

(10)

(A = Irn+h (,)

-j^ [* X

_ r)|(2 ,)r

rl(n

o

+^ (11)

7_ (w+i)

(s)

>

r

/f„ + , (.) =

y

r- 2

^-^

r

o;T

(13)

*•,«-£)'.-.

(14)

iTo (*)

«\ n (15)

x (-)"(«- m- 1)1 «- ^ s' ^.W-jS, m!(w»-m

- - log (£*) 7

2

(16)

ô (*) = --

(17)

J, (*«""<)

(18)

Z

(19)

W

(20)

J„ (*)

The

^

+ 2

/1 2 \n+am

eo

„

<*)

.

<")n+1

+

r!( W

-r)!(2^J'

- ^î) L^r r!(n-r)!(2^ + (-)»e-

(12)

f

'

{log (**)

v.

, ,

* f

e*

008 '

-

{log (2* sin2 0)

^ r!(n ^

r)!(2(8)rJ

,

* <«,+ 1),

W (m + - & 1)

+ 7}

(n +

m +1

)],

dO,

= emv7ri Iv (z),

(*"*) = {!,(*),

e—« JT, (,) #„(*)}

JT^ (,) +

7„ +1 (z)

tti

^^

/, (,),

1/*,

K

v

(z)

=

1/z.

integral involved in (16) has been discussed

by Stokes

(cf.

§ 3"572).

The integrals involved in (9) and the series in (14) were discussed by Riemann in his memoir "Zur Theorie der Nobili'schen Farbenringe," Ann. der Physik und Chemie, (2) xcv. (1855), pp. 130

power

— 139, in the special case in which

series for 7

v=0; he

also discussed the ascending

(2).

The recurrence formulae have been given by Basset, Proc. Camb. Phil. Soc. vi. (1889), 19; by Macdonald, Proc. London Math. Soc. xxix. (1899), pp. 110—115; and by pp. 2

—

Aichi, Proc. Phys. Math. Soc. of Japan, (3)

11.

(1920), pp.

8—19.

Functions of this type whose order is half an odd integer, as in equations (10) and (12), were used by Hertz in his Berlin Dissertation, 1880 [Oes. Werke, 1. (1895), pp. 77—91]

and he added êt another notation

to those described in § 3*41.

:

BESSEL FUNCTIONS

3-8] 3'8.

Thomson's functions ber

(z)

and

bei (z)

81

and

their generalisations.

A class of functions

which occurs in certain electrical problems consists of Bessel functions whose arguments have their phases equal to \tr or f ir.

The

may be

functions of order zero were

defined

ber (x)

(1)

first

examined by W. Thomson *

;

they

by the equation f

+ i bei (x) = J

(xi »Ji)

=I

(x

^/i),

where x is real, and ber and bei denote real functions. For complex arguments we adopt the definitions expressed by the formulae ber (z) ±

(2)

i

bei (z)

-J

(zi

Vi

*)

= h (* V ±

*')•

Hence we have

berW=1 _|i| + |g_...,

(3)

Extensions of these definitions to functions of any order of the third kinds have been effected

by

Russell:}:

first,

second and

and Whitehead§.

The functions of the second kind of order zero were defined by Russell by a pair of equations resembling (2), the function IQ being replaced by the function thus

K

,

ker

(5)

(z)

±

i

kei (z)

=K

(zy/±

i).

Functions of unrestricted order v were defined by Whitehead with reference to Bessel functions of the first and third kinds, thus (6)

ber, (z)

± i bei, (z) =

(7)

her, (z)

±

i hei„ (z)

Jv (ze** *), 1

= H™ (ze**™).

It will be observed that||

ker (z) =

(8)

— \tr hei

(z),

kei (z)

— \ir her (z),

in consequence of § 3'7 (8).

The

following series, due to Russell, are obtainable without difficulty

ker (*)

(9)

*»

- log (±z)

.

ber (z) +

fr bei

(z)

* Presidential Address to the Institute of Electrical Engineers, 1889. [Math, and Phys. Papers, in. (1890), p. 492.] f In the case of functions of zero order, it is customary to omit the suffix which indicates the order.

Mag. (6) xvii. (1909), pp. 524—552. § Quarterly Journal, xui. (1911), pp. 316 342. Integrals equal to ker (z) and kei (z) occur in a memoir by Hertz, t Phil.

—

II

(3)

xxn. (1884),

p.

450

\_Ges.

Werhe,

i.

(1895), p. 289].

Ann. der Physik und Chemie,

:


82 kei (z)

(10)

=-

log (\z)

.

bei (*)

[CHAP.

Ill

- \ir ber (*)

It has also been observed by Russell that the ber2 («)+bei 2 («) have simple coefficients, thus

first

few terms of the expansion of

but this result had previously been obtained, with a different notation, by Nielsen m § 5-41) the coefficient of ($z)* in the expansion on the right is l/[(m !) 2 (2«i) !]. ;

(cf.

.

Numerous expansions involving squares and products

of the general

functions have been obtained by Russell; for such formulae the reader

memoir and

referred to Russell's

also to a paper

is

by Savidge*.

Formulae analogous to the results of §§ 3*61, 3"62 have been discussed by Whitehead it is sufficient to quote the following here ;

ber_„ (z)

(12)

— cos vtr

= cos vir her_„(,z) = cos vir hei_ v (z) = sin vir bei_„ {z)

(13)

(14) (15)

.

ber„ (z)

.

bei„ (z)

.

.

—

sin vrr

+ sin inr herv (z) — sin vtr her„ (z)

+ cos vir

.

— bei„ (z)], [her„ (z) — ber„ (z)],

.

hei„ (z),

.

.

[hei„ (z)

hei v

(z).

The reader will be able to construct the recurrence formulae which have been worked out at length by Whitehead.

The

functions of order unity have recently been examined in some detail

by B. A. Smith f. The

3*9.

definition

of cylinder functions.

Various writers, especially SonineJ and Nielsen§, have studied the general theory of analytic functions of two variables 9$¥ {z) which satisfy the pair of recurrence formulae (i)

«rM(l)+»H»(') =

7»rW,

(2)

*_.<#) -tfH.»(*) =

W(*).

which z and v are unrestricted complex variables. These recurrence formulae are satisfied by each of the three kinds of Bessel functions. in

Functions which satisfy only one of the two formulae are also discussed by Sonine in his elaborate memoir a brief account of his researches will be given in Chapter x. ;

* Phil.

Mag.

(6)

xk.

(1910), pp.

49—58.

f Proc. American Soc. of Civil Engineer*, xlvi. (1920), pp. 375

—

425.

t Math. Ann. xvi. (1880), pp. 1—80. §

Handbuch

der Theorie der Cylinderfttnktitmen (Leipzig, 1904), pp.

1,

42

et seq.

BESSEL FUNCTIONS

3*9]

Following Sonine we shall

any function ^(z), which satisfies both of now be shewn that cylinder functions

call

the formulae, a cylinder function.

83

It will

are expressible in terms of Bessel functions.

When we

combine the formulae (1) and

we

find that

= z^Âz), mW (jr) - v<9w (z) = - z<#¥+1 (z),

z^:{z) + v%\{z)

(3) (4)

and

(2),

so, if

^

be written

for z (d/dz),

we deduce

that

+ v)<$v (z) = z'@v _ (z), <* - v) Wr (z) = - zVv+1 (z). (*

(5) (6)

1

It follows that

(&»

-

I*) 3?„ (Z)

= (Sr - V) {ZK-L (*)} = ^(^-I/+l)«^„_ (^) = -*#,(*). 1

u

.

that

to say

is

^„ (*) = a v Jv (z) +

Hence where a„ and

we

independent of

6„ are

substitute in (3)

ay Jy _ and

x

z,

we

find that

(z)

+ fcX-i {z) s

bv

Yv (z),

though they may depend on

<*„_,

Hence ay and

bv

z,

we must have

must be periodic functions of v with period unity

conversely, if they are such functions of

When

Jv _, (*) + 6„_, !;_, (z),

since /„_! (z)/ F^_ 2 (^) is not independent of

so,

v.

v, it is

;

and,

easy to see that both (1) and

(2) are satisfied.

Hence the general

%

(8)

where

-er^v)

may be

It

solution of (1)

and

«r„(i/)

(*)

=

and

«i (») Jv (z)

(2) is

+

cr2 („)

are arbitrary periodic functions of v with period unity.

observed that an equivalent solution S?„ (z)

(9)

F, (z),

is

= ., („) JETr « («) + «r4 (») #,<*> (s).

A

difference equation, which is more general than (1), has been examined by Barnes, Messenger, xxxiv. (1905), pp. 52 71 ; in certain circumstances the solution is expressible by Bessel functions, though it usually involves hypergeometric functions.

—

is used by Nielsen to denote Jv (z), Yv (z), ffvW () more general functions discussed in this section. This procedure is in accordance with the principle laid down by Mittag-Leffler that it is, in general, undesirable to associate functions with the names of particular mathematicians. The name cylinder function is derived from the fact that normal solutions of Laplace's

Note.

and

HW V

The name

(z)

cylinder function

as well as the

equation in cylindrical coordinates are

e**>Jm {kpf r

*

sin

(cf.

§ 4-8

and Modern Analysis,

§ 18*5).

m

^


84 Some

writers* following Heine t

who

called

Jn (z)

[CHAP.

a Fourier- Dessel function,

call

Ill

Jn {z)

a Fourier function.

Although Bessel coefficients of any order were used long before the time of Bessel §§ 1-3, 1*4), it seems desirable to associate Bessel's name with them, not only because it has become generally customary to do so, but also because of the great advance made by Bessel on the work of his predecessors in the invention of a simple and compact notation (cf.

for the functions.

Bessel's

name was

associated with the functions by Jacobi, Journal fiir Math. xv.

(1836), p. 13 [Ges. Math. Werke, vi. (1891), p. 101].

que usus

"Transcendentium 7 « naturam variosill. Bessel in commentatione fc

in determinandis integralibus definitis exposuit

celeberrima."

A more recent controversy on the name to be applied to the functions is to be found in a series of letters in Nature, lx. (1899), pp. 101, 149, 174; lxxxi. (1909), p. 68. * E.g. Nicolas, Ann. Sci. de f Journal fiir cylinder function.

VEcole norm. sup.

Math. lxix. (1868),

p. 128.

(2) xi. (1882),

Heine also seems

supplement, to be responsible for the

term

CHAPTER IV DIFFERENTIAL EQUATIONS Daniel Bernoulli 's solution of Riccati's equation.

4*1.

The

solution given

by Bernoulli* of the equation

-^ = azn + by 2

(!)

consisted in shewing that

0;

when

the index n has any of the values

—4,-4: — § — &•

—

—J-2

-

-1-2--

—

—is.-

1«

while a and 6 have any constant values f, then the equation is soluble by algebraic, exponential and logarithmic functions. The values of n just given are comprised in the formula

means of

/9\ \*)

=—

n

where

equation,

method

it is first

of solution is as follows If n be called the index of the proved that the general equation J of index n is transformable :

into the general equation of index

N, where

N=- n + 1'

(3)

it is

also proved that the general equation of index

into the general equation of index

The

=-

Hence, by

(4),

Hence by (3), the equation of index tinued by using the transformations set of soluble cases given above,

comprised in the general formula

—475.

is

transformable

4.

obviously integrable, because the

is

the equation of index

-1

is

(3)

and

and

it

integrable.

is

- 4 is integrable.

If this process be con-

(4) alternately, we arrive at the easy to see that these cases are

(2).

quaedam muthematicae (Venice, 1724), pp. 77—80 Acta Eruditorum, 1725, The notation used by Bernoulli has been slightly modified and in this analysis

Exercitationes

pp. 473

n

-

n

Riccati equation of index zero

variables are separable.

n

where

v,

v

(4)

*

1

m is zero or a positive integer.

Bernoulli's

and

4m 2m ±

;

;

not restricted to be an integer.

is

t It is assumed that neither a nor h be separable. %

That

is,

the equation in which a

is

and

zero.

b

If either

were zero the variables would obviously

have arbitrary values.

THEORY OF SESSEL FUNCTIONS

86

[CHAP. IV

Daniel Bernoulli's transformations of Riccati's equation.

4*11.

Now

that the outlines of Bernoulli's procedure have been indicated,

we

proceed to give the analysis by which the requisite transformations are effected.

Take

standard equation of index n and

§ 4*1 (1) as the

„

The substitutions

[Note. n.

The

are possible because

—1

Yis

not included

among

the values of

was not inserted by Bernoulli the effect of that the transformed equation is more simple than if it were omitted.]

factor

presence

n+ 1

= Z, y = + !-«"*-

make the substitutions

is

in the denominator

;

its

The equation becomes 1

dY

-T7t Y*dZ

that

= « +T

b

Y*zn

'

is

where

N = - n/(n + 1)

Again in

§

41

(1)

;

and

make

this is the general equation of index

N.

the substitutions

The equation becomes

where

=—n—4

v

;

and

this is the general equation of index

v.

transformations described in § 4'1 are therefore effected,

The

equation

is

soluble in the cases stated.

But

and so the

this procedure does not give the

solution in a compact form.

The limiting form of Riccati's equation, with index

4*12.

the processes described in $4'1, 411 are continually applied to oo in which the index tends, when not soluble by is - 2. The equation with index - 2 is consequently

When

m—

Riccati's equation, the value to §

41

(2),

number

a finite

To

of transformations of the types hitherto under consideration.

solve the equation with index

write y

- 2.

= v/z,

and the equation becomes z

unci this is

Hence,

- 2, namely

-.-

dz

= a + v + bv2

;

an equation with the variables separable. in this limiting case, Riccati's

of elementary functions.

equation

is still

soluble by the use

4-11—4'13]


This solution was implicitly given by Euler, p. 185.

If

we

write

(cf.

§ 414) y = _ _ J?

,

which

is

homogeneous, and consequently

Inst. Cole. Int. n. (Petersburg, 1769), § 933,

the equation which determines

+

dz^

87

z*

r,

is

-U

>

immediately soluble.

it is

Euler does not seem to mention the limiting case of Riccati's equation explicitly, although he gave both the solution of the homogeneous linear equation and the transformation which connects any equation of Riccati's type with a linear equation. It will

appear subsequently

4*7— 4'75)

that the only cases in which terms are the cases which have now been to say, those in which the index has one of the values (§§

Riccati's equation is soluble in finite

examined

;

that

is

-f, -|;

0;

and

also the trivial cases in

-f, -§;

which a or b

...,

_2

,

(or both) is zero.

This converse theorem, due to Liouville, is, of course, much more recondite than Bernoulli's theorem that the equation is soluble in the specified cases.

4'13.

A

Euler* s solution of Riccati's equation.

method of constructing a solution of Riccati's equation in the was devised by Euler* and this method (with some slight changes

practical

soluble cases

in notation), will

now be

explained.

First transform Riccati's equation, § constants as follows

41

(1),

by taking new

variables

and

:

(!)

V

= -v/b,

the transformed equation

ab

= -c\

(2)

cT

+r,2

~ &zn~^ = °

and the soluble cases are those in which 1/q Define a new variable w by the equation „

(3)

so that the equation in

d

/A \

w

w

2q

-

2

;

'

an odd

is

integer.

= c^-i + I<^, w dz

is _

dw

-^ + 2cz*-> -^ + (q - 1 ) cz*-«-w = 0.

(4)

A

2

=

n

is

solution in series of the last equation

is

00

w = z-m-v 2 A r z~Qr

,

provided that' Ar_+l

Ar *

Nov. Comm. Acad. Petrop.

[1764], pp.

154—169.

_ (2qr + q +

1) (2qr

8qc(r vm. (1760—1761)

+

+q-

1)

l)

[1763],

pp.

3—63; and

ix.

(1762—1763)


88

[CHAP. IV

and so the series terminates with the term A m z~ qm if q has either of the values ± l/(2m + 1); and this procedure gives the solution* examined by Bernoulli. The general solution of Riccati's equation, which is not obvious by this method, was given explicitly by Hargreave, Quarterly Journal, vn. (1866), pp. 256—258, but Hargreave's form of the solution was unnecessarily complicated; two years later Cayley, Phil. Mag. (4) xxxvi. (1868), pp. 348—351 [Collected Papers, vn. (1894), pp. 9—12], gave the general solution in a form which closely resembles Euler's particular solution, the chief difference between the two solutions being the reversal of the order of the terms of the series involved.

Cayley used a slightly simpler form of the equation than (2), because he took constant multiples of both variables in Riccati's equation in such a way as to reduce it to

fc W-^~

(5)

= 0.

Cayley s general solution of Riccati's equation.

4'14.

We

2

have just seen that Riccati's equation

reducible to the form

is

dz

and we shall now explain Cayley 'sf method of solving equation, which is to be regarded as a canonical form of Riccati's

given in this

§

413

(2);

equation.

When we make

the substitution %

r)

= d (\ogv)/dz,

the equation becomes

*?_<-,*-.,,_<);

(1)

if U^ and Uz are a fundamental system of solutions of this equation, the general solution of the canonical form of Riccati's equation is

and,

where

d and C

are arbitrary constants

2

respect to

U

Y

and

U

2

in a finite form, v

so that the equation satisfied

w

= w exp

by

w

is §

proceeding in ascending powers of i

differentiations with

z.

To express

in

and primes denote

?-l q(q-l)

U

x

to

(cz q

/q),

4'13 sfl

write

(4).

(g-l) (3g-l) *

-^*_

,

be exp

A

solution of this equation

is

^^q(q-l)2q(2q-l) —Vx ^

czq+

/,>

_ and we take

we

c

iq

(y-l)(3g-l )(og-l) q{q-l)2q(2q-l)Sq(Sq-l)

(czq/q) multiplied

by

*

this series.

the index n of the Riccati equation is - 2, equation (4) is homogeneous, Mag. (4) xxxvi. (1868), pp. 348—351 [Collected Papers, vn. (-1894), pp. 9—12]. the memoirs by Euler which were cited in § 4*13. cf. § 1-1. X This is, of course, the substitution used in 1702 by James Bernoulli *

When

t Phil.

;

Cf. also

;


4*14]

Now

equation (1)

by changing the sign of c, and

unaffected

is

89 so

we take

?(9-l)

L

?(0-l)2$(2gr-l) - l)(5i- 1)_ Z + + _il_- 1JJ3 ? q(q- l)2q(2q- l)Sq(Sq-lf -J' terminate when q is the reciprocal of an odd positive "I

and both of these

series

Since the ratio

integer.

Ci

the exponential function exp (2oz9 /<7)

J72 is

:

multiplied by an algebraic function of

U U

%

lt

zv, it cannot be a constant form a fundamental system of solutions of (1).

If q were the reciprocal of an odd negative integer, equation (1) in the form

;

and so

we should

write

*®-"W.)^(./.)-a whence

it

follows that

d where

^ and y

2

are constants, and

V F = z exp( + 2

lt

The

series

afljq)

1

± -?-—;.

czfl

+

which have now been obtained

detail in §§ 4'4

—

+ l)(Sq + l) CV2 ,..] l)2q(2q + l) + q(q (q

.-.

will

be examined in

much

greater

4'42.

The reader should have no difficulty in constructing the when it is soluble in finite terms.

following solutions of Riccati's

equation,

Equation

(i)

(ii) (iii)

(dr,/dz)

Values of

+ t,*=l

(dtj/dz)

rf

C72

/

+ rP=Z-W

+ i =z- i

,

(l+3z l s )exv(±3zW) (1

+ W* + ^z2 6 ) exp ( + 5z '

Equation

(i)

x

exp (±z)

(dr,/dz)+Ti*=z-*i 3 {dr)ldz)

U

Values of

I

z

exp

(

1/5

)

Vlt V%

± 1 /*)

(ii)

(dT,;dz)+q*=z-M

z(l

+ 3Z- 1'3) exp ( ± 3z~ »/3)

(iii)

(dT)/dz)+t)Z=z-™

z (1

+ 5«-

1 /5

+^«-2/*) exp ( ±5z-»/s)

be noticed that the series Di, U^ (or Vu V2 as the case may be) are supposed term before the first term which has a zero factor in the numerator 4-42 and Glaisher, Phil. Trans, of the Royal Soc. clxxii. (1881), p. 773.

It is to

,

to terminate with the see §

w.

B. F.

4

THEOBY OF BESSEL FUNCTIONS

90

Among

[CHAP. IV

who have studied equation (1) are Kummer, Journal fur Math. xn. 144—147, Lobatto, Journal fur Math. xvn. (1837), pp. 363—371, Glaisher (in the memoir to which reference has just been made), and Suchar, Bull, de la Soc. Math, de France, xxxn. (1904), pp. 103—116; for other references see § 43. the writers

(1834), pp.

The reader

when q—0, the equation (1) is homogeneous and immeand that the second order equation solved by James Bernoulli (§ 1*1) is obtainable by taking q—2 in (1), and so it is not included among the soluble cases. will observe that

diately soluble;

4*15.

Schlafli's canonical form

The form (i)

Zt This

of Riccati's equation.

of Riccati's equation which was examined

=

ift

Schlafli *

was

~ 1ra~lu

*-

reduced to the form of

is easily

by

4*13(2) by taking

§

—t~a/a

as a

new

independent variable.

To

solve the equation, Schlafli wrote

at

and arrived at the equation

F(a,t)=X

Iff

1=9

t

w!r(o + m + l)'

the general solution of the equation in y

y

The

solution of (1)

is

u

;

1

F(a,

t)

is

+ c*t-aF(- a,

t).

then

— Ct^Fia + 1,

t)

+ c2 F{- a - 1,

t)

^F^^ + cJ-HFi-a,*)

The connexion between rendered evident

=c

m

Riccati's equation

and

Bessel's equation is thus

but a somewhat tedious investigation is necessary (§ 4*43) between Cayley's solution and Schlafli's solution.

to exhibit the connexion

Note.

The

function ,

:

a

z,

defined as the series 1

1+- + H2' .

1 a3 a2 e(z+l)^2.3'z(z + l)(z+2y'"'

which

is evidently expressible in terms of Schlafli's function, was used by Legendre, Elements de Oeometrie (Paris, 1802), note 4, in the course of his proof that « is irrational.

Later the function was studied (with a different notation) by fragment in his Math. Papers (London, 1882), pp. 346 349.

Clifford; see

a posthumous

—

*

Ann. di Mat.

(2)

i.

(1868), p. 232.

The reader

will see that

James

Bernoulli's solution in

be associated with Schlafli's solution rather than with Cayley's solution, t This notation should be compared with the notation of § 4-4.

series (§ 1*1) is to

;


4*15, 4*16]

91

Jv (*) = (\zy F (v, - \z*),

It is obvious that

and it has recently been suggested* that, because the Schlafli-Clifford notation simplifies the analysis in the discussion of certain problems on the stability of vertical wires under gravity, the standard notation for Bessel functions should be abandoned in favour of a

—

notation resembling the notation used by Schlafli-Clifford a procedure which seems comparable to a proposal to replace the ordinary tables of trigonometrical functions by tables :

of the functions

„-o(2»)!' nZo(2n + l)\'

4*16.

Miscellaneous researches on Riccati's equation.

A solution of Riccati's equation, which involves definite integrals, was given by Murphy, Trans. Camb. Phil. Soc. in. (1830), pp; 440

and,

if

a be written

for l/(r»+2)

y =4*

f

w

where

and

h~^[(f> (A)

<6(A)=e*A-° ^

/

—443.

The equation which he considered

A' d (log y)jdt 1

exp

e~ h ha

(*i/«/A)

+ <£ (1/A)

- 1 dh= I

—, n =oa(a

Jo

for u, his solution

exp

—

(when

is

ABa2 =l)

is

(A*V«)] dh,

,. ^2)..,(a + n) + l)(a +\, t

-,

be written for A in the second part of the integral, then the last expression given reduces to wit multiplied by the residue at the origin of h~* <£ (A) exp (* 1/0/A), and the

If 1/A

for

y

connexion between Murphy's solution and Schlafli's solution

An

(§ 4*15) is evident.

was published by Challis, Quarterly Journal, vn. (1866), pp. 51 which shewed how to connect two equations of the type of § 4*13 (2), namely investigation

—53,

in one of which l/q is

an odd positive integer, and in the other it is an odd negative This investigation is to be associated with the discovery of the two types of

integer.

solution given in § 4*14.

The equation which

new

is easily

variables,

formed

it

t- H

f-

bz" u 2

- czm — 0,

transformed into an equation of Riccati's type by taking «»-« + and z*u as was investigated by Rawson, Messenger, vn. (1878), pp. 69 72. He transi

—

into the equation

dv -jt

— y+

a+a

bz m

-a

+a 2 y -cz" =Q,

by taking bu=czaly; two such equations are called cognate Riccati equations. A somewhat similar equation was reduced to Riccati's type by Brassinne, Journal de Math. xvi. (1851), pp. 255—256.

The connexions between the various types of equations which different writers have adopted as canonical forms of Riccati's equation have been set out in a paper by Greenhill, Quarterly Journal, xvi. (1879), pp. 294 298.

—

* Greenhill, Engineering, cvh. (1919), p. 334 see also Engineering, cix. (1920), p. 851.

;

Phil.

Mag.

(6)

xxxvm.

(1919), pp.

501—528


92 The reader should

[CHAP. IV

by Siacci, Napoli Rendiconti, (3) vn. a monograph on Riccati's equation, which apparently contains the majority of the results of this chapter, has been produced by Feldblum, Warschau (1901), pp. 139

— 143.

also consult a short paper

And

Univ. iVach. 1898, nos.

and 1899,

The generalised Riccati

4*2.

An

5, 7,

no. 4.

equation.

obvious generalisation of the equation discussed in § 4'1

is

^ = P + Qy + Rf,

(1)

where P, Q, R are any given functions of z. This equation was investigated by Euler*. It is supposed that neither P nor R is identically zero for, if ;

P or R is zero,

either It

the equation

is easily

integrable by quadratures.

was pointed out by Enestrom, Encyclop4die des namely

Sci.

Hath. n.

16, § 10, p. 75,

that a

special equation of this type

nxxdx — nyydx-k-xxdy—xydx was studied by Manfredius, De constructions aequationum differentialum primi gradus (Bologna, 1707), p. 167. "Sed tamen haec eadem aequatio non apparet quomodo construibilis sit, neque enlni videmus quoinodd illam integremus, nee quomodo indeterminatas ab invicem separemus."

The equation order,

reduced to the linear equation of the second

(1) is easily

by taking a new dependent variable u defined by the equation f

»--£*-£-"

(2)

The equation then becomes d2 u /o\ Conversely,

if in

dR) du

„_

_

_ du du ^__ + lh2 _ + p,„«o, 2

/a\

p p lt p2 ,

are given functions of

u

(5)

the equation to determine y

z),

we

= e^' dz

write

,

is

^^.fly-y., y

W

(6)

which

1

the general linear equation of the second order,

(4)

(where

\r\

dz is

of the

same type

as (1).

p** p The complete

'

equivalence of the generalised

Riccati equation with the linear equation of the second order

is

consequently

established.

The equations 1896, pp.

1

—33.

of this section have been examined by Anisimov, Warschau Univ. Nach.

[Jahrbuch

iiber die Fortschritte

der Math. 1896, p. 256.]

— —

Nov. Comvi. Acad, retrop. vin. (1760 1761) [1763], p. 32 ; see also a short paper by W. W. Johnson, Ami. of Math. in. (1887), pp. 112 115. t This is the generalisation of James Bernoulli's substitution (§ 1-1). See also Euler, Inst. *

Calc. Int. n. (Petersburg, 1769), §§ 831, 852, pp. 88, 104.

y


4*2, 4*21]

4*21.

93

Eider's theorems concerning the generalised Riccati equation.

been shewn by Euler* that, if a particular solution of the is known, the general solution can be obtained by two quadratures if two particular solutions are known the general solution is obtainable by a single quadrature f. And it follows from theorems discovered by Weyr and Picard that, if three particular solutions are known, the general solution can be effected without a quadrature. It has

generalised Riccati equation ;

To prove the

first result, let

y be a particular solution of

g-P+fc + JV. and write y = y

+ 1/v. The

equation in v

is

dv

^ + (Q+2Ry )v + R =

of which the solution

exp {f(Q

v

and, since

is

+ 2Ry ) dz) + jR exp {/(Q + 2%,) dz)

v = l/(y — y

To prove

0,

),

the truth of the

first

theorem

is

.dz

= 0,

manifest.

the second, let y and yx be two particular solutions, and write

y-yi The

result of substituting (y x y<>-yi

*?

w

,

and,

and

when we

P + Qy

in the equation is ,

substitute for (dyjdz) and (dy /dz) the values

+ Ry*,

the last equation

w = c exp

so that

we

y

w-1 dz~

wTz where

for

!_ dy*_ P >Q yw- .vo + p (yxw - y ** w-l V w - 1 )

d}h

(w-lfdz^w-ldz

w — y )/(w — 1)

is

•

P + Qy + Ry* 1

reduced to

=Ry»- Ry^ {f(Ry

— Ry ) dz], x

c is the constant of integration.

see that

y

is

To prove the

Hence, from the equation defining w, expressed as a function involving a single quadrature. third result, let y

y% be a third solution, and reduce to y2 Then y let

let c'

and y, be the solutions already specified, be the value to be assigned to c to make

.

yjrJf.o

y-yi and

this is the integral in a

* Nov.

Comm. Acad.

form

=l C

free

y*-y<> '

y.2

-y

'

x

from quadratures.

Petrop. vni. (1760—1761) [1763], p. 32.

t Ibid. p. 59, and ix. (1762—1763) [1764], pp. 163—164. Math. xl. (1850), p. 361.

See also Minding, Journal fur

;


94

[CHAP. IV

It follows that the general solution is expressible in the

Hence giving

C

it is

evident that,

the values

ylt yit y8 yt be any four solutions, obtained by

if

,

C C C G lf

2

,

3

form

t

,

respectively, then the cross-ratio

(yi-y*) (y*-y*) (yi-y*)(y*-y*) is

independent ofz; for

it is

equal to

(0 -(7# )(Ci 1

-(Q"

In spite of the obvious character of this theorem, been noticed until some forty years ago*.

it

does not seem to have

Other properties of the generalised Riccati equation may be derived from properties of the corresponding linear equation (§ 4*2). Thus Raffy f has given two methods of reducing the Riccati equation to the canonical form

these correspond to the methods of reducing a linear equation to its normal form by changes of the dependent and independent variables respectively. Various properties of the solution of Riccati's equation in which P, Q, R are rational functions have been obtained by C. J. D. Hill, Journal fiir Math. xxv. (1843), pp. 23 37 Autoune, Comptes Rendus, xcvi. (1883), pp. 1354 1356; cxxvm. (1899), pp. 410 412; and Jamet, Comptes Rendus de V Assoc. Francaise (Ajaccio), (1901), pp. 207 228; Ann. de la

—

Fac. des Sci. de Marseille, xn. (1902), pp.

1

—

—21.

R has

The behaviour of the solution near singularities of P, Q, Nieuw Archie/ voor Wiskunde, (2) vi. (1905), pp. 209

—

—

been studied by Falken-

— 248.

hagen,

The equation

of the second order

whose primitive

C1C1

is

of the type

+ C2& + C3C3'

where c x c2 , c3 are constants of integration (which is an obvious generalisation of the primitivo of the Riccati equation), has been studied by Vessiot, Ann. de la Fac. des Sci. de ,

Toulouse, ix. (1895), no. 6 and by Wallenburg, Journal fiir Math. cxxt. and Comptes Rendus, cxxxvii. (1903), pp. 1033 1035.

—

(1900), pp.

210

217-

* Weyr, Abh. biihm. Ges. Wiss. (G) vm. (1875—1876), Math. Mem. i. p. 30 ; Picard, Ann. Sci. de VEcole norm. sup. (2) vi. (1877), pp. 342 343. Picard's thesis, in which the result -is contained, is devoted to the theory of surfaces and twisted curves a theory in which Riccati's

—

—

equation has various applications. t

Nouv. Ann. de Math.

(4)

n. (1902), pp. 529—545.


4*3]

95

Various transformatiwis of Bessel's equation.

4*3.

The equations which we

are now about to investigate are derived from by elementary transformations of the dependent and inde-

Bessel's equation

pendent

The

variables. first

type which

we

shall consider is*

p -*,.BiE±»

(i)

dz*t

where

c is

z

an unrestricted constant.

The equation

in physical investigations, and, in such problems,

p

is

is

of frequent occurrence

usually an integer.

The equation has been encountered in the Theory of Conduction of Heat and the Theory of Sound by Poisson, Journal de VlZcole Polytechnique,- xn. (cahier 19), (1823), pp. 249—403; Stokes, Phil. Tram, of the Royal Soc. 1868, pp. 447—464 [Phil. Mag. (4) xxxvi. (1868), pp. 401—421, Math, and Phys. Papers, iv. (1904), pp. 299—324]; Rayleigh, Proc. London Math. Soc. iv. (1873), pp. 93—103, 253—283 [Scientific Papers, i.

pp. 138, 139].

The

the Earth; see

Ellis,

special equation in

its

p=2

(1899)',

occurs in the Theory of the Figure of

Camb. Math. Journal, n. (1841), pp. 169—177, 193—201.

Since equation (1) ,

which

may be

d? (uz-i)

general solution

d

written in the form (uz-i)

,

is

(2>

u

= z^p+i

(ciz).

Consequently the equation is equivalent to Bessel's equation when p is and no advantage is to be gained by studying equations of the form (1) rather than Bessel's equation. But, when is an

unrestricted,

p

tions of (1) are expressible "in finite

termsf"

(cf.

frequently desirable to regard (1) as a canonical form. various types of solutions of (1) will be examined in

integer, the solu-

§

3

4),

The

and

detail in §§

The second type

it

relations

is

then

between

441—-44:}.

of equation

is derived from (1) by a transformation of the dependent variable which makes the indicial equation have a zero root The roots of the indicial equation of (1) are + 1 and p -p, and so we write u = vz-p we are thus led to the equation •

z dz

dz*

of which the general solution

W

v

CV

U

'

is

= zP + i<@pH (ciz).

Mem. della R. Accad. delle Sci. di Torino, xxvi. (1821), pp. 519—538, andPaoli di Mat. e di Fis. della Soc. Italiana delle Sci. xx. (1828), pp. 183—188.

* See Plana,

Mem.

t This was known to Plana, who studied equations has just been made.

(1)

and

(5)

in the paper to

which reference

—


96 Equation

(3),

which has been studied in detail by Bach, Ann. Sci. de Vjtcole norm. sup. 47 68, occurs in certain physical investigations; see L. Lorenz, Ann.

—

(2) in. (1874), pp.

—21 [Oeuvres

und Chemie, (2) xx. (1883), pp. 1 and Lamb, Hydrodynamics (Cambridge,

der Physik 396];

in the

I. (1898), pp. 371 Solutions of equation (3)

Scientifiques,

1906), §§ 287

— 291.

(cf. §§ 5*6, 9 '65) have been examined by Catalan, Bulletin xxxi. (1871), pp. 68 73. See also Le Paige, ibid. (2) xli.

form of continued fractions

de VAcad. R. de Belgique, (1876), pp.

(2)

—

1011—1016, 935—939.

Next, we derive from

normal form. becomes in its

(3),

by a change of independent

variable,

We write z — %qjq, where q = l/(2p +1),

an equation

the equation then

Jr-c^-ô,

(5)

and

[CHAP. IV

its

solution

is

v

(6)

When

a constant factor

is

= (Wqyi™<$im) (cip/q). absorbed into the symbol

^?,

the solution

may be

taken to be

PîKwiciplq). Equation

Mem. (4)

(5),

which has already been encountered

-

in § 4 14, has

been studied by Plana, 519—538; Cayley, Phil. Mag. Papers, vn. (1894), pp. 9—12]; and Lommel,

della B. Accad. delle Sci. di Torino, xxvi. (1821), pp.

xxxvi. (1868), pp. 348—351 [Collected ilber die BesseVschen Functionen (Leipzig, 1868), pp. 112

Studien

The system

of equations which has

now been

— 118.

constructed has been dis-

cussed systematically by Glaisher*, whose important memoir contains an interesting account of the researches of earlier writers.

The equations have been studied from a different aspect by Haentzschel f who regarded them as degenerate forms of Lame's equations in which both of the invariants g 3 and g3 are zero.

The following papers by Glaisher should also be consulted Phil. Mag. 433—438 Messenger, vm. (1879), pp. 20—23 Proc. London Math. :

pp.

pp.

;

;

(4) xliii. (1872),

Soc. ix. (1878),

197—202. It

may

be noted that the forms of equation (1) used by various writers are as follows:

k(k + l)

cPy

d*R n(n + l) n D _J__i __ _ p2 /2= Rf .,

(Poisson),

S-*-*^*

(Wisher).

has been encountered by Greenhill J in his researches on the stability of a under the action of gravity. When the cross-section constant, the special equation in which
Equation

(5)

vertical pole of variable cross-section, is

Bessel functions of order +-J.

—

Tram, of the Royal Soc. clxxii. (1881), pp. 759 828 and Dublin Math. Journal, ix. (1854), pp. 272 290. t Zeitschriftfiir Math, und Phy*. xxxi. (1886), pp. 25 33. % Proc. Camb. Phil. Soc. iv. (1883), pp. 65—73. * Phil.

bridge

—

—

;

see also a paper

by Curtis, Cam-

:


4-31]

97

Lommel's transformations of Bessel's equation.

4*31.

Various types of transformations of Bessel's equation were examined by

Lommel on two

occasions; his earlier researches* were of a

type, the later f were

much more

somewhat

special

general.

In the earlier investigation, after observing that the general solution of

&y i?-— s + *-° 2i/-l dy

/i\ (1) .

n

,

y- *"#,(*).

"(2)

Lommel proceeded by general solution

which

it will

be

direct transformations to construct the equation

is gfi"~

at

i^v

(y^), where

a,

7

#,

whose

His

result,

special cases.

It will

are constants.

sufficient to quote, is that the general solution of

z*^ + (2a-2/3v+l)z^ + {&y*z*i + a(a-2/3v)}u = i

(3) is

u

(4)

When /3=0,

= z»*-*<@v (yz*).

the general solution of (3) degenerates into

tt=g-«(Cl +c2 log3);

and when y=0, unless

degenerates into

(dv is zero.

The be

it

solution of (3)

sufficient to

(7)

z

(8)

2

explicitly

by Lommel

S+ ^+

(1

-"

(1

_ v) g_i

)

S + 5"*=0; tt=0;

^+/3»y»«*-»«=0;

(9)

2

(10)

(11)

An

was given

in

numerous

quote the following for reference

*- + V,W')«-**.{iV*). u=

*Wmw (y*).

l^ + „„0;

«-**#S (Jld),

**^ (*«*).

<^ ±zu =0-

u=**#4 (3*»),

*^j (§»«»).

account of Stokes' researches on the solutions of equation (11) will be given in

§§ 64, 10-2.

•

* Studien liber die Bessel'schen Functionen (Leipzig, 1868), pp.

(1871), pp.

475—487.

f Math. Ann. xiv. (1879), pp. 510—586.

98

— 120

;

Math. Ann.

in.


98

[CHAP. IV

Lommel's later researches appeared at about the same time as a memoir by Pearson*, and several results are common to the two papers. Lommel's procedure was to simplify the equation f

2v-ld{y/ X (z)}

d*{ylx(z)\

+(s)

d{ir{z)Y of which the solution

^-li^£) + (2^-l)t>) + 2 ^)l^ (*"<*)

Now

+ (2 ,-i)£^ + 2^x>)_aGf) Tf ^

define the function

It will

(2)

i.l

y=

0.

by the equation

^W-^WfeWI'^W^

we eliminate x ( 2 )> ;

<£

W

be adequate to take

(H)

v

'

reduction the equation becomes

l*'(*)

If

X

?-*(*) {*(*)}' *,{*<*)}.

(13)

+

+ y(*)~

is (§ 4*3)

( 12 )

On

d+iz)

ck»

<£(*)

it is

dz

1 .

apparent that the general solution of

2<£(*)

|_4 ((^j

y

=o

is

As a (17

>

special case, if

a?

+

[a

we take

(z)

= 1,

it is

seen that the general solution of

» ° *^r - « ito +«+<»)-»+ *i i+wf J -

IS

(18)

y

-V{* («)/*' CO}. #, {*<*)}•

Next, returning to (13), we take

x (*) = {* (^)}M_V and we find J

that the general

solution of

is

(20) *

y «{*<*)}* ®W*<*)).

Messenger, ix. (1880), pp. 127—131.

f

The

Inactions

x

(•*)

and

\p (z)

are arbitrary.


4-32]

The

99

following are special cases of (17):

^ + (^~^)y = H + ^^ =

(21)

(22)

0;

y -#,(«•),

0;

y-*^^>

The independent researches of Pearson proceeded on very similar lines except that he started from Bessel's equation instead of from the modified form of it. The reader out in his paper.

A

will find

many

special cases of equation (17)

partial differentia] equation closely connected with (7) o2 « 0Z l

du v

'

and

(8),

worked

namely

du

n

VZ

Ct

—

has been investigated by Kepinski, Math. Ann. lxi. (1906), pp. 397 405, and MyllerLebedeff, Math. Ann. lxvi. (1909), pp. 325—330. The reader may verify that Kepinski's formula

is

a solution, when/(w>) denotes an arbitrary function of

The int.

de

I'

special case of the equation

Acad, des

4'32.

Sci.

de

when

i/=

Cracoirie, 1905, pp.

—1

198

w.

was also investigated by Kepinski, Bull,

—205.

Malmsten's differential equation.

Twenty years before Lommel published his researches on transformations of Bessel's Malmsten* investigated conditions for the integrability in finite terms of the

equation,

equation

« which

3+;£-(^> is

obviously a generalisation of Bessel's equation

;

and

it is

a special case of

^ 4*31

(15).

To reduce the

equation,

Malmsten chose new

variables defined

by the formulae

where p and q are constants to be suitably chosen.

The transformed equation

We choose p

is

and q so that we take

this

may

reduce to the equation of § 4*3

(1)

considered by

Plana, and therefore so that p

2pq-q + l+qr=0,

= - ir — fan.

The equation then reduces

to

cPuV d? *

(m + 2)q = 2,

4A

L(»»

+ 2) 2+

2 2 g {4*+(l-r) }-l ~]

4£*

—

_P

Camb. and Dublin Math. Journal, v. (1850), pp. 180 182. The case in which «=0 had been previously considered by Malmsten, Journal fUr Math, xxxix. (1850), pp. 108 115.

—


100

By

§ 4*3 this is intcgrable in finite

terms

if

to»{4« + (l-r)«}-i-*(» +

where n

is

an integer

l),

so that

;

m + 2= + ^+(l-r)>}

(2)

»+£

The equation

4'4.

is

[CHAP. IV

A =0 and

also obviously integrable in the trivial cases

is

The notation of Pochhammer for

m=

of hypergeometric

series

-2.

type.

A compact notation, invented by Pochhammer* and modified by Barnesf, convenient for expressing the series which are to be investigated. We shall

write

now and subsequently

= o(a+ l)(a+

(a)„

The notation which p* q (au a2

ap

...,

,

Pl

,

p2

...,

,

(a

...

be used

will

,

2)

=

(a)

1),

1.

in general,

is,

pq

+«-

-

z)= ±

,

rt=0

n

J

j-^ KPi)n \Pi)n

•

• •

zn

.

\Pq)n

In particular,

< g >"

« I

-»

n=0W!(p)n CO

.71

=0«!(p)n' *F,(«,/9;

The

,;*)=*

functions defined by the

-^7^^

(

first

three series are called generalised hyper-

geometric functions. It

may be

noted here that the function

t

F

t

(a

;

p

;

z) is

a solution of the

differential equation

and,

when p

is

not an integer, an independent solution of this equation

^"".^(a-p +

l; 2 -p;

is

z).

It is evident that

Various integral representations of functions of the types \Fi

by Pochhammer, Math. Ann. *

Math. Ann. xxxvi.

(1890), p.

t Proc. London Math. Soc. of the suffixes

p and

84

;

xxxvm.

F

i , 0^*3

(1891), pp. 227, 586, 587.

(2) v. (1907), p. 60.

q before and after the

,

have been studied

174—178, 197—218.

xli. (1893), pp.

The

Cf. § 4-15.

modification due to Barnes

is

the insertion

F to render evident the number of sets of factors.

:

.


4'4, 4*41]

4'41.

We

Various solutions in

shall

now examine

series.

various solutions of the equation

—

— c*u = p(pr + \)- u,

d2 u

-j—

„

dz*

z*

and obtain relations between them, which in Pochhammer's notation. It is

supposed

for

101

will for

the most part be expressed

the present that p is not a positive integer or zero, is unaltered by replacing p by — p — 1, it is

and, equally, since the equation

supposed that p It is already

is

not a negative integer.

known

(§4-3) that the general solution*

is

$p+

f sfi

\

(ciz),

and

this gives rise to the special solutions z*+*

.

F, (p

+

|

\c*z«-)

;

The equation may be written

z~p

;

-oF^-p;

in the forms

dz

dz*

which are suggested by the

\c*z*).

v

z*

fact that the functions e ±e* are solutions of the

original equation with the right-hand side suppressed.

When ^

is

written for z (d/dz), the last pair of equations become

(^

-p -

1) (^

+ p)

.

(ue**) ± 2cz% (uei*2 )

= 0.

When we solve these in series we are led to the following four expressions for u zP+^.tF^p + l; 2p + 2; -2cz); zp+i e-^^F^p + 1; 2p + 2; 2cz)

sT*'e**

(1)

«*. ,F, (p

(2)

+

1

;

2p + 2

;

,

z^ zp+3

;

-2p; -2cz);

to

.

,

is

x

x

(p

+

1

;

2p + 2

(1) has

have to consider the cases when 2p from

§ 3*1 that

1

obvious on replacing

a special investigation

integer.

t Journal fllr Math. xv. (1836), pp. 138—141.

is

left

are

And the expressions

....

we must have

- 2cz) = e~ n F

Kummerf. When

2cz).

Since none of the two sets of

z-'P, ....

not an integer,

1

values of p, the truth of (2)

* It follows

is

z1 ~p,

(- p

ef.Ai-p; -2p; -2cz) = e-". F (-p; -2p;

These formulae are due

We now

tr*,

l

two expressions on the

direct multiplication of series, the

expansible in ascending series involving z?+1

on the right similarly involve powers are the same when 2p

F

zr* era .xFx (-p\ -2p\

;

Now, by

l

is

2cz)

been proved

p by —p — 1

an

2cz)

;

for general

in (1).

integer.

also necessary

when p

is

half of an odd


102

When p

[CHAP. IV

..., the solutions which contain z~p by series involving logarithms (§§3*51, 3*52), only one solution which involves only powers of z. By the

has any of the values £, f f ,

,

as a factor have to be replaced

and there

is

previous reasoning, equation (1)

When p

has any of the values

holds.

still

a comparison of the lowest powers

0, 1, 2, ...

of z involved in the solutions shews that (1) that there are no relations of the form z-p J\(\-p\ \
x

We and

(2)

2cz)

+ k2 zr+*

not obvious

+ §; \&z*) F {p + %; \c*z% 1

1

k2 are constants which are not

lcit

it is

= z-ve*\F (-p\ -2p; -2cz) + k zP+\F ^z-Pe-^F.i-p; -2p;

where

holds; but

still

(p

1

zero.

have to give an independent investigation of which depends on direct multiplication of series. shall consequently

(1)

In addition to Rummer's researches, the reader should consult the investigaby Cayley, Phil. Mag. (4) xxxvi. (1868), pp. 348—351 [Collected Papers,

Note.

tions of the series

(1894), pp. 9—12] and Glaisher, Phil. Mag. (4) xliii. (1872), pp. Trans, of the Royal Soc. clxxii. (1881), pp. 759—828.

vii.

4'42.

433—438;

Phil.

Relations between the solutions in series.

The equation

*Vi(p+l; 2p+2; which forms part of equation general formula due to (i)

-2cz)

= e-ez F 1

441,

(1) of §

1

is

(p

+ l; 2p + 2;

2cz),

a particular case of the more

Kummer*

0-«Wp-«;

i^(«; p;

/>;

-£),,

which holds for all values of o and p subject to certain conventions (which will be stated presently) which have to be made when a and p are negative integers.

We first suppose that p is not a negative integer and then the coefficient of n £ in the expansion of the product of the series for e* and (p — a p — £) is

&

mto(n

- m)\

ml(p)m

;

;

" Wp)n .V^*' {p " ")m (1 " P ~ n)

"

nl(p) n («)»

=

n\(p)n if

we

first

'

use Vandermonde's theorem f and then reverse the order of the factors and the last expression is the coefficient of fn in X X (a; p; £*).

F

in the numerator;

The

result required is therefore established

when a and p have general complex

values J. *

Journal fur Math. xv. (1836), pp. 138

— 141;

see alsoJBach, Ann. Sci. de I'ticole norm. sup. (2)

in. (1874), p. 55.

t See, e.g. Chrystal, Algebra, n. (1900), p. 9. $ Another proof depending on the theory of contour integration has been given by Barnes,

Tram. Camb.

Phil. Soc. xx. (1908), pp.

254—257.

.


4'42]

When

p

is

103

a negative integer, equation (l)is obviously meaningless unless

and a < p

also o is a negative integer

The

j.

!

\

|

interpretation of (1) in these

circumstances will be derived by an appropriate limiting process. First let a be a negative integer

that the preceding analysis

terminating

F

The

valid.

+N

while

x

(p

x

p

;

When

M

is

p not be an integer, so p\ £) is now a

let

^(—N;

series

— £)

;

N

N

+ 1, p + p + with the later factors of the sequences iFt (p

— N) and

is an infinite series which conterms followed by terms in which the earlier factors p + N, + 2, ... in the sequences in the numerators can be cancelled

series,

N+ 1

sists of

is

(=

p,

+

p

1,

p

+ 2,

in the denominators.

...

these factors have been cancelled, the series for

+ iV p

F

X

N

(—

X

— £)

;

Hence we may proceed of (1)

N

N

may then be

p

and where

£)

;

. . .

to the limit

when p

-*.

—

if,

and the limiting form

written*

tJFxO-JV;

(2)

;

= — M,

are both continuous functions of p near p any of the integers N, + 1, + 2, ;

in which the symbol

"1

-Mi Olê^F^N-M; -M; means that the

-£)1, term in

series is to stop at the

£*', i.e.

the last term in which the numerator does not contain a zero factor, while 1 means that the series is to proceed normally as far as the term and then it is to continue with terms in f Jf+1 f->/+2 ... the vanishing factors in numerator and denominator being cancelled as though their ratio were one of equality.

the symbol in % M

~N

,

With (3)

this convention, it is easy to see that

Ai-N; -M;

?)1=i^,(-iV;

+ ( - )M ~ N When we (4)

replace iV

by

N MKM~Ay.

As an ordinary

-

If

£) 1

^ (-

combined with

iV;

1

^ (^ " ^ +

;

3/

+

2

;

?).

we have

£,

= iJFi
1

Jf

;

-M; -?)1 1 ;

3/

+

2 ;

-

t).

we have

^(M-N+l; M+2; (5)

** +I

f)

k")! <- P™'* (^ +

case of (1)

this result

-M;

M — N and £ by —

^(N-M; -M; + ( ->"

and from

,

,

,

=

f)

«(A +l; and

(4)

^ (JV-

if;

(2), (3)

- Jf 01 - rf ;

r

if

+ 2; -

£),

we deduce that

-

if;

-

This could have been derived directly from (1) by giving p value, and then making p tend to its limit.

£) 1.

—a

(instead of o)

an integral

* Cf. Cayley, Messenger (old series), v. (1871),

462],

aDd Glaisher, Messenger, vm.

(1879), pp.

pp.77— 82

20—23.

[Collected Papers,

vm.

(1895), pp. 458—

.


104

We

next examine the equation

0*J

(6)

which

X

(p

+1

2p

;

+

2

-

;

2cz)

=

& (p +

forms the remainder of equation (1) in §4*41,

Kummer*. If we suppose

that 2p

(-2r

(p

+

m= o(n-m)\ml(2p

Now

n

—1

2W

2

expansion of (1 1

f(o+)

2~

(1

.

.

ix»

+

— 2t)~p~

l

(1

\&z% is also

due to

left in (6) is

1 2m Q> + u»(-»--2;>-i)n-w ml(n-m)l

(-)*

=

(2p

+

2) n m=0

l)m (-

n

- 2p - l)n-,„

2) m

»Cm (^ +

f;

and which

not a negative integer, the coefficient of (cz) n in

is

the product of the series on the

^

[CHAP. TV

— t) n+2p+1 and ,

- 20"P_I (1 - t)n+2P +1 tr"-

so it

is

dt

=

equal to /«>+)

1 1

the coefficient of tn in the

is

0-

Y~\

~ 0~p_1 w~ n~

1

dM,

= t/(l — t) and the contours enclose the origin but no other singularities integrands. By expanding the integrand in ascending powers of u, we

where w of the

see that the integral

is

zero if n

is

odd, but

(p

it is

4- l)i

equal to v,,

.,

when n

is

even.

Hence

it

follows that

„=o2

and

this is the result to

When we make p

m .n!(p + f) n (cz)

%

2n

'

be proved.

tend to the value of a negative integer,

— N, we

find

by

the same limiting process as before that lim

,F,

(p +

1

;

2p + 2

;

- 2cz) = F (l-N; l

l

2

- 22V; - 2cz) 1

It follows that

&Q-N; we change the we find that If

(7)

2 tfx (f

icV) = ^.

1

F

1

(l-iV; 2-22V;

- 2c*) 1

signs of c and z throughout and add the results so obtained,

- N icV) = e«. ;

^

(1

- N;

2

+ e-^.

—

- 2tf; - 2cz) 1

~1

F (l-N; 2-2N; 1

2cz)~\,

* Journal fUr Math. xv. (1836), pp. 138 141. In connexion with the proof given here, see Barnes, Tram. Camb. Phil. Soe. xx. (1908), p. 272.


4*43]

105

the other terms on the right cancelling by a use of equation This is, of (1) J_y+ j (icz) in finite terms with a different notation.

course, the expression for

For Barnes' proof of Rummer's formulae, by the methods of contour

inte-

gration, see § 65.

4*43.

Sharpe'8 differential equation.

The equation

'g+g + C + ^-o.

(1 >

which is a generalisation of Bessel's equation for functions of order zero, occurs in the theory of the reflexion of sound by a paraboloid. It has been investigated by Sharpe* who has shewn that the integral which reduces to unity at the origin

is

y=C\

(2)

+A

cos {z cos

log cot \0)d0,

J

where (3)

1

=

"cos

f

(4

log cot \0) dd.

Jo

This

is

the appropriate modification of Parseval's integral

= tanh

vestigate its convergence write cos

y

(4)

=G

,

and

r C°SW + * tanh<

*>)

coshrf>

Jo

It is easy to see from this form of the integral that

values of

A

for

which \I(A)\<

1,

it

(§ 2-3).

To

in-

becomes

dd> ^

it

converges for (complex)

andf 2

C=-

cosh

hvA.

ir

The

integral has been investigated in great detail by Sharpe and he has given elaborate rules for calculating successive coefficients in the expansion of y in powers of z.

A simple

form of the solution (which was not given by Sharpe)

y-« ±to ,^(jTi»^; The reader should have no * Messenger, x. (1881), pp.

1;

T

2iz).

difficulty in verifying this result.

174—185

;

xn. (1884), pp.

66—79

;

Proc. Camb. Phil. Soc. x. (1900),

pp. 101—136.

t See,

e.g.

is

Watson, Complex Integration and Cauchy's Theorem (1914), pp. 64

— 65.


106

[CHAP. IV

Equations of order higher than the second.

4*5.

The

construction of a differential equation of any order, which

soluble

is

been effected by Lommel* its possibility depends on the fact that cylinder functions exist for which the quotient <$„ (z)/^ v (z) is independent of z.

by means

of Bessel functions, has

;

Each of the functions Jn (z) and Yn (z), of integral order, possesses ® (z), H„ property [§§ 2*31, 3*5]; and the functions of the third kind p

H

possess it (§ 3*6 1),

Now when

whether v

is

an integer or

gr^'^fr V«)- (£7)w *

the cylinder function on the right

This v

= n+\

and

Hence

From z^ n

the case either

is

if

(i)

not.

is

}( "- m,

of order

an

if v is

^,-m(7 V*)>

—v

m = 2v.

if

and

integer, n,

m = 2n,

or

(ii)

if

m = 2n + 1.

^n denotes either Jn or Yn

this equation

Yn (y \lz) are

we obtain Lommel's

W

where 7 has any value such

we have

result that the functions z* n

J"y_ (frry dz™ ~ zn that 2n = (— )n cm
7 giving 7

,

Jn (y \Jz)

t

solutions off

,o\

By

(z)

the form

§ 3*9 (5) is written in

C1 )

this {2)

all

'

,

so that

= ic exp (riri/n).

possible values

(r

we obtain 2w

= 0,

1, 2,

. .

.

solutions of (2),

,

n-

1)

and these

form a fundamental system. Next,

r

if tfn+i

and hence

z* n+

denotes

lH

{l)

H

{1)

n+i

,

we have

n+i (7 «Jz) is a solution of

dzm+1

\P)

where 7 has any value such that 72n+1

y and the solutions *

Studien

tiber die

#Hn+i) -«*»+*> **#„+*, so that

= -ic exp

so obtained

zn+i

'

= cm+1 e- (n+ * {riri/(n

),ri

+ £)},

,

so that

= 0,

(r

1, 2,

. .

.

,

2rc)

form a fundamental system.

BesseUschen Functionen (Leipzig, 1868), p. 120

;

Math. Ann. u. (1870),

pp. 624—635.

f

The more

general equation

has been discussed by Molins, MSm. de VAcad. des

Sci. de Toulouse, (7) vin. (1876), pp.

167— 189.


4-5]

For some applications of these 317—320.

pp.

107

results, see Forsyth, Quarterly Journal, xix. (1883),

In view of (1), which holds when m is an integer, Lommel, Math. Ann. n. (1870), p. 635, has suggested an interpretation of a "fractional differential coefficient." Thus he would

exp(±yV«)

mean 90o

to

The

(yijz).

JgJ length by Heaviside in various papers.

idea has been developed at

Lommel's formulae may be generalised by considering equation § 4*31, after writing it in the

some

(3) of

form

(* + a) (^

+ a - 2j3i>) u = - 'ptftPu,

the solution of the equation being u = zpv- a(@ v by induction that, with this value of u,

(
For

it is

easy to verify

n-\ II

(^

+ a - 2r/3) (^ + a - 2/3p - 2r£)

a

r=0

= (-) n/92nc2ns 2n

<,

w,

and so solutions of »-i

n

4)

(^

+ a - 2r£) (^ + a -

- 2r/8) u = (-)n/3^cmz^ u

2/3v

u = z^"-^v ( 7*0), 7 = c exp (riri/n).

are of the form

where

By

giving 7 these values, system.

we

(r

= 0,

1,

. . , .

n-

1)

obtain 2n solutions which form a fundamental

= 2, equation (4) reduces to + a ) (^ + «_ 2£)(^ + a 2pv) (* + a - 2/3* - 2/3) u = £W»u.

In the special case in which n (Sr

This equation resembles an equation which has been encountered by Nicholson* in the investigation of the shapes of

Sponge Spicules, namely
(5)

that

is

If

2 Au.d u\

$(3-l)(S + 4/*-2)

to say

we

[

identify this with the special

of values for a, £,

j*,

v

($

form of

.

+ 4p-3) u=*-i*u. (4)

we obtain the following four distinct

a

M

P

* Proc.

Royal Soe. xcm.

B

A

(1917), pp. 573

2

1

*

%

-1

~\

•2.

;

1

2

* 3

1

10

—

519. See also Dendy and Nicholson, Proc. Royal the special cases of (5) in which /*=0 or 1 had been solved

(1917), pp. 506

—587

V

JL

1

Soc. lxxxix.

previously by Kirchhoff, Berliner Monatsberiehte, 1879, pp. 815—828.

Chemie,

sets

:

(3) x. (1880), pp.

501—512.]

[Ann. der Physik und

=

:


108

[CHAP. IV

These four cases give the following equations and their solutions

««**{#!« +^ 4 («)},

(6)

^=«;

(7)

~ S} =

(8)

£{*V S}— *«

(9)

£ t S} =2

4

«=^

22tt;

{?

,2

{^2 (2^)+^2(2tV^},

1

«-*'*{^(«»l )+^t-(t il ».

J

«-^{^»(«»~*) +*«•(«•-*)}.

° tt;

These seem to be the only equations of Nicholson's type which are soluble with the aid in the case fi 2, the equation (5) is homogeneous. Nicholson's general

of Bessel functions

equation

is

;

associated with the function / 3-2/x

2 + 2/x

V4-2/*'

4-2 M

3

z*-** \

1+2/i

4-2M

'

'

(4-2M )V'

Symbolic solutions of differential equations.

4*6.

Numerous mathematicians have given

solutions of the equation § 4*3 (1)

namely

in symbolic forms,

when p

is

a positive integer (zero included).

These forms

are intimately connected with the recurrence formulae for Bessel functions. It has

been seen

(§ 4*3)

that the general solution of the equation

and from the recurrence formula

§

is

39 (6) we have

***Wcu) - (- ays*** (J£f [r*Vh (ciz)}. Since any cylinder function of the form z

(ae°

where a and be written

/3

are constants,

it

A

+ Pe-^ls/z,

/

d

ae + &ey —. 6*

w

where

a'

= a/c, #'= -fife.

may

c*

r—

[jdz)

modification of this, due to Glaisher*,

(3)

is

= z*** (Ay^(a'** + P'e- *),

This

6

may be

seen by differentiating CLeP+ffe-*2 once.

Royal Soc. clxzii. (1881), p. 813. It was remarked by Glaisher that Earnshaw, Partial Differential Equations (London, 1871), See also Glaisher, Quarterly Journal, xi. (1871), p. 269, formula (9), and p. 270.

* Phil.

equation p. 92.

,

u=zP+1

2>

(ciz) is expressible as

follows that the general solution of (1)

.

<

*
Tram, of

(3) is

the

substantially given by

4 * 6]


109

Note. A result equivalent to (2) was set by Gaskin as a problem* in the Senate House Examination, 1839; and a proof was published by Leslie Ellis, Camb. Math. Journal, n. (1841), pp. 193—195, and also by Donkin, PhU. Trans, of the Royal Soc. cxlvii. (1857), pp. 43—57. In the question as set by Gaskin, the sign of c 2 was changed, so that the solution involved circular functions instead of exponential functions.

Next we

shall prove the symbolic theorem,

due to Glaisherf, that

W4Y.JL

(4)

zp+1

\zdz)

In operating on a function with the operator on the right, is multiplied by I/* *- 2 before the

that the function

5

it is

supposed

application of the

operators z3 (d/dz). It

is

convenient to write

and then

to use the symbolic formula

(5)

in

/('*) (e"°Z) .

which a

is

a constant and

Z is

=

e*°

+ a) Z,

./(*

any function of z.

The proof S, as is

of this formula presents no special difficulties when/(<>) is a polynomial in the case in the present investigation. See, e.g. Forsyth, Treatise on inferential jrww»m*

Equations (1914), §

33.

It is easy to see from (5) that

= •"-*•<*- 2p + 2)<*-

2j,

+ 4)<* -

2p

+ 6). ..*,

when we bring the

successive functions er» (beginning with those on the left) past the operators one at a time, by repeated applications of (5).

We

now

reverse the order} of the operators in the last result, and by a reversal of the previous procedure we get

= «<*-«• [<* + 2p - 2) (* + 2p - 4) (* + = e -(p+i)» [(4*$) (e**>)... (eie *).e-w--> 9 . . .

2) * «-<*-*>«] .

]

= J-[( 23 lYJL~\

z^

1

\_\

dz) z*-*]

'

Th e P bIem W<18 * he 8eCODd Part f uestion 8 Tu e«day ° 1 afternoon, Jan. n Cambridge University Calendar, 1839, p. 319.

\

,

^

-

L

JJ:;iZ ££S£ t

It

(1876)>

pp 240

~243

'

349

"350

'

was remarked by Cayley, Quarterly Journal, xn.

Glaisher, that differential operators of the form *•+'

;

8,

and PhiL Tram of '

1839; see the t,ie

Rmjal Soc

(1872), p. 132, in a footnote to a paper

± s -,

i.e.

»-„, obey

-

by

the commutative law.

'


110

and

:


When we

be proved.

If

we

replace

[CHAP. IV

p by p +

1,

we

transform (2) and (3) with the aid of (4) and (6), (] ) is expressible in the following forms

find that

we

see that

the general solution of {1)

u

v /Q 8

l / -dy+WtP + fftr * uss **\*di)

The

«*-»

dz)

>

1

'

solutions of the equation . 2pdv - -E-j-— cH = 0,

2 d --v

,

which correspond to

[(3) of § 4-3], <9 >

(2), (3), (7)

•-Ûb) v = ^ (-^J

i

+1

(10)

12 >

»

A different and

=

(aV*

and (8) are

—

+ 0'e-a

),

dya f + fier"

1^,

/nx

<

z

—^

>

<

~zp+A

?!—

^s)

(7) is due to Boole, Phil. Trans, of the Equations (London, 1872), ch. xvn. on Differential Royal IX. (1854), p. 281. pp. 423—425; see also Curtis, Cambridge and Dublin Math. Journal, II. Journal, (1841), pp. 169, Ellis, Camb. Math. first Leslie given by The solution (9) was 193, and Lebesgue, Journal de Math. xi. (1846), p. 338; developments in series were

more

direct

Soc. 1844, pp. 251, 252

obtained from

it

;

by Bach, Ann.

method of obtaining Treatise

Sci.

de

I'jtcole

norm, sup.

Similar symbolic solutions for the equation

John Hopkins University

A

Circulars, vi. (1886

transformation of the solution

pp. 364—371,

(9),

—

~-c

2

(2) in. (1874), p. 61.

z%>~i

/3 v==c

7), p. 29.

due to Williamson, Phil. Mag.

IV

^\dc'c)

(

ae"

+ 3e ~ '

*)-

derived from the equivalence of the operators -

functions of

(4) xi. (1856),

C

1

is

were discussed by Fields,

is

(13)

This

v=0

r)

~-

,

if) - g-

,

when they operate on

cz.

We thus obtain the equivalence of the

following operators

-^'[(M^—m-i} it

being supposed that the operators operate on a function of

is

then manifest.

cz;

and Williamson's formula

,


4*7] 4*7.

in

Liouville's classification

111

of elementary transcendental functions.

Before we give a proof of Liouville's general theorem (which was mentioned §412) concerning the impossibility of solving Riccati's equation "in finite

terms

"

except in the classical cases discovered by Daniel Bernoulli (and the we shall give an account of Liouville's* theory of a class of functions known as elementary transcendental functions; and we shall introduce a convenient notation for handling such functions. limiting form of index -2),

For brevity we write f h (z) =

l(z)

=

=

e(z)

= ez

e 1 (z)

log z,

l2

ea (z)

,

*if(*)-*f(*)-ff(*)dM,

A

(z)

= I (I (z))

(z)

= I (l2 (*)),

e3 (z)

= e(ea (z)),

l3

}

= e(e(z)),

*/<*)- 9

9,/(#)-9{9/(#)},

. .

.

....

{*/(*)},

function of z

is then said to be an elementary transcendental function.% as an algebraic function of z and of functions of the types where the auxiliary functions (z), yjr(z), %(z) are

if it is expressible lr

e r y}r(z),

(z),

expressible in terms of z and of a second set of auxiliary functions, and so on; provided that there exists a finite number n, such that the nth set of auxiliary

functions are

all

The order

algebraic functions of

z.

of an elementary transcendental function of z

is

then defined

inductively as follows:

Any

(I)

(II) If

algebraic function of z

fr (z)

is

of order zero§.

denotes any function of order

r,

then any algebraic function

of functions of the types

*M*\ (into

which at

(III)

Any

least

r,

one of the

function

possible order.

order

efr {z),

is

,fr (z), first

fr (z),

three enters)

/„(*),.../,(,) is

said to be of order r

+

1.

supposed to be expressed as a function of the lowest is to be replaced by r (z), and it is a function of f

Thus elfr (z)

not of order r

+ 2.

In connexion with this and the following sections, the reader should study Hardy, Orders of Infinity (Camb. Math. Tracts, no. 12, 1910). The functions discussed by Hardy were of slightly more restricted character than those now under consideration, i

since, for

his purposes, the

symbol

not required, and also, for his purposes, postulate the reality of the functions which he investigates. r is

it is

convenient to

It may be noted that Liouville did not study properties of the symbol s in merely remarked that it had many properties akin to those of the symbol I.

T Journal de t It

is

Math. n. (1837), pp. 56—105 in. (1838), pp. 523-547 supposed that the integrals are all indefinite.

t " Une fonction fiuie § For the purposes of

;

;

iv.

detail,

(1839), pp.

but

423—456

explicite."

this investigation, irrational

not be regarded as algebraic functions.

powers of

:,

such as z" of course must ',


112 4*71.

The

in

theorem* concerning linear differential equations.

Liouville's first

investigation of the character of the solution of the equation

which

% (z)

a transcendant of orderf

is

has established the following theorem (I) has

If equation

m > n,

[CHAP. IV

a

n,

has been

made by

Liouville,

who

:

solution which is

a transcendant of order

m+

where

1,

then either there exists a solution of the equation which is of order\ n,

or else there exists a solution, u i} of the equation expressible in the form

=
U:

(2)

where

M (z) is

/

such that

of order

ft,

and

the order of

<£M

(z) does not exceed

/a,

and

fi is

n^p^m.

m + 1, let it be fm+i(z) then an algebraic function of one or more functions of the types lfm (z) sfm ( z )> efm (z) as well as (possibly) of functions whose order does not exceed m. Let us concentrate our attention on a particular function of one of the three types, and let it be called 6, ^ or ® according to its type. If the equation (1) has a solution of order

/m+i ( z )

(I)

m + 1,

t

We shall first shew how to prove that, if (1) has a solution of order then a solution can be constructed which does not involve functions of

the types 6 and For,

S-.

= F(z, 0), where F is an algebraic

if possible, let /, )l+ i (z)

and any function of z (other than is algebraically independent of 0.

Then /qx {6)

it is

itself) of

order

m+1

function of

which occurs

in

;

F

easy to shew that

*F

/ n F X K) " Tz*~

+ (_!_

W+ dz>

2

fm

dz

df™ (z) dz

(z)

dfm {z) \*d*F

\fm {z)

it

;

is

\d_

d*

F

d6dz

f_l_ dfm (z)}

dP^ldz\fm (zj

1

dz

W-'-xW

]

being supposed that z and 6 are the independent variables in performing

the partial differentiations.

The expression on the right in (3) is an algebraic function of 6 which vanishes identically when 6 is replaced by lfm (z). Hence it must vanish identically for all values of 6

zero would express lfm

orders do not exceed

m

;

tor if it did not, the result of

together with transcendants of order

ex huputhesi, algebraically independent of '

Journal de Math.

t This phrase

is

than

w,

which

it

to

whose which are,

iv. (1839), pp.

m+ 1

6.

435—442.

used as an abbreviation of " elementary transcendental function of order u."

% Null solutions are disregarded less

equating

(z) as an algebraic function of transcendants

is

;

if

u were of order

contrary to hypothesis.

less

than «, then - --^ would be of order


4'71]

by

113

In particular, the expression on the right of (3) vanishes when + c, where c is an arbitrary constant and when this change ;

expression on the

^

is

That

therefore zero.

When we

c) '-

„.

,

.

.

-F(z,0 + c). x (z), ,

to say

is

differentiate (4) partially with regard to

+ c)

dF(z,

c,

we

find that

+ c)

o-F(z,

'"

'

oc

'

dc>

are solutions of (1 ) for all values of c independent of z. If performing the differentiations, these expressions become

dF(z,

d3F(z,

)

d0

) "'

d&

'

which are consequently solutions of Fe>

replaced

made the

of (3) changes into

left

d-F(z,0 + which

is is

we put

=

after

'"'

For brevity they

(I).

c

will

be called

-Fee, ....

Now

either

F

and

Ft

form a fundamental system of -solutions of (1) or

they do not. If they do not,

we must have*

F = AF, where

A

is

On 0. F=
independent both of z and

integration

we

find that

,

m

where 4> involves transcendants (of order not exceeding + 1) which are algebraically independent of 0. But this is impossible because e Ae is not an algebraic function of and therefore and e form a fundamental system ; of solutions o/'(l).

F

Hence where in

A

Fee

and

is

F and F

by an equation of the form

FM = AF, + BF, Now this may be

regarded as a linear equation

expressible in terms of

B are

constants.

(with constant coefficients) and

F = O^* + where

,

and

F

2e0«

3>2 are functions of

e

its solution is

or

/= e»* {4>

x

+ 4>a 0},

the same nature as

4>,

while a and

/9

are

the roots of the equation

a?-Aa;-B = 0.

F which is an algebraic function and then F is a linear function of 0.

The only value a

=$=

;

Similarly, if it

must be a

of

fm+1 (z)

of

involves a function of the type ^,

is

we can prove

linear function of ^. * Since

F mutt involve

0,

Ft cannot be

obtained when

identically zero.

that


114

[CHAP. IV and ^,

It follows that, in so far as fm+x (z) involves functions of the types it

involves

them

linearly, so that

fm+i(z)^%0 where the functions of order

m+1

1

yffp q (z) are of

m + 1 at most, and

order

,

them

involved in

...%,%,

lt 0.,,

write

(z)02 (z)...0p (z).^ 1 (z)%(z)...^ q (z).yfrPiq (z)

Take any one of the terms function of

we may

Ox {z)0t {z)

are of the type ©.

in

...,

fm+l (z)

and

let it

which be

P (z).* 1 (z)

...

...

is

of the highest degree, qua

* Q (t).irPiQ (z).

Then, by arguments resembling those previously used, "

d_d_ d0!

is

a solution of (1)

But ^p

t

'"

d_

d_

d0p'd&

d*S 2

1

follows that

it

_a_

"*

J

S^

/**.(*)

yffpîz) is a solution of (1).

i.e.

a function of order not exceeding m, or else it is a which involves functions of the type © and not of

Q (z) is either

m+1

function of order

and ^.

the types

In the former order,

;

d_

d02

>

the only functions

and in the

case,

we

repeat the process of reduction to functions of lower

latter case

we

see that

some solution of the equation

is

an

algebraic function of functions of the type ©.

We

have therefore proved

that, if (1) has

of order greater than n, then either solution which

is

$* ( z )> where /M (z)

We

(II)

it

a solution which

is

a transcendant

has a solution of order n or

else it

has a

an algebraic function of functions of the type efîz) and is of order /* and <£M (z) is of an order which does not exceed ft.

shall

next prove that, whenever (1) has a solution which is a n, then it has a solution which involves

transcendant of order greater than

the transcendant efîz) only in having a power of

We concentrate

it

as a factor.

our attention on a particular transcendant

© of

and then the postulated solution may be written in the form where is an algebraic function of © and any function (other than of order /* + 1 which occurs in G is algebraically independent of ©.

e/j, (z),

;

Then

it is

rf*G

the form

G (z,

©),

© itself)

easy to shew that

&G

rPG

+ e[/„"

«+

d*G

{/,'(*))•]

|g-G.x<*>.

The expression on the right is an algebraic function of © which vanishes when is replaced by e/îz), and so it vanishes identically, by the arguments used in (I). In particular it vanishes when © is replaced by c ©, where c is independent of z. But its value is then

^

'-G(*,ce).x(«),


4*71]

115

so that

^

d*G(z,c%);

/a\ (6)

When we

~. , „, -G(z,c&). X (z) = .

differentiate this with regard to

&G (z,

dG(z,c&)

we

c,

0.

find that

c0) '•'

dc

9c»

'

'

are solutions of (1) for all values of c independent of expressions become ft

^ d*G(z,e)

dG(z,S)

d@

a@ 2

*

Hence, by the reasoning used in

we have

(I),

we put

If

z.

c

= 1, these

••"

'

®G9 = AG or else

&G*B = A®GB + BG, A

where

and

B are

constants.

In the former case $>,«*

where

4>,

Q>lt

+ O,**

8

G — Q% A

,

and

&{&

or

1

in the latter

+ &t \og&} =

G

has one of the values

&{&

l

+

& f?(?)},' t

are functions of z of order jl+ 1 at most, any functions of

order fi+1 which are involved being algebraically independent of

while

;

7 and $ are the roots of the equation x (x In any

G

is

case,

the

G

sum

either contains

-

1)

- Ax - B = 0.

only by a factor which

of two expressions which contain

is

a power of

or else

only in that manner. In the

latter case*,

G(z, c%)-c*G(z, 0) is

only by a factor which

a solution of (1) which contains

By

repetitions of this procedure,

+

we

see that, if 0,,

is

2 , ...

a power of 0. r

are

all

the

which occur in the postulated solution, we can derive from that solution a sequence of solutions of which the sth contains ... 0,; and the rth 0,, 0„, ... 0, only by factors which are powers of n 2 member of the sequence consequently consists of a product of powers of 0i, r multiplied by a transcendant which is of order ft at most; this solution is of the form transcendants of order

/*

1

,

£„(*)exp|2 7»log*V> which

is

of the form

^M (z)

.

ef^ (z).

* If $j is not identically zero

;

if it is,

then

4> 0* is 2

a solution of the specified type.


116 4*72.

We

Liouville's second theorem concerning linear differential equations.

have just seen that,

the equation

if

(!) [in

|CHAP. IV

;£i-«X<*)

which x ( z )

is

°f order n] has a solution which

scendant of order greater than

where

/*

^ n.

then

n,

/*

an elementary tran-

more than one solution of

If the equation has

solution for which

is

must have a solution of the form

it

this type, let a

has the smallest value be chosen, and let

Liouville's theorem,

which we

order of d (log uj/dz

is

equal

now

shall

prove,

is that,

for

be called

it

«,.

this solution, the

to n.

Let

d

log

«i

dz

and then If

is

t

N=n,

satisfied

by

t,

of order

p

at

most

the theorem required

tys-i i z),

is

proved.

t

If

be N, where iV < /*.

N>n,

then the equation

namely

has a solution whose order iV

Now

C'

the order of

let

;

=

greater than

is

n.

an algebraic function of at least one transcendant of the types s/jr-i (z), efN _ x (z) and (possibly) of transcendants whose order does t is

not exceed

N-

1.

We

call

the

first

three transcendants

0,

^,

@ respectively.

contains more than one transcendant of the type 0, we concentrate our attention on a particular function of this type, and we write If

t

t

By arguments

= F(z,0).

resembling those used in § 4*71, we find that,

if

N>

n,

then

F(z,0 + c) is

also a solution of (2).

The corresponding expJJF(*,

and

this

is

a solution for

all

differentiation with respect to

^-c is

also a solution of (1);

c,

solution of (1)

+ c)dz,

values of c independent of

we

[ex V

jF(z,0 + c)dz]j^

u2 = u JF l

dz,

so that

diu

*Tz-

z.

Hence, by

find that the function u^ defined as

and we have

U

is

da,

U

*dz~

„ = U F„ *

4-72,4-73]


But the Wronskian of any two where C If

is

solutions of (1)

is

117

a constant*; and so

a constant.

C= 0, Fis independent of 0, which is contrary to hypothesis ;soG^ 0, and m,

Hence

=

J(C/F,).

an algebraic function of and similarly it is an algebraic function of all the functions of the types and ^ which occur in t. Next consider any function of the type which occurs in t we write m, is

;

;

t

= G (z,

&), and, by arguments resembling those used in § 471 and those used earlier in this section, we find that the function u., defined as 3 ris

a solution of (1)

;

r

exp }&(z, c
i

and we have

so that

„

This Wronskian

is

1

__

a constant,

i(j

_=

C and lt

!

,,.eG 9

.

so

«,-V{W(?.)l. Consequently u

an algebraic function, not only of all the transcendants of and ^, but also of those of type which occur in t and therefore

the types

x

is

;

«! is

of order

iV.

This

is

contrary to the hypothesis that u 1

is

of order

fi

+

l,

where f n.

The of

contradiction shews that

d (log Uj)/dz 4

#

73.

We

is n.

And

N cannot be greater than n

this is the

hence the order theorem to be established.

Liouville's theoi-emf that Bessel's equation

shall

now shew

;

has no algebraic integral.

that the equation

has no integral (other than a null-function) which first reduce the equation to its normal form

is

an algebraic function of

z.

We

by writing

y=uz~^, p

= ±v-%-

* See e.g. Forsyth, Treatise on Differential Equations (1914), § 65.

t Journal de Math. iv. (1839), pp. 429—435 ; vi. (1841), pp. 4—7. Liouville's first investigation was concerned with the general case in which x (z) is any polynomial the application (with various modifications) to Bessel's equation was given in his later paper, Journal de Math. vi. ;

(1841), pp.

1—13,

36.

,


118

This

is

[CHAP. IV

of the form

» -"%<*>

dhi

,

x

where

X (.).£(£±l)-i.

(2)

an algebraic integral then (1) also Let the equation which expresses this integral, u,

If possible, let Bessel's equation have

has an algebraic integral.

;

as an algebraic function of z be

= 0,

64(u,z)

(3)

where 64

is

a polynomial both in u and in z

;

and

it is

supposed that 64

ik

irreducible*.

Since u

is

a solution of (1)

we have

64uu 64z* - 2>64uz 64u 64z + 64M 64U* + 64u3 uX (*) =

(4)

The equations of (3) satisfy

(3)

and

(4)

have a common

root,

0.

and hence

all

the roots

(4).

and (4) (qua functions of u) would than 64 itself, and this would be a factor other reducible, which is contrary to be would Hence 64

For, if not, the left-hand sides of (3)

have a highest common polynomial in u and in z. hypothesis.

Let

the roots of (3) be

all

«,, «», tt/

is

M

...

+ W *+ 2

a rational function of z ; and there for which this sum is not zerof.

Let any such value of

uM

s be taken,

• • •

is

Then,

.

if s is

any positive integer,

+ Ujg*

at least one value of s not exceeding

and

let

»»=i

Also

W

let

where r =

r

1, 2, ... 5.

= s (s - 1) Since «„

. . .

** 2

(8

,

- r + 1) j^ «.-* (-^)'

... m,,

are

all

solutions} of (1),

it is

easy to

prove that

dWo

<*\

^=W^ + r(s-r+l) X (z)W .

(6)

*

That

-w r

1

is to say,

l ,

Because

=

1, 2, ...

«- 1)

in z or in both u and -J^ has no factors which are polynomials in u or would be zero.

t If not, all the roots of (3) +

(r

(4) is satisfied

by

all

the roots of

(3),

qua equation in

u.

«.

4 *73]


W

Since

a rational function of

is

so that

where

W=

An

and

Bn>q

_

X A n z»+2

Kz -

« 9 ) in

'\H>

and X are integers, n assumes summation and aq ± 0.

Let the highest power of l/(z l

~

BW

are constants, k

integral values only in the last

It follows

expressible in partial fractions,

z, it is

*

119

- aq)

which occurs in

by an easy induction from (5) and p+r where r = r is lj(z - a g ) 1, 2,

W

,

W

be lj(z

-

(6) that the highest

positive

aq ) p

.

power of

... s.

Hence there is a higher power on the left of (7) than on the right. This contradiction shews that there are no terms of the type 2? n n>9 (z - a q )~ in a and so

W

W

= X A n z\ n=

We may now have a

last

term

assume that if it

From (5) and (6) which occur in

-it

A K ^ 0,

because this expression for does not vanish identically.

it is lt

must

easy to see that the terms of highest degree in z ... are* 2 3

W W W W ,

W

,

,

A K z\ XA K z K ~\ A K sz\ XA K (Zs-2)z k -\.... By

a simple induction

in

W,r h

shew that the term of highest degree A K z*.l.3...(2r-l).s(s-2)...(s-2r+2). it is

possible to

An induction of a more complicated nature term of highest degree in v+l is

is

W

then necessary to shew that the

^z*->2.4,...(2r).(s-l)(s-3)...(s-2r+l). 2F (h-ls;l-±sl

where the

suffix

r

+1

indicates that the first r

+1

l) r+1

,

terms only of the hyper-

geometric series are to be taken. If s

odd, the terms of highest degree on the left and right of (7) are X - 2 and X respectively, which is impossible. Hence vanishes whenever s is odd. is

W

of degrees

When That and

s is even, the result of

say

is to

so,

by Vandermonde's theorem, 2.4.6... .. ,

expression on the

* It is to be

t

The

x

XAi.sl^-XAi.sl^d, -te i_to XA^sl^Q,- %s; £_£*; J) = 0, XA K .sl

The

equating coefficients of zx ~ in (7)

left

,

—

s

.

=

1.3.5...(«-1) vanishes only when X is

remembered that the term of highest degree

analysis given by Liouville, Journal de Math.

l)is

is

iu

zerof.

x (2)

is

-

1.

seems to fail at this he apparently overlooked the possibility of \ vanishing. The failure seems inevitable in view of the fact that J* 2 (z) + J _ n _ h (*) is an algebraic function of *, by § 34. The +{ subsequent part of the proof given here is based on a suggestion made by Liouville, Journal de Math. iv. (1839), p. 435; see also Genocchi, Mem. Accad. delle Sci. di Torino, xxm. (1866), Comptes Rendiu, lxxxv. (1877), pp. 391—394. pp. 299—362 point, because

;

vi.

(1841), p. 7,


120

We

have therefore proved that, when n is expressible in the form

s is even,

s is odd,

W

[CHAP. IV

vanishes,

and

that,

when

W

2 A ny8 z- n

,

where

A

0)g

does not vanish.

From Newton's theorem which

expresses the coefficients in an equation

M must be even,

sums of powers of the roots, it appears that and that the equation £€ (u, z) = is expressible in the form in terms of the

4 if

u M + 2 u M~"

(8)

r=l

$

where the functions T$ r are polynomials

When we

r

(1/*)

= 0,

in \jz.

solve (8) in a series of ascending powers of

each of the branches of u

is

l/'z,

we

find that


m=0 a positive integer and, in the case of one branch at least, c does not vanish because the constant terms in the functions r are not all zero.

where n

is

^

And

the series which are of the form

2 cm z- m,n wi=0

are convergent* for all sufficiently large values of

When we

z.

substitute the series into the left-hand side of (1),

we

find that

the coefficient of the constant term in the result is c and so, for every branch, c must be zero, contrary to what has just been proved. The contradiction thus obtained shews that Bessel's equation has no algebraic integral. ,

4*74.

We

On

the impossibility

in a position to prove Liouville's

now

are

of integrating Bessel's equation in

finite terms.

theorem f that Bessel's

equation for functions of order v has no solution (except a null-function) which is expressible in finite terms by means of elementary transcendental functions, if

As

2i> is

in § 4-73,

where x ( z )

Now

not an odd integer.

and P = ± v ~ and we have

= ~ 1 + P (P + *)/**

write

d

(log u)/dz

=

t,

normal form

%'

* + , + i-*k±!>-a

» * Goursat,

Bessel's equation to its

we reduce

Cours d* Analyse,

ii.

t Journal de Math.

vi. (1841),

pp.273—281. Many treatises manner from an algebraic equation. 1—13, 3G.

(Paris, 1911),

convergence of a series derived in this pp.

tacitly

assume the


4*74]

Since

x iz)

°f order zero,

is

which

of order zero,

is

i.e.

from

it follows

has an integral expressible in

finite terms,

§ 4"72 that, if Bessel's equation then (2) must have a solution

must have an algebraic

it

121

integral.

If (2) has an algebraic integral, let the equation which expresses this t, as an algebraic function of z, be

integral,

a(t,z) = 0,

(3)

where S4 Since

an irreducible polynomial in

is

t is

a solution of

and

z.

we have

S4z +{ x {z)-t^s4t = Q.

(4)

As

(2),

t

in the corresponding analysis of § 4*73, all the branches of

t

satisfy (4).

First suppose that there are

more than two branches of t, and let three of them be called tlt t2 t^, the corresponding values of u (defined as exp jtdz) being uly v?, us These functions are all solutions of (1) and so the Wronskians ,

.

dus

du*

duj

dtu

du2

dz

dz

dz

dz

dz

it is

lf

dus

duz Ua

rf

t3

—£

not zero, because,

is

2

3.

,

easy to verify that

n and

dz

G C2 C

are constants, which will be called

Now

dux 2

if it

~d~

were

.

~ U* Ua

~"

'

zero, the

^'

equation (3) would have a

pair of equal roots, and would therefore be reducible.

Hence G

x

± 0,

and so UtU3

= dfcU -

Therefore uû^ (and similarly m3 Mj and uû^)

But

Wl

and therefore ux

is

=A

2 ).

is

an algebraic function of z.

/^-^,

an algebraic function of z. This, as we have seen t has not more than two branches.

in § 4-73,

cannot be the case, and so

Next suppose that t has two branches, so that S4 (t, z) is quadratic Let the branches be U ± V V, where U and V are rational functions, of z. substituting in (2)

we

find that

K) Let

{V'

V be

+ 4>UV=0.

factorised so that

V=Az*Tl(z-a q y
A

is

constant, Kq

and \ are

integers,

and

k,{

and a q are not

zero.

in

t.

By


122

From

the second

member

of (5)

follows that

it

\z

and then by substituting into the

[CHAP. IV

4

q

(z

— aq

member

first

)

we have

of (5)

^ + f4i^ + {s +2 4(r^)F + ^n(,-a,r'- X

<«>

Now

(,),o.

consider the principal part of the expression on the left near a q

is

evident that none of the numbers x q can be less than

of

them

is

greater than

—2

it

must Kq

so that Kq is

or

the numbers

tc

q

— 4, which

and,

=s

T!/f q

—

It

.

any one

V/>

are both excluded from consideration.

are equal to

if

satisfy the equation

t

Hence

all

2.

we consider the principal part near V must cancel with the — 1 in % (z), so

Again, if

power in

— 2,

oo

,

we

that

see that the highest

\ = — 2 Kq

.

9

It follows that is

xJV

and consequently £€ (t,

is rational,

z) is reducible,

which


Hence

t

many

cannot have as

as two branches and so

Accordingly, let the expression for

t

x

must be

rational.

7? -°n, g

A n zn + 2

t= 2

it

in partial fractions be

r«(*-o»)*'

An

where

and 5„ i9 are constants, k ancL.X are

values only in the last summation and aq If

If

we

l/(z

we

substitute this value of

t

in (2)

integers,

n assumes

positive

we

see that

£ 0. we

find that

consider the principal part of the left-hand side near aq

— aq )

cannot occur in

t

to a higher

power than the

first

and that

Bljq -&hq = 0, Bhq =l.

so that

we consider the principal parts near and oo we find *=1, (A^y-JL-t-pip+l); X~0, 4,' = -l. Since p = ± v — ^, we may take A- = — p without loss of generality. Similarly, if

,

that

Y

It

then follows that

u

= z-Pe ±i?U(z-aq ). 9

Accordingly,

must have a

if

we

replace

u by z~ p e ±u w

in (1),

solution which is a polynomial in

this polynomial does not vanish.

we

z,

see that the equation

and the constant term

in


123

When we substitute 2 cmzm for w in (7) we find that

the relation connecting

4'75]

successive coefficients

is

m (m — 2p — 1) cm ± 2icm-! (m — p - 1) = 0, and so the

p

is

series for

w cannot terminate unless in — p — 1

can vanish,

i.e.

unless

zero or a positive integer.

Hence the hypothesis that Bessel's equation positive integer;

and

this is the case

v — \ is zero or a an odd integer.

numbers +

and only

if,

soluble in finite terms leads

is

of necessity to the consequence that one of the

if,

2v

is

Conversely we have seen (§ 34) that, when 2v is an odd integer, Bessel's equation actually possesses a fundamental system of solutions expressible in The investigation of the solubility of the equation is therefore finite terms. complete.

Some

applications of this theorem to equations of the types discussed in § 4*3 have

been recorded by Lebesgue, Journal de Math.

4*75.

On

By means

the impossibility

xi. (1846), pp.

we can

fr az« + It has

where

been seen

n=2q — 2;

(§ 4*21)

in general, not integrable in finite terms.

it is,

that the equation

is

reducible to

and, by §4-3, the last equation

the only poss-ible cases in

terms are thpse in which q are precisely the cases in which n

finite

discuss Riccati's equation

brf

equation for functions of order l/(2g) unless

Hence

— 340.

of integrating Riccati's equation in finite terms.

of the result just obtained,

with a view to proving that

338

is

reducible to Bessel's

q = 0.

which Riccati's equation is integrable in or 1/q is an odd integer ; and these

is zero is

equal

to

4w 2w +

l

—2

or to

(i»-0,

1,

2,

...)

Consequently the only cases in which Riccati's equation is integrable in finite terms are the classical cases discovered by Daniel Bernoulli (cf. § 4'1 1) and the limiting case discussed after the manner of Euler in § 412.

—

This theorem was proved by Liouville, Journal de Math. vi. (1841), pp. 1 13. It seems impossible to establish it by any method which is appreciably more brief than the analysis used in the preceding sections.


124 4*8.

The seen

[CHAP. IV

Solutions of Laplace's equation. first

appearance in analysis of the general Bessel coefficient has been be in connexion with an equation, equivalent to Laplace's

1*3) to

(§

equation, which occurs in the problem of the vibrations of a circular membrane.

We

shall

now shew how

Bessel coefficients arise in a natural

manner from

Whittaker's* solution of Laplace's equation

^ + ^ + ^=

(1)

The

solution in question

V—

(2)

in which

f denotes

is

f(z + ixcosu

\

J

°-

-f

smu, u)du,

iy

—n

an arbitrary function of the two variables involved.

In particular, a solution

is

ek

(z

+ ixcosu + iysinu) cog

mu

J

in

which k If

any constant and

is

we take

m is

any integer.

cylindrical -polar coordinates, defined

x = p cos this solution

^

y = p sin

by the equations

,

becomes

cosmM ^ M=::e*z

eiipcog(u-*)

0** |

J

—n

eik* CMV cos

I

— it

J

= 2ekz

m (v +

)

dv,

e^ 008 " cos mv cos m dv,

I

J o

= 2iri m e** cos m<$> Jm (kp), .

by

In like manner a solution

§ 2-2.

gk

I

J —

and

this

is

(z

is

+ ix cos u + iy sin u)

sm mu

^

IT

equal to 2vim e kz sin m
.

Jm (kp).

Both of these solutions are

analytic near the origin.

Again, nates,

it is

if

Laplace's equation be transformedf to cylindrical-polar coordi-

found to become

d^V dp' *

Monthly Notices of

the

+

ldV p dp

+

1

d*V

p*

d*

R. A. S. lxii. (1902),

d

2

F

dz*

pp.

'

617—620; Math. Ann.

lvii.

(1902),

pp. 333—341. t

The

simplest

method

W. Thomson, Camb. Math.

of effecting the transformation is by using Green's theorem.

Journal,

iv.

(1845), pp.

33—42.

See


4-8, 4-81]

and a normal solution of

which ekz

this equation of

125

a factor must be such that

is

lcTV Vd
to

independent of 2

factor of

V

and, if the solution

is to be one- valued, it must be equal Consequently the function of p which is a must be annihilated by

— m where

,

m

is

an integer.

dp 3

and therefore line

p —

We

it

p dp

\

must be a multiple of Jm (Jcp)

if it is

to be analytic along the

0.

thus obtain anew the solutions e**

.

sin

m
.

Jm (kp). r/ '

\

These solutions have been derived by Hobson* from the solution Maxwell's method of differentiating harmonics with respect to axes.

^*J

(kp)

by Clerk

Another solution of Laplace's equation involving Bessel functions has been obtained by Hobson (ibid. p. 447) from the equation in cylindrical-polar coordinates by regarding d/dz as a symbolic operator. The solution so obtained is cos .

sin

where/(s)

m+ 'V«(pz)M>

is an arbitrary function but the interpretation of this solution when <@ involves m a function of the second kind is open to question. Other solutions involving a Bessel function of an operator acting on an arbitrary function have been given by Hobson, Proc. London Math. Soc. xxiv. (1893), pp. 55—67 xxvi. (1895), pp. 492—494. ;

;

4'81.

We K

Solutions of the equations of wave motions.

now examine

shall

}

the equation of wave motions

dx2

in which

t

+

dy3

+

dz 3

~&

dt 2

'

represents the time and c the velocity of propagation of the waves,

from the same aspect. Whittaker'sf solution of this equation

V=

(2)

J -IT

where

/ denotes

JO

f(x sin u cos v + y sin u

is

sin v

+ z cos u 4- ct,

u, v)

dudv,

an arbitrary function of the three variables involved.

In particular, a solution

-r

is

gik(xainucosn + yainuainv+zcoiiu + et)

f,

J? /„

„\

dU fJv

i

where

F denotes an

* Proc.

London Math.

u and

v.

Soc. xxn. (1892), pp.

431—449. 342—345. See also Havelock, Proc. London Math. pp. 122—137, and Watson, Messenger, xxxvi. (1907), pp. 98—106.

t Math. Ann. (1904),

arbitrary function of

lvii. (1902), pp.

Soc. (2) n.


126

The is

physical importance of this particular solution

the general solution in

which the waves

Now let the polar coordinates of angular coordinates of the direction

[CHAP. IV

lies in

the fact that

have the same frequency

all

and

it

kc.

yfr) be the which the passes through the polar axis is the direction (6, ) and the plane y\r = .s-axis. The well-known formulae of spherical trigonometry then shew that

cos

a>

y, z)

(a?,

be

(r, 0,

cos u

take the arbitrary function F(u, v) to be notes a surface harmonic in (u, v) of degree n we ;

We

is

= Sn (0,

V)

a surface harmonic* in

for

),

yfr.

Now

Sn (U,

let (m,

new axes

+ sin sin u cos (v — = sin a> sin sin u sin (v —

= cos

)

where Sn

),

(u, v) referred to

8n (u, «)sin u, where Sn may then write

de-

O), i/r),

l

of degree n.

(a>, yfr)

thus get the solution

Vn = e**

1

e*rc08 *- Sn (0,

(*

;

to,

J

8n

Since

Sn (0,

is

<*> 5

ty) sin wdtodty.

a surface harmonic of degree n in

<0,

t)

+ £

- ^« (0,

4>)

•

(to, sfr),

we may

write

Pn ( C0S »)

{4 n<»> (0,#) cos myfr

+ BnM (0,

)

sin mi/rj

Pnm (cos

o»),

m=l

where 4„

(0, 0),

A„< w) (0,

<£)

and

Bn

< TO)

(0,

)

are independent of

Performing the integration with respect to

-^r,

we

w and

^r.

get

Vn = 1ire A n (0, 4>) r^trmm Pn (cos o>) sin a dm ikel

= (2*)U»e^^Â n (0,4>) by

§

332.

Now the equation of wave motions is unaffected if we multiply x, y, z and by the same constant factor, i.e. if we multiply r and t by the same constant unaltered so that A n (0, ) may be taken to be inand factor, leaving dependent f of the constant k which multiplies r and t. t

;

Hence lim (k~ n Vn ) to say, r n

A n (0,

motions, and

is

* This follows

<£) is

is

a solution of the equation of wave motions, that

a solution (independent of

t)

of the equation of

consequently a solution of Laplace's equation.

from the

fact that Laplace's operator is

an invariant

for

is

otherwise obvious, because

Sn may be taken independent

wave

A n (0,

)

changes of rectangular

axes.

t This

Hence

is

of k.


4-82] is

127

n. If we assume it to be permissible any such harmonic, we obtain the result that

a surface harmonic of degree

A n (#,

)

to be

eikct ?-iJH+i* (kr)

a

to take

Pnm (cos 6) C0S md>r 8111

of the equation of wave motions*; and the motion represented by this solution has frequency kc.

is

solution,

To

assumption that

A n (0,

may

construct the normal solution of the equation of

be any surface harmonic of degree wave motions

which has factors of the form e*et

factor

justify the

the

;

The

md>. r

.

sin

form Pnm (cos 0) and

)

the factor which involves r

is

n,

we

which involves 6 must then be of annihilated by the operator

*("*) -<•+»+*'. so that

be analytic at the origin

this factor is to

if

it

must be a multiple

Theorems derived from solutions of

4*82.

the equations

of

Jn +± {hr))^,:

of Mathematical

Physics. It is possible to prove (or, at any rate, to render probable) theorems concerning Bessel functions by a comparison of various solutions of Laplace's

equation or of the equation of wave motions.

Thus,

if

we take the

function

e^t/o [k>s/(p*

by making a change of origin

+a 2

lap cos

)},

to the point (a, 0, 0),

we

of Laplace's equation in cylindrical-polar coordinates.

a factor and expect

it to

it is

analytic at all points of space.

(kp)

+ 22 (A m cos m + Bm sin m) Jm (kp) = Wl

Assuming the

We

a,

l

.

J

we observe that the function under an even function of 0, and so Bm = and, from the symmetry ^4 m is of the form c m Jm (ka), where cm is independent of p and a.

consideration

and

It is therefore natural to

be expansible in the form

AJ

in p

see that it is a solution This solution has e** as

possibility of this expansion,

is

;

thus get Jo {kx'Xp*

+a — 2

lap cos

<£)}

= 2 em cm Jm (kp) Jm (ka) cos m. m=0

If

we expand both

sides in powers of p,

coefficients of (k-pa cos

<£)

m we

* Cf.

,

a and cos

get

Bryan, Nature, lux. (1909),

p.

309.

,

and compare the


128

and

so

[CHAP. IV

are led to the expansion*

we

{W(ps + a*-2ap

Jo

= 1

e

m=

m Jm (kp) Jm (ka) cos

if

we take e ik{ct+z \ which is a solution

of the equation of wave motions,

moving in the direction of the with frequency kc and wave-length 27r/k, we expect

and which represents a wave

+ to

oc to

x

-

axis of z from this expression

be expansible + in the form h

(^) \K)J where cn

is

a constant

;

e

2

iket

c n i*

„=o

=

*

(If)*

we compare the

so cn

=

n

+

\

;

CninJn+i {kr)

coefficients of (kr cos 6)

n \-^ and

JnH (kr)P n (cos 0),

so that

eikrco *°

If

m,

in § 112.

more formal proof will be given

of which a

Again,

cos

we

1T)

n

Pn (cos

on each

side,

e)

-

we

find that

2»+*r(n + f)"2».(n!r

are thus led to the expansion}: i

e

of

ikrc se

= fiE) i

(n

\krj „=o v

which a more formal proof

will

+ l)i»Jn+ i(kr)P n (cos0),

be given in

§ 11 5.

Solutions of the wave equation in space of p dimensions.

4*83.

analysis just explained has been extended by

The

Hobson§

to the case of

the equation

dx?

'"

dx£

'

dxp*

c2 dt-

A

normal solution of this equation of frequency kc which function of r and t only, where r

=

V(#i

2

+

a^*

+

. . .

+ xp

is

expressible as a

-),

must be annihilated by the operator a?2

and

r

9r

+

^

ikct so such a solution, containing a time-factor e

e *

+

This

is

due

to

iket

<& ilp -*

Neumann, Theorie

,

must be of the form

(kr)l{kr)*^K

der Bessel'schen Functionen (Leipzig, 1867), pp. 59

—

65.

harmonics do not occur because the function is symmetrical about the axis of X This expansion is due to Bauer, Journal filr Math. lvi. (1859), pp. 104, 106. § Proc. London Math. Soc. xxv. (1894), pp. 49—75. t

The

tesseral

z.


4-83]

Hobson of rank p

;

129

-

f

describes the quotient ^\{ P-i)(kr)l(kr)^fh %) as a cylinder function such a function may be written in the form

<$(kr\p).

By

using this notation combined with the concept of jp-dimensional space, in proving a number of theorems for cylinder functions of

Hobson succeeded integral order

and of order equal to half an odd integer simultaneously.

As an example

of such theorems

we

J {k v^r + o 2

2

shall consider

2ar cos

convenient to regard

for

p),

) \

where

an expansion

as being connected with

xp by the equation xp — r cos This function multiplied by e**ct is a solution of the wave equation, and when we write p = r sin it is expressible as a function of p, t and of it is

.

,

,

no other coordinates.

Hence «** J is

+ a? -

V(r2

{k

2ar cos

)

j

p)


dp2 that

to say,

is

p

'

dx£

dp

by the operator

& ,

J>~1

9

r

dr

dr*

(ff-2)cos4> d

.

r*sin

1

y

r*d
d

Now normal functions which are annihilated by this operator are of the form "

where

Pn (cos

j

p)

is

the coefficient* of on in the expansion of

- 2a cos

(1

By

the reasoning used in § 4*82,

J \k V^r + 2

a*

—

2ar cos

)

~ (/fca^-^/fcr)^-

Now

So

infer that

p)

1

^

AnJn+ip~ ( kr ) ^h-Jp-i (ha) Pn (cos 1

n

A n = 2**-

!

1

r (n + |p) (»

An

= {2»

+fc>-»

+ %p - 1) T (|p -

T (n + ip»« 1

*

).

that, in Gegenbauer's notation,

Pn (cos

j

p) S3 C*

p_ * (cos

0).

\p).

coefficients of

;

2"

*

we

+ cpy-*P.

expand all the Bessel functions and equate the n on each side we find that

)

2*"+ iP-»

so that

\

00

1

(k?ar cos

pn (cos
(hr)ir->

2"T(n + ^-l) nlV($p-l)

'


130

We

thus obtain the expansion

(r

An

+ a -2arcos_ *

{fcy'( r2

Jfr-,

2

+a

2

i!

analytical proof of this expansion, which holds for Bessel functions of

orders (though the proof given here

all

[CHAP. IV

is

valid only

when p

is

an

integer), will

be given in § 114.

Batemav's solutions of the generalised equation of wave motions.

4*84.

Two

systems of normal solutions of the equation

'

*

+

dx1i

+

dxf

2 9a;,

+

c2 dt*

dx4*

have been investigated by Bateman*, who also established a connexion between the two systems. If

we

take

new

variables p, a, %,

yjr

defined by the equations

xx — p cos 2£,

x3 — a cos -^r,

x2 = p sin %,

Xi

— a sin ^r,

the equation transforms into

'

*

A

{dp 2

+

2

p 3x

+

2 )

[do*

+

'

d*

o*

3> is

c* dt*

dyfr»\

normal solution of this equation with frequency kc

J„ where

+

p dp

is

(kp cos 0>) «/„ (&) a* (Mc+^+fc*),

any constant.

Further,

if

we

write

p=*r cos so that(r, ^,

-^r,

<£)

0,

= r sin <£,

form a system of polar coordinates, equation (2) transforms

into

O

&V SdV r dr

dr*

+

l&V r* d
+

c ot-tfm

tV dtp

r* 1 2

r» cos <£

Now

&V + 9x

2

1

3»

dr3

3

F

ld*V '

r2 sin2 4*

normal solutions of this equation which have

are annihilated

92

W*

e* (MX +"*

+ * c<)

c2

^'

as a factor

by the operator d_

r dr

* Messenger, xzxiu. (1904), pp.

182—188; Proc. London Math.

Soc. (2) hi. (1905), pp.

111—123.

,


4-84]

131

and since such solutions are expressible as the product of a function of r and a function of they must be annihilated by each of the operators

d

y

3 9

4X(X+1)

dr2

rdr

r*

'

+ 4 \(\+l)\2 + (cot-t*n)^r ^

where X

*

** cos2 ^>

l

9<£

3^>

sin 2

'

a constant whose value depends on the particular solution under

is

The normal

consideration.

solutions so obtained are

now

easily verified to

be

of the form

J^ +1 (kr) cos»$ sin"
(kr)-1

x

2

^- + \ +

F (*— - X, X

1

„+ l

;

;

sin 2

+

\

«*o»x+*++**>.

It is therefore suggested that

J^ (kr cos is

cos

)

Jv (kr sin

sin

)


2 a A (kr)~

l

J^ (kr)

cos*<£ sin"<£

.

(~^ - \ ^^ + X +

2JP,

1

;

v

+

I

;

sin 2 )

where the summation extends over various values of X, and the coefficients a* depend on X and , but not on r or . By symmetry it is clear that aA where

= 6 X cos* sin" ® 2 F .

b K is

independent of

t

(^-J^ -

X,

^£^ + X + 1

;

v

+

1;

sin 2 \

,

<£.

It is not difficult to see that

X = Ha* +

")

+ ».

(n

= 0,

1, 2, ...)

and Bateman has proved that

We

shall not give

Bateman's

proof,

which

differential equations,* but later (§ 11*6)

J„ (kr cos

cos 4>)

Jv (kr sin

sin

)

by a

we

is

based on the theory of linear

shall establish the

direct transformation.

expansion of

CHAPTER V MISCELLANEOUS PROPERTIES OF BESSEL FUNCTIONS Indefinite integrals containing a single Bessel function.

5*1.

In this chapter we shall discuss some properties of Bessel functions which have not found a place in the two preceding chapters, and which have but

one feature in common, namely that they are

obtainable by processes of a

all

definitely elementary character.

We

some

shall first evaluate

The recurrence formulae (1)

J

To

§ 3.9 (5)

V>%

[" 2-" +1

(2)

indefinite integrals.

(*)

and (6) at once lead to the

dz

#„ (z) dz

results

= *H-» #v+i (z)> = -z-"+> r(? v_

(z).

y

generalise these formulae, consider

^z^f{z)%%(z)dz; be equal to

let this integral

z"+>

where

A

The

and

(z)

B {z)

{A

(z) <$ v (z)

+ B (z)

are to be determined.

result of differentiation is that

z"+>f(z)

V, (z) =

<&v+1 (z)},

jH-i {

A

>

(z ) <® v (z)

+ A (z)

^

2l

I

+ z^ In order that take

A (z) =

B'

(3)

it

$

(

(z)-A(z)<@ v+1

v

[B' (z) <@ v+1

(z)\ )

(jr)

+

B (z) <$

v

<*)}.

A (z)

(z),

and B (z) may not depend on the cylinder function, we and then /(s) .5 A' (z) +

Hence

z

^

2

z

A(z)

+ B (z).

follows that

** +1

\b"(z)

+

^~

B' (z)

+

J"

B (*)| #„ (*) dz

= *+»

{B' (z) <@ v (z)

+ B (z) <@ v+1 (*)}.

This result was obtained by Sonine, Math. Ann. xvi. (1880), p. 30, though an equivalent formula (with a different notation) had been obtained previously by Lommel, Studien fiber die BesseVschen Functionen (Leipzig, 1868), p. 70. Some developments of formula (3) are due to Nielsen, Nyt Tidsshrift, ix. (1898), pp. 73—83 and Ann. di Mat. (3) vi. (1901), pp.

43—46. For some associated

integrals

which involve the functions ber and

—340.

Quarterly Journal, xlii. (1911), pp. 338

bei, see

Whitehead,

MISCELLANEOUS THEOREMS

5-1, 5-11]

The

following reduction formula, which

133

an obvious consequence of

is

(3),

should be noted: (4)

I

V+

ir

^„ 0) ds

= - (ft - v*)f'* - &r (z) dz li

L

2

+ |> +1 ^v+1 (z) + (/*-») **#„ (*)].

5*11.

The

Lommel's integrals containing two cylinder functions.

two Bessel functions are those (2), namely

simplest integrals which contain

derived from the Wronskian formula of

Jr

/

\

(z)

v

T»

/

\

J'_„ (z)

§

312

r / \ T - Jv (z)Jv

'

/

\

(z)

I/7T 2 sin-—

=

,

which gives K

,

) similarly,

from

7T

zJv (z) J_„ <>) ~

v

'

(z) «/_,, (.g)

.

2 sin

10g

J

*tt

v

'

(z)

§ 3*63 (1), z

,ox

dz

[

^ K

J

2sini/7r

dt

/"*

v

9 Kl)

and

zJ*(z)

J

['

}

J

_ir

dx _ zYv\z)~

Fy (>)

it

Jv (z)

2

Y

v

(z)'

The reader should have no difficulty in evaluating the similar integrals which contain any two cylinder functions of the same order in the denominator. The formulae actually given are due to Lommel, Math. Ann. iv. (1871), pp. 103 116. The reader should compare (3) with the result due to Euler which was quoted in § 1*2.

—

Some more

interesting results, also

due to Lommel*, are obtained from

generalisations of Bessel's equation. It is at

once verified by differentiation that, d*y

then

n

«

d?n

~

if

y and y rt

j\P -Q)ynd^y^-v^. *

Math. Ann.

xiv. (1879), pp.

520—536.

satisfy the equations


134

Now

where

apply this result to any two equations of the type of

denote any two cylinder functions of orders

*^V> î,

and

(z)

(z)

and Z

[

(7)

is

(z) to

yjr

(2) are arbitrary

i/r

This formula

[CHAP.

functions of

fi

and v

{{^-P)z-^^\^{kz)^

v

(17).

If

we have

Iz.

As a

special case, take

It is then found that

^

(lz)dz

= ,fa(kz) d^-^(lz) dz

431

z.

too general to be of practical use.

be multiples of z, say kz and

§

respectively,

V

d

dz

= z {k^ +1 (kz) W. (Iz) - 1&* (kz) # r+1 (lz)} -(/*-») ^M (kz) <@ v (Iz). The expression on the left simplifies

still

further in two special cases (i)fi—v,

(ii)k=*l.

If

we take

(8)/

,

%

fi

=

(*.)

This formula

v, it is

found that

i. { u) d* =

may be

'<«ww»>yw»w (Wi

verified

when k =

It becomes nugatory numerator is a constant.

If this constant is omitted,

when

by differentiating the expression on the I,

for

the denominator

an application of

is

right.

then zero, while the

l'Hospital's rule

shews that,

I -*- k,

(9)

The

z<@» (kz)

J"

§V (kz) dz^-^

#/ (kz)

- ks&r (kz) W^(kz) - ^V (kz) ^ M+1 (kz)}.

result of using recurrence formulae to

right of (9)

remove the derivates on the

is Z

(10)

{kz <^M+1 (kz)

l z<&t (key®* (kz) dz

= \z* {2^ (kz) &» (kz) - ^_

1

(A*)

#M+1 (kz)


5*11]

135

Special cases of these formulae are

fs <$,? (kz) dz = %z* {%* (kz) -

(11)

^

#„+,

(for)

- \* {(l " ^) *V <*«) + ^m' [*?<&,, (kz)

(12)

§LM (kz) dz = ±*»

+ function of order

To obtain a is

—

t 'rt

e~'

2

<**)}

^

{2^M (kz) iLM (fa) +

the latter equation being obtained by regarding

(**)}

^

+1

,

(fa)

f _M _x (fa)

(fa)iU +1 (fa)},

^ M (kz) ,

as a cylinder

/*.

different class of elementary integrals take

k=

I

in (7)

and

it

found that

(

/y <*.)*.

i3)

(*•)

£ - - fat^w^y^w^wt ff,(fa)ff,(fa) ,

The

result of

making

j/

-*- /*

| V, (fa) tf,(fa)

(14)

in this formula is

^ = g j^ -

The

^

-^-

8 +1 (fa)

(*«)

}

(fa)| 3^ M+1 g-

—

last equation is also readily obtainable

|

^^ = 2f^

—

\
——

Z

OI&

^(fa)^M (fa) 2-

.

by multiplying the equations

V^(,)«0, V^ d by -

+

li

(z)

1

,

-^^(z) z

respectively, subtracting

and integrating, and then

re-

placing z by kz.

As a

special case

(is)

J

An

we have

w (fa)gKfa)-^(fa)gu

jj(kz)^ -g{/

alternative

method of obtaining

this result will

+1

(fa))+~j

-(fa)L

|l

be given immediately.

Results equivalent to (11) are as old as Fourier's treatise, La Theorie Analytique de la Chaleur (Paris, 1822), §§ 318 319, in the case of functions of order zero but none of the other formulae of this section seem to have been discovered before the publication of

—

;

Lommel's memoir. Various special cases of the formulae have been worked out in detail by Marcolongo, Napoli Rendiconti, (2) in. (1889), pp. 91—99 and by Chessin, Trans. Acad. Sci. of St Louis, xii. (1902), pp. 99—108.


136 5"12.

[CHAP.

integrals containing two cylinder functions;

Indefinite

V

Lommel's

second method.

An

alternative

method has been given by Lommel* for evaluating some By this method their values are obtained in a

of the integrals just discussed.

form more suitable for numerical computation.

The method

£

[z»

adding the two results

consists in

V, (z) K. (z)} = - z>

{<&> (z)

M

+9

tf r+1 (z)

+ (p +

^

{zo S? M+1 (z)

& v+

>

CO}

=*

[V, (z)

(z)

^

v

(,)}

+ V) Z>~ #„ {Z) V$ v (Z), 1

/M

W v+X (z) + ^M+1 (z) £„ (z)}

+ (p- H,-p-2)z'>-> <@»+1 (z) W v+1 (z), so that (p

+ ft + v) j'z'-

1

^

(z) <@ v

(z)dz

+ (p-p-v- 2)

= Z> and then giving

{<&»

*

J

(jf)

z"- 1 <@r+l (z)

%

v (jr)

W

„ +1

+ ^M+1 (0

(z)

^

dz

+1 (,)},

special values to p.

Thus we have e

(1)

J

(2) J

As

z-»->-^+1 (z) & p+1 (z) dz

^

2(^ + y +l) l

^

(z)

*'

{z)

+

^

+l ( *>

~7^

i^(f)»,(f) +

— ^g

{», (*) + *V«

^(«)?r Wdf- f

^

<*>!»

^(f)?rH (i)i,

)

special cases of these

jV*- 9 V+1 <#) df 1

(3)

rz

(4)

/

Again,

if

^

^

+1 r

(*)
p be made zero,

=

it is

^_ W

<*)}.

»v+»

(*)

+ *V+i (*)}•

found that

- ^M (*) ^, (*) + ^M+l (*) »H1 (*)> so that,

by summing formulae of this rz

(5)

0*

+ „)J

^

type,

we get

J~ (,) V„ (,)

£-

rz (

M + „ + 2n) J

- ^M («) ^r (#) + 2 2 V,^ («) * Math. Ann. xiv. (1879), pp.

A*

^M+n (*) *f,+» (*) ~

^+

,»

(#)

530—536.

+ r^+» (*) ^*+» (*)•


5'12-5*14]

In particular,

if fi

137

= v = 0,

/'*.(#)*.(*)*

(6)

=

where n

1, 2, 3, ....

But there seems

to

be no simple formula

for

%(z)<& (z)^,

/<

For a I.

special case of (1) see Rayleigh, Phil.

Mag.

[Scientific Papers,

(5) xi. (1881), p. 217.

(1899), p. 516.]

5*13.

Sonine's integrals containing two cylinder functions.

The

analysis of § 5-1 has been extended to the discussion of conditions that

may be expressible

in the

A (*)

by Sonine, Math. Ann.

xvi. (1880), pp.

—33,

30

form M {0 <»)}W W {+ (»)>+* <*)

^M+1 ft

(*)}

f\,

{*(*)}

+^(*)^{*(*)}«fr+1 {^(*)}+^(*)»IH4 {*(*)>^ {^W}. 1

but the results are too complicated and not

sufficiently

important to justify their insertion

here.

5*14.

Schafheitlin'8 reduction formula.

A reduction

formula for \' *<&*(£) dz,

which

is

a natural extension of the formula

Schafheitlin* and applied by

him

§ 5*1 (4),

has been discovered by

to discuss the rate of

change of the zeros of

<&9 {z) as v varies (§ 156).

To obtain the formula we observe that

= [- ***<&. (z) <@: (*>] + Now, by a (fi

* 1

{*»+»#/» (*)+(,*

+

1

>

**+»

%%

(z) 'w:

(*» dz.

partial integration,

+ 3) f *#*•<&* (z) dz « |>+» #/» (*)]

+2

J

V» <@: (z) {*&:

(z)

* Berliner Sitzungsherichte, v. (1906), p. 88.

+ (*» - *»>

k

(^)j cte,


138

[CHAP.

V

and so

+

(fi

V+

1)

2

= |>+»

dz

<&,'* (z)

J

W

+

(z)]

2

1

J'**

(z*

-

2 i/

)

#„ (z) <&,' (z) dz.

Hence, on substitution, (/*

+

1) [

V {? -

= |>+3 ^V +

= |>+

3

2

'

2

v

2

#„» (z) dz

)

- (ji +

(*)

2 [

^

1/

V+ (*)

3

1)

^+

2

^

(*)

<@v (z) <@: (z) dz

- (/*+

1)

^+

2

#/ (*)]

+

^„(*)

{(/a

+

l)2

M j"V+»#

-

r

(*)

#/ (2) + z* +3 €* (z) + {£(/* + 1)2 -"'l* * 4

1

^V(s) ek

^ ^)] 3

Z

z»+*'@vi (z)-(ti

-(p+3)j By (fi

rearranging

+ 2)

**• #,»

1) {£

(/*

+ 1) 2

v*\

J'rW(t)dz.

find that

= (n+ 1)

(5) (2c

J*

+ * |>+' and

we

+

{*#.' (*)

(i/

2

-\

(jjl

+

1 )2 } [

V

r

#„2 (*)

<2e

- * (^ + 1 ) #„ (*)} + *»+» <* - v + J (a* + 2

2

2

^„8 (*)],

1 )»}

this is the reduction formula in question.

Expansions in

5*2.

We

shall

now

result of §

To

27

The general theory

of such expansions

is

reserved

at once suggests the possibility of the expansion

to Y .l<£±**p!±$ j^.1

(1)

which

2*7.


some of the simplest expansions of the type ob-

discuss

tained for (\z)m in § for Chapter xvi.

The

series

is

due to Gegenbauer* and

is

valid

when

ji is


establish the expansion, observe that

ni

n=»

is a series of analytic functions which converges uniformly throughout any bounded domain of the 2-plane (cf. § 3'13); and since

v-m

-^ {(\z)-» J»+ M (z)} it is

= ^7^ {nJ^+m-i (z) - (A* + ") ^m+2h+i (*)},

evident that the derivate of the series

„

n

J

s r(u + w)

Lm-o *

n

-

T

,.

now under

»r(At+n + »-o

Wiener Sitzungsberichte, lxxiv.

(2),

w

consideration

i)

r

(1877), pp.

1 J

-

124—130.

is


5*2, 5*21]

and so is

the' sum is

unity; that

is

When we make z--0, we see that

result

established.

is

reader will find that

it is

not

a crude method of proving the

5*21.

that when the expansion on the the coefficients except that of z* vanish ; but

difficult to verify

right in (1) is rearranged in powers of this is

the constant

to say

and the required The

a constant.

139

z,

all

result.

The expansion of a Bessel function as a

series


The expansion (i)

($z)-'J¥ (z)

= az)-»r( v + v

Qi+2n)r(g+n)

5 .to

is

is

i-fi)

nir(F + l-/*-»)r(ir + n +

l)

T jM+2n

W .

(

a generalisation of a formula proved by Sonine* when the difference v — ft are not negative a positive integer ; it is valid when fi, v and v — /j.

integers.

It is

most

easily obtained

by expanding each power of z

in the expansion

of i\zY~"J,(z) with the aid of §5'2, and rearranging the resulting double series, which is easily seen to be absolutely convergent. It is thus found that

(*zY

*' »

\z )

~

*»

i

n/

—

TT\

(-)» m!r(i, +m+ ~i: o

_ 2

*

(fj,

+ 2m + 2p)r(n + 2m+p)

?

l) pto

«/*-h*W

(~) m v Qt + 2n)r(/i-hm + w ) J, (^^T)! ",tom!r(ir + m + l).t.

«W

_ v - I

v ".?.

i

2

(-)Tfr + m + n) V(p + n)r(v+l-fi)

which

we put

is

,Q\r

/\

+ 2 ")^™«' »!I> + l-„-,0I> + n+l) ("

v —fj.

,

•

by Vandermonde's theorem If

|

+ m, we

;

and the

result is established.

find that

Sonine's form of the result, and

is

readily proved

* Math. Ann. xvi. (1880), p. 22.

by induction.


140

By

a slight modification of the analysis, we

[CHAP. V

may prove

that, if

+ 1;

+ 2n) J^+îz).

k

is

any

constant, (8)

(**)-*(*)-*.?.,,#£& x 2F,(/x

+ n, -n;

y

h?) (fi

This formula will be required in establishing some more general expansions in § 11-6.

5*22.

+ h)~^ v Jv {\/(z + h)},

It is evident that (z

for all values of the variable,

with

+ h)^ v Jv [\/{z + h)}.

Lommel's expansions of(z

§ 3-21 (6) ;

qua function of z + h, is analytic and consequently, by Taylor's theorem combined

we have ao

(1)

(,

+ h) -4.Jp y (z +h)] . J

Again, (z

+ h) "Jv {
provided that h |

(2)

(,

j

<

I

+hpj

9

z

is

km

*

tJm

*

{g

-* Jw Wz)]

analytic except

when

z+h = 0;

and

so,

we have

, \

W(z + h)}=Z

m-0

^ !*- [z»J

v

(V*)}

"*J ***

These formulae are due to Lommel*. If we take v — — \ in (1) and = ^ in we deduce from § 3*4, after making some slight changes in notation, i/

(2)

£) COsV(^-2^)= 2 ~,Jm- h {z),

equation (4) being true only when J*|
Proofs of (3) and (4) by direct expansion of the right-hand sides have been given by Glaisher the algebra involved in investigations of this nature ;

is

somewhat formidable.

—

* Studien iiber die Bessel' schen Functionen (Leipzig, 1868), pp. 11 16. Formula (1) was given by Bessel, Berliner Abh. 1824 [1826], p. 35, for the Bessel coefficients. t Quarterly Journal, xn. (1873), p. 136 ; British Association Report, 1878, pp. 469—470. Phil. Tram, of the Royal Soc. clxxii. (1881), pp. 774 781, 813.

—


5-22]

We

shall

now enumerate

various modifications of (1) and (2).

In (1) replace z and h by

and kz z and then

z*

,

jv {W(i + *)} = a +

(5)

141

(-) w (i^) ?

wi

V

m

m=0

y+w (^)

>

-

and, in particular,

If

we

+ &)*"

divide (5) by (1

l

+

(i/

In like manner, from

and then make k -* —

1)

we

1,

find that

m!

TO=0

(2),

Jv [z
(8)

provided that k |

If

- 2* 5 fcJîf^J^ (*).

/„ (z V2)

(6)

we make k

|

<

1.

-*-

-

1

+ 0, we

find,

[(1+*)KMW0 *-»•- 1+0 lim

by Abel's theorem, (

+*)}]« i

~ )W(

)W>

w^

m=0

,/y _

w (^),

-

provided that the series on the right

when

v

is

an integer. If v

^

./„_

is convergent. The convergence is obvious not an integer, then, for large values of to,

is

—

m (*)

sm (to - *) 7t

= _( 4f^sin^ +1 7TTO*

Hence the condition satisfied,

when

v

for

the convergence

is

any

l

convergence

is

absolute.

I

{

{1

+

(1/w)}

'

R (v) > 0,

is

Consequently,

and

when

the condition

if

R (v) > 0,

and

is

also

integer,

(9)

m=« In like manner,

if

R(v) >

- )m{h * )m ml

— 1,

and

Jv- m (z) = Q.

also

^

Jv (zJ2) = 2-i» X

(10)

when

m-0

W!r

v is

any

integer,

we have

Jv -m (z).

It should be observed that functions of the second kind for functions of the first

kind in

(1), (2), (5)

and

may be substituted provided that h < z (8) j

\

\

\

and k < \

J

(11)

(i2)

1

;

so that

(*

+ A)-*'

(*

K,

M* + h)} -

2 N-L*l

2

-^m Yv+m (^z),

+ &)»» rr {v(r+ A)} = i <**>%<-«> r^^/*), m=0

da)

r„

{*

vd + *>] = (i +

(14)

Y,

\z

V(l

Wi!

w

§

<-rl^>

+ k)} = (1 + jfc)-* 5 m=0

&*£ F„_ TO!

Yv+m (zl

lrt

(*).


142

[CHAP.

V

These may be proved by expressing the functions of the second kind as a linear combination of functions of the first kind by proceeding to the limit when v tends to an integral value, we see that they hold for functions of ;

integral order.

By combining the first kind, we

—(14) with the corresponding

(11)

we may

see that

results for functions of

substitute the symbol

^ for the

symbol

Y throughout. formulae were noted by Lommel, Studien, p. 87. Numerous generalisations be given in Chapter xi. It has been observed by Airey, Phil. Mag. (6) xxxvi. 242, that they are of some use in calculations connected with zeros of (1918), pp. 234

These

of

them

last

will

—

Bessel functions.

When we combine

and

(5)

(13),

and then replace V(l

when |\«-1|<1,

that,

(

9. (\m) = X* 2

(15)

~ )m(V

1)W(

", m.

m-0

and, in particular,

when X

is

=

v

/.(x.)

d«)

^

+ k) by

we

X,

find

^h™ «,

unrestricted,

H-fr'-^fr^ w

I

These two results are frequently described* as multiplication theorems

for

Bessel functions. observed that the result of treating (14) in the same way as taken equal to an integer n)

may be

It

(when

v is

-(n-l)l

(17) x

(2/ 2 )»=tt 2 ra=0

'

(8) is

that

i-Jfm Yn . m {z). •

alternative proof of the multiplication formula has bee.n given by Bohmer, Berliner Sitzungsberichte, xiii. (1913), p. 35, with the aid of the methods of complex integration

An

see also Nielsen, Math.

Ann. lix. (1904),

p.

and

108,

(for

numerous extensions of the

formulae) Wagner, Bern MittheHungen, 1895, pp. 115—119; 1896, pp. 53—60. special case of formula (1), namely that in which v = l, was discovered by [Note.

A

Lommel seven

years before the publications of his treatise

;

see Archiv der Math,

xxxvn.

(1861), p. 356.

His method consisted in taking the integral

—

/

over the area of the circle £* + $»=

The

co« (£r cos

/

1,

and evaluating

result of integrating with respect to

fir — 2ttJ_i|_ i

j

(£r cos sin vs

„

M + r)r sin 0) .

_i

\

TO =o

by two

different methods.

d£ a rsin.0 .

^ ° OS 6) 8i " {V(1 ~^' r SiD &) ^~6

(-rtr«nw* n

-

m=o (

(

it

is

-

_W0-f)

^ * See, e.g. Schafheitlin,

t)

-\JQ-P>

" \ /_, C0S *r

+ ijr sin 6) d£dt)

(2»i+l)!

COfi{$rcoa6h{1

J -i

- )m ($r sin 6?m Jm + i(r cos 0) (rcos0) m + 1 ml

^

)m , id$

'

Die Theorie der Besselscken Funktionen (Leipzig, 1908),

p. 88.


5*23, 5*3]

and the

result of changing to polar coordinates (p,

1

5~

fn

f

/

I

—v J

4bW J

1

= If

we compare

r2 cos2

=

cos {rp cos (0 — $)} pdpdUp

o

and r2

^ ff

these equations

we

cos tfr)

— 1

*W

)

/"" /

/

is

f

1

cos (rp cos ^) pdpd

I

— ir

J

^d^^-jjl- ?)* cos (£r) <*£=./, (r)/r.

obtain (1) in the case

i/

=l

with

z

and h replaced by

sin 2 *.]

The expansion of a Bessel function as a

5*23.

143

series


From formula

§ 5*22 (7), Lommel has deduced an interesting series of Bessel functions which represents any given Bessel function.

If

and v are unequal, and

ft

The repeated arrange

it

fi is

not a negative integer,

j,

m — n, and

(v

ft)

=

(v

This formula was given by Lommel, Studien

22—23, in the special case M =0; by putting *=0, it is found that

*, Fo (,)- Ji

(.)

log <*)

-£te±l>

— n)l

J

o

I

(v

o^ m!

1)

1868), pp.

re

then we have

| n.-^^ w = r^+1) — M r + m+ 1

(2)

we may

series is absolutely convergent; consequently

by replacing p by

m=o U-ow! ym and hence, by Vandermonde's theorem,

a)

we have

+n+

l)j

* +TO

'


v

Functionen (Leipzig,

differentiating with respect to v

and then

C^> (^~^M+m W ,

,

and,

*{V' (»l +i)+V''(-/*)-'
,

when /i=0, we have Lommel's formula

This should be compared with Neumann's expansion given in § 3'57.

An

53.

An

addition formula for Bessel functions.

extension of the formula of § 2*4 to Bessel functions of any order

(1)

./r

where z |

|

<

j

£

{,

v

(*

+ <)- 2 JîftJ^z),

being unrestricted.

This formula

is

due

to Schlafli*

the similar but more general formula

%\{z + t)=

(2) is

due

2 »»= — 06

^(0/.(i)

to Soninef. *

is

MatA. Ann. hi. (1871), pp. 135—137.

( JWd.

xvi. (1880), pp.

7—8.

;

and


144

[CHAP.

It will first be shewn that the series on the right of (1) convergent series of analytic functions of both z and t when

r,

A

R,

numbers

are unequal positive

in ascending order of magnitude.

m is large and positive, J ^m (t)Jm {z) is comparable m-V sin vtt {\Ry {r/Ry* V(

When

a uniformly

R^\t\Â,

|*|
where

is

V

v

with

)

.

and the convergence of the series for (1

term

is

— r/R)".

.

comparable with that of the binomial and negative (=— w), the general

series is

When m

large

is

comparable with

(-)"(*A)'(jAr)"

T(v + n+l).n\ and the uniformity of the convergence

follows for both sets of values of

m by

the test of Weierstrass.

Term-by- term differentiation

is

consequently permissible*, so that

J*-m {t) J m {z)]

'

- (t) Jm {z) == J "m {t)Jm {z) Qt " l~z)ji/v m m ?-J ,

'

,

*

i

=H 2

{

" m= — ao " 1

J,-™-! (0

- Jv- m+1 (t)} Jm (Z)

-^* 2 J -m(t){Jm- (z)-Jm+1 (z)}, v

m= -oo

and

1

seen, on rearrangement, that all the terms

it is

on the right cancel, so

that

d-a)J./~< >'-<'>- a ,

30

Hence, when z |

of z

and

t

which

\

<

\t \

1),

t

m=

is

an analytic function

-oo

as a function of z

are identically equal.

+t

only, since its derivates

If this function be called

then

F(z + t) =

If

2 Jv -m (t) Jm (z)

the series

is expressible

with respect to z and

F {z +

,

we put

z

= 0,

I

Jy-nit)

Jm (z).

m= — oo we see that F(t) = Jv (t), and the truth of (1) becomes

evident.

Again,

if

the signs of v and J-. v (z

m in

(3)

this result

is

X

(1),

we

see that

Y _m (t)Jm (z). v

1»=-00 * Cf.

(z),

-oo

combined with Y,(z + t)=

we have

(-)m J- v+m (t)Jm

+ t)= X «l =

and when

(1) be changed,

Modern Analysis,

§ 5*3.


5*4]

When

this is

The reader

combined with

(1),

will readily prove

145

equation (2) becomes evident.

by the same method

that,

(4)

Jv (t - z) » I Jv+m (t) Jm (*),

(5)

^„ (*-*)= I Wr+miftJmiz),

(6)

T„

when

j

z

\

<

t

!

|,

«= — oo

Of these

results, (3)

(Leipzig, 1868),

when

Yv+m (t)Jm (z).

(<-*)= X

was given by Lommel, Studien iiber die BesseVschen Functionen an integer while (4), (5) and (6) were given* explicitly by Graf,

v is

;

Math. Ann. xliii. (1893), pp. 141 given in Chapter xi.

— 142.

Various generalisations of these formulae

will

be

Products of Bessel functions.

5*4.

J„. (z) Jv (z) has been given by various sometimes stated to be due to Schcuholzerf, who in 1877, but it had, in fact, been previously published (in 1870) More recently the product has been examined by Orr§, while

The ascending series for the product writers

;

the expansion

published

by

it

SchlafliJ.

is

Nicholson has given expansions ||

(cf. §

5'42) for products of the forms

J„(z)Yn (z) and

Ym (z)Yn (z).

In the present section we shall construct the differential equation

We

by the product of two Bessel functions, and solve it (§ 541) obtain the expansion anew by direct multiplication of series. in series.

Given two

if

differential equations in their

satisfied

shall

then

normal forms

y denotes the product vw, we have y['

= if'w + 2v'w' + vw" = -(/ + J)y + 2v'w

/

where primes indicate

differentiations

,

with respect to

z.

* See also Epstein, Die vier Reehnungsoperationen mit BeueVsehen Functionen (Bern, 1894), [Jahrbuch tiber die ForUehritte der Math. 1893—1891, pp. 845—846]. t Ueber die Auswerthung bettimmttr Integrate mit Hillfe von Ver&ndetungen det Integrationsweget (Bern, 1877), p. 13. The authorities who attribute the expansion to SchSnholzer include Graf and

Oubler, Einleitung in die Theorie der BesseVschen Funktionen, u. (Bern, 1900), pp. 86—87, and Nielsen, Ann. Sci. de Vltcole norm. sup. (3) xvin. (1901), p. 50 ; Handbueh der Theorie der Cylinder/unktionen (Leipzig, 1904), p. 80. According to Nielsen, Nouv. Ann. de Math. (4) u. (1902),

obtained some series for products in the Iserlohn Programm, 1862. % Math. Ann. hi. (1871), pp. 141—142. A trivial defect in Schlafli's proof is that he uses a contour integral which (as he points out) converges only when B(/t+r + l)>0. p. 396, Meissel

§ Proe. ||

Camb. Phil.

Soc. x. (1900), pp.

93—100.

Quarterly Journal, xliii. (1912), pp. 78—100.

'


146

[CHAP. V

d ~ \y" + (J + J) y] = 2v'V + 2v'w"

It follows that

dz

= - 2Ivw' - 2Jv'w y'" + 2 (/ + /) y + (/' + J') y = (/ - J) (v'w - vw'). the special case when I—J,y satisfies the equation y + 4/y' + 2/>=0;

and hence Hence, in

/'/

(1)

but, if

I±J,

qr+w+wv+J-),

*

(2) This

easy to shew by differentiation that

it is

is

the form of the differential equation used by Orr Rendm, xci. (1880), pp. 211 214.

To apply these to a

normal form

z

9!> v

(z) as

is

simpler to

a

;

(1),

see

be reduced

both Orr and Nicholson effect the reduction by taking

new dependent variable, take a new independent Z, >

but, for purposes of solution in series,

variable

d

d

_

it

by writing

-^

dz~dO~*'

$%£$ + (*• - *») /„ (*) = 0.

so that

Hence the equation d_

dd

(3)

in connexion with

results to Bessel's equation, the equation has to

~e Z„-j>

is

;

^

—

Appell, Comptes

that

_ _ (/ _

}

satisfied

by J^ (z) Jv (z), when

ft*

^ v*,

0+2(2^-^-i, )^| + 4^y| + (^-^) 8

2

is

y = 0,

to say

[&-2(p* + v*)** + ( fi*-v>Y']y + 4e»(*r + l)(% + 2)y =

and the equation

satisfied

by

J

v

(z)

J±„ (z)

>

is

*(*»-4»»)y + 4«"(ar + 1)0 = 0.

(4)

Solutions in series of (3) are *»

where a — ±

±

/*

v

2

{-)m cm

z™

and

+ 2m — !)(« + 2m) ctn _i (a + ft + + 2m)(a + p -v + 2m)(a- p+ v + 2m)(a- fi-p + 2m) If we take a = /* + v and 4 (a

°m

1>

^-2^+"r(/*+i)r(i/ +

we

i)'

obtain the series «o

(_)w

Qgy,+p+m, Y(

fl

+ y+ 2m + 1)

w?o m!r(/* + v + m + l)r( A* + w+l)r(i/+w+l) and the other series which are solutions of (3) are obtained by changing the signs of either

/i

or v or both

/*

and

v.

"


5*41]

By

Jv (z)

it

are not negative integers and

if

considering the powers of z which occur in the product

is

easy to infer that,

fi*

^ »,

m w

147

if 2/*,

2v and 2

(/* + v)

«/M (z)

then (~)m

r t .\ T u\- v J|lW,/rW

(^Y +¥+im r (^ + » + 2m + 1)

"".r i»ir(fi + ir+m+i)r(/»+iii + i)r(»+«t+i)"

In like manner, by solving (4) in

series,

we

when

find that,

2i> is

not a

negative integer, then

W and,

W -wr /«!r(2

'

when

»>

+ m-Hl){r(j/ + m+l)}

2

'

not a negative integer, then

is

J„(s)J_„(s)=

(7)

v

(-)m (±zYm (2m)\

2

2

=0 (w!)

r (i/ + m + 1) T(- v + m +

1)

reasoning which resembles that given in § 4'42, it may be shewn that (6) holds when v is half of an odd negative integer, provided that the quotient T (2v + 2m +• l)/r {2v + «i + 1) is replaced by the product (2v + + l)m

By

m

.

Products of series representing Bessel functions.

5*41.

It is easy to obtain the results of § 5*4

by

direct multiplication of series.

This method has the advantage that special investigations, for the cases in which fi* = v* and those in which y. + v is a negative integer, are superfluous.

The

coefficient of (-)"*

(yY +v+sm

in the product of the two absolutely

convergent series

.

(_y ( fr)M— m =o»»! r((i + m+ 1) m

(-)w (**)"+w

g

n=nn\T(v + n + l)

1

is tt

^

= n!

-

Y(v

+n+ l).(m-n)! T(/i-¥m-n + l)

_l r (/* + w!ir/

m!

T(/*

2

w+ Ii\ x«,xn (v + m + l,)*=o tR m 1) Ir<\

+ m+

(n

l)

p (i> + w +

+v+m +

Cw

(-»'-»»)w_n .(-/A-m)

fl

l)

l)m

m!r(/t + w+l)r(i/+m + l)'

when Vandermonde's theorem Hence,

n\ (1) and

for all values of

/*

is

used to

and

g t /.-\ t¥ {z)-^ (.\ j^z)J

sum the

v,

(-)m

m]

(i^ +,,+aw (^ + + m+l)TO r{fJt + m + i )r ( v + m + iy i>

this formula comprises the formulae (5), (6) '

finite series.

I

\

and .

(7) of § 5*4. -----

H--'

'

*~ S


148

[CHAP.

V

This obvious mode of procedure does not seem to have been noticed by any of the was given by Nielsen, Math. Ann. lii. (1899), p. 228.

earlier writers; it

cos z and J {z)e,mz were obtained by Bessel, Berliner Abh. 1824, and the corresponding results for Jv (z) con z and Jv {z)smz were deduced from Poisson's integral by Lommel, Studien uber die BesseVschen Functionen (Leipzig, 1868), pp. 16 18. Some deductions concerning the functions ber and bei have been made by Whitehead, Quarterly Journal, xlii. (1911), p. 342.

The

series for

[1826], pp.

38

J

{z)

—39,

—

J (bz),

More generally, if we multiply the series for J^(az) and an expansion in which the coefficient of (— w orb" (fcY +v+am m

v

we

obtain

is

Qitn—2n A2n

T (y + n + 1) .(m - n)l T (ft + m-n + l) 2 m a _ a tB\(-m,-iA-m\ v + 1 6 /a ) ml r<> + m + l)r(i/ + l)

„,o»!

;

and so J, (az) Jv (bz) -

(2)

V2

'

r ^ ^J (~)w QaiY" . fi (- m, - /i - m x | t»! r(/i+TO+l) m=0

;

and

+1

i/

;

fr/q')

be simplified whenever the hypergeometric

this result can

series is

expressible in a compact form.

One another

case of reduction is

the case

is

b = ia,

= a, which that ft* = v*.

the case b

provided

has already been discussed

In this case we use the formula*

and then we see that

w

/o\

r/ v

w/ ;

\

v

'

wt=0

v

- %

r(|m

(-)m (âzy" +m cos fymr + i)r(i/ + |w + i)r(i/+in (-)m (\azf" +im

-

=0 w! T (v +

(5)

If (6)

we take a = el"* ber„» (z)

in (3)

+ bei„

a

(z)

m + 1) r

(i/

+ 2w +

we

find that

- 2

ptt

it***"*

an expansion of which the leading terms were given in § * Cf.

1)'

(-f (ifliycM(i» + ht)T

fart- v

/fa*)/

+

Kummer, Journal fUr Math.

xv. (1836), p. 78,

3*8.

formula

(53).

i)


5*42]

The formulae

(3), (4), (5)

—

149

were discovered by Nielsen, Attidella R. Accad. dei Lined,

(5)

xv. (1906), pp. 490 497 and Monatshefte fiir Math, und Phys. xix. (1908), pp. 164—170, from a consideration of the differential equation satisfied by Jv (az)J±v (bz).

Some types

have been given, Quarterly Journal, xli. (1910), p. 55, for products of the but they are too cumbrous to be of any importance.

series

Jy3 (z) and J¥% (z) t/_„ (z),

By

giving «•«-•

(7)

fi

the special values ± | in

easy to prove that

J^ (z sin 6) = iig (2 sin 0)-*^ r *2y + n)
The special case of by Hobson*.

which 2v

this formula in

is

6).

an integer has been given

Products involving Bessel functions of the second kind.

5*42.

The

(2), it is

series for the products

J {z)Yn {z), Jm (z)Yn (z), IL

and

Ym {z)Yn {z)

have been the subject of detailed study by Nicholson f; the following outline of his analysis with some modifications.

We

i/

an

have irJ, (z)

where

is

is

to be

Yn (z) = 1

made equal

to

{

J* (,) Jw («)}

- (-)» i {/, (z) J. ¥ (*)},

n after the differentiations have been performed.

Now l~

{

JM (,) J, (*)} - log {\z). J» (z) J¥ (z) v

I"

(~)r (^>t+,,+8r

T(p + v + 2r + l)

and

r (fi - v + 2r + 1) rO-i/ + r+l)r(/* + r+l)r(-i/ + r + l) fi/r (p - y + 2r -H 1) - ^ (/* - v + r + 1) - yfr (+ r + 1)} (-)" (^y4-"*"*

r=0 Lr:

x

We

1/

n-l

divide the last series into two parts,

oo

2 and X

.

In the former part we

have

* Proc. London Math. Soc. xxv. (1894), p. 06 8 je also Cailler, ilf£m. d« ia Soc. de Phys. de Geneve, xxxiv. (1902—1905), p. 316. t Quarterly Journal, xliii. (19i2), pp. 78 100. The expansion of Jq(z) Yq (z) had been given previously by Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), p. 21. ;

—


150

while in the latter part there r is replaced in this part

no undetermined form to be evaluated.

is

by n + r,

it is

n +* r

(A* + n + r + | (-Y {hzY+ r!(n+r)ir(^ + r+l) r -o

{2 log (£*)

When

+ r+l)

r!r
9

x

V

seen that

wJ,(b)Yu <*)—%

(1)

[CHAP.

+

2^r (p

l) r

+ n + 2r + 1)

-^r(^ + n + r+l)-^r(/* + r +

l)

- -^ (?i + r+l)-yjr(r + 1)}. The expression on the vn

by

= 0,

1, 2, ...,

m on The

right

and so the

the right in

irjm

(z)

Yn {z)

is

fi

at

/tt

=m

where

obtained by replacing

fi

(1).

Ym (z) Yn (z)

series for

a continuous function of

is

series for

can be calculated by constructing series

for

~

[ d-J± h (z)J±v (z)

dfldv

|_

jn-m, v-n

The details of the analysis, which is extremely laborious, have been given by Nicholson, and will not be repeated here.

in a similar manner.

Jv (z).

The integral for J^ (z)

5*43.

A generalisation of

Neumann's

integral (§ 2*6) for

Jn

2

(z) is obtainable

by

applying the formula*

eo**-+-

cos fr

Jo to the result of § 5

that

-

41 ;

v)

0d0 = 2iL+¥+M

+ m + 1)r(v + m + 1) provided the integral has this value when m — 0, 1, 2, ,

. .

.

,

R (jM + v) > - 1.

•It is

then evident that (

J„(*)J„(*) =

- 2 m -o.'o + *>) > — 1, 7r

so that

(

r(fJ

when

1

J?

(/u.

/M (s) J„ (z) = -

T

^s(fi-v)dd0, ,w .. + m ..n m!r(/*+v + l)

f

V,^, (2* cos 0) cos (fi-v)0d0\

TJfl

the change of the order of summation and integration presents no serious difficulty. * This formula is due to p. 263.

Cauchy

;

for

a proof by contour integration, see Modern Analysis,


5-43-5-51]

151

R — n) > — 1, then JM (z) Jn (z) = *i~L P'j^ (2z cos 6) cos + n) 0d0, T

If n be a positive integer and (2)

(jj,

(fi

Jo

and

this formula is also true if

/*

and n are both integers, but are otherwise

unrestricted.

Formula

(1)

was given by

Schlafli,

Math. Ann. in. (1871),

p. 142,

when n±v

are both

integers; the general formula is due to Gegenbauer, Wiener Sitzungsberichte, cxi. (2a), (1902), p. 567.

The expansion of (\zY+ v as a

5*5.

series

of products.

A natural generalisation of the formulae of Neumann (§2*7) and Gegenbauer (§

5 2)

is

that

-

(**r

(1)

r{/M

+ v+1 y m.

m-0

The formula

is

true if

proof applies only

From

§ 5*2

fi

R(fi

if

and

not negative integers, but the following

v are

+ v + l)> — 1.

we have

^^

(.««>-.- 1 0>+>+«»)ro. + > + m) J If

we multiply by

cos

(fi

— v)0 and

integrate, it is clear from § 5*43 that

2^+»

:p\«*".»c«o.-»)M»- 2 0>+» + ««)ro.+>+») Jo

and the result p and v the

of

The formula lxxv.

beriehte,

5*51.

An

ml

to-0

by evaluating the integral on the left for other values may be established by analytic continuation.

follows result

is at

;

once deducible from formulae given by Gegenbauer, Wiener Sitzungs-

(2), (1877), p. 220.

Lommel's

series

of squares of Bessel functions.

expansion derived by

Lommel* from

the formula

+h)
j>

i8

so that (*)
The

532

;

results of this section will be

MUnchener Abh.

=

2

(*

+ 2») JV2„ (*)1

J

found in Math. Ann. 548—549.

xv. (1886), pp.

ii.

(1870), pp.

632—633

;

xiv. (1878),

-

;

[CHAP. V


152

Hence, by

§ 5*11 (11),

**

(1)

2

{./

we have

„_ x (z)

- J^ (z) J„ (z)} = 2

(i;

n-0

+ 2n) JV „ (*), 2

by adding on terms at the on taking zero as the lower limit when R (v) > is superseries, it may be seen that the restriction R,(v) > ;

beginning of the fluous.

If we take in turn we have (§ 3*4)

= \, v = f

v

-= 2

(2)

T

by taking i>=

1,

we

\z* {J *

(4)

results so obtained,

+ *)JV»(*).

(n

n=0

™*£.

(3)

while,

and add and subtract the

,

£ (-r
see that

(jr)

+ J»» (5)} = I

(in

+

«-0

Another formula of the same type

2

1) An+x

(*)•

derived by differentiating the series

is

J\ + » (s)

€n

»-o for it is evident that

a ^-

°°

°°

2 e„J\+n (*) = 2 2

«2 n=0

e„ «/"„+„ (*)«/'„+„(*)

»=0

00

= 2

€n

t/"„

+n (s) (Jv+n-i (^)

— Jv+n+i (*)}

»t=0

= ivJ?(z)\z, and so. when iJ(v) >0, we obtain a modification of Hansen's formula (§2 namely 00

2

(5)

f*

en

5),

fit

J\+n (z) = 1v\ Jf(i%. Jo

«=o

An

-

*

important consequence of this formula, namely the value of an upper

bound

By

for

j

Jv (x)

taking

\,

will

v— \, v 2 m=o

be given in

it is

13 42.

found that

e„ Jt«n+i ,(*)\

2 = - P sin**1 dt " •

8

I

tJo f |_

and

§

2 sin2 *]*

2 f*

7T

7T.I

t

Jo

.

_ ,cft *

so

(6)

2 n=0

«/'

n+J (*)

= -£i(2*), T

where, as usual, the symbol Si denotes the "sine integral." This result is given by Lommel in the third of the memoirs to which reference has been made.


5-6]

153

Continued fraction formulae.

5*6.

Expressions for quotients of Bessel functions as continued fractions are deducible immediately from the recurrence formula given by §32(1); thus, if

the formula be written

«/,_,(*)

%zJ v+x (z)j v'

-

Jv (z) it is

at once apparent that

^

-

1

*/„_,(*)

-

1

1

\z

\z-j{(v+m-l)(v\-m)}

— This formula

K

is

2v/z

v

for integral values

that, if Q„ {z)

J v+m\ z )

-2(v + l)/z-...-2(v + m)lz - Jv+m (z)

These results are true Bessel*

v

easily transformed into

J ^ (z)

'

J + m+1 (z)/{v + m)

—

1

for general values of v\

of

v.

= Jv+l {z)l{\z Jv (z)},

An

(1)

'

was discovered by

equivalent result, due to Schlomilchf,

is

then

Other formulae, given by LommelJ, are

Jv+1 (z) (4)

Jv (z)

2(i/

^+2 (*) _

W

,k\

The

z

z*

zJv+m+1 (z)

z2

J*

+ l)-2(i/ + 2)-2(i/ + 3)-...-2(»/ + m)-

J, +m (z)

zJv+m + {z) i! + 2fe±l) 2(i/ + l)-2(* + 2)-...-2(> + m)- J ¥+m (s) *'

i

i

«/„<»

Bessel functions in

all

these formulae

'

x

may

'

obviously be replaced by any

cylinder functions. It

was assumed by Bessel

that,

when

m -*-

,

the last quotient

may be

neglected, so that

AA^L-^ll

,«x (b)

J_i(*)~"

1

i^l\ V (v

-

+ l)}

$*/{(»+ !)(»+

2)}

1

1

* Berliner Abh. (1824), [1826], p; 31. Formula (2) seems not to have been given by the earlier writers; see Encyclopedic des Sci. Math. n. 28, § 58, p. 217. A slightly different form is used by

Graf, Ann. di Mat. (2) xxin. (1895), p. 47. integral values of t Zeitschrift fiir Math, und Phys. n. (1857), p. 142; Schldmilch considered v only.

J Studien

fiber die

BesseVschen Functionen (Leipzig, 18G8), p. 5 see also Spitzer, Archiv der (1858), p. 332, and Gunther, Archiv der Math, und Phy*. lvi. (1874), ;

Math, und Phys. xxx. pp. 292—297. W. B.

F.

6


154

[CHAP.

V

It is not obvious that this assumption is justifiable,

though it happens to and a rigorous proof of the expansion of a quotient of Bessel functions into an infinite continued fraction will be given in § 9'65 with the help of the theory of " Lommel's polynomials." be

so,

The reason why the assumption

[Note. the fraction

pm /qm tends

tends to that limit

;

to a limit as m-»-oo

this

may be

,

is

not obviously correct

it is

is that,

not necessarily the case that

even though

aw ^">

^m+1

seen by taking

pm =m+amm,

q m =m,

a,„=-l.]

The reader will find an elaborate discussion on the representation of Jv (z)/Jv -i (z) as a continued fraction in a memoir* by Perron, Munchener Sitzungsberichte, xxxvn. (1907), 504; solutions of Riccati's equation, depending on such a representation, have pp. 483

—

been considered by Wilton, Quarterly Journal, xlvi. (1915), pp. 320—323. The connexion between continued fractions of the types considered in this section and the relations connecting contiguous hypergeometric functions has been noticed by Heine, Journal fiir Math. lvii. (1860), pp. 231—247 and Christoffel, Journal fur Math, lviii. (1861), pp. 90—92.

5*7.

It

Hansen's expression for J,(z) as a limit of a hypergeometric function.

was stated by Hansen f that J.

(i)

W -iin r a^

..i'1 J

(»

>

M;»+i;-^).

We shall prove fi

this result for general (complex) values of v and z when tend to infinity through complex values.

If

\—

= 1/t),

1/8, fi

(m -f

the

—

/,

K

*

}

\ and

l)th term of the expansion on the right

=-,

n

[(1

is

+ rh) (1 + rv )\

a continuous function of 8 and rj and, if 8 r) are arbitrary positive (less than 2 z j"1 ), the series of which it is the (wi + l)th term converges uniformly with respect to 8 and 77 whenever both 8 ^ 8 and 17 ^ i» For the term in question is numerically less than the modulus of the (m + 1 )th This

is

,

;

numbers

|

|

I

|

-

|

term of the (absolutely convergent) expansion of

& W*., ^

r^,+\)

and the uniformity of the convergence Since the convergence * This

(1908), pp.

memoir 85—88.

is

is

>

v

+ 1 Kô), >

follows from the test of Weierstrass.

uniform, the

sum

of the terms

is

a continuous

the subject of a paper by Nielsen, Milnchener Sitzungsberichte, xxxvni.

t Leipziger Abh. n. (1855), p. 252 ; see also a Halberstadt dissertation by F. Neumann, 1909. [Jakrbueh ilber die Fortschritte der Math. 1909, p. 575.]


5-7, 5*71]

function of both the variables

the

/

and

($, rj)

at (0, 0), and so the limit of the series

of the limits of the individual terms

sum 4

ô\r(v + l)

\B

155

that

;

4

r}'

is

m . m!r(i/ +

/

is

to say

m+l)

this is the result stated.

Bessel functions as limits of Letfendre functions.

5*71.

known that solutions of Laplace's equation, which are analytic near the origin and which are appropriate for the discussion of physical problems connected with a sphere, may be conveniently expressed as linear It is well

combinations of functions of the type rn

rn

Pnm (cos 0) cos m; sin .

normal solutions of Laplace's equation when referred to polar

these are coordinates

Now

Pn (cos0),

(r, 0,

).

consider the nature of the structure of spheres, cones and planes

associated with polar coordinates in a region of space at a great distance from

the origin near the axis of harmonics.

The spheres approximate to planes and

the cones approximate to cylinders, and the structure resembles the structure associated with cylindrical-polar coordinates

;

and normal solutions of Laplace's

equation referred to such coordinates are of the form e ±k*Jm

It is therefore to be expected that,

way

in such a

that r sin

(§ 4*8)

(kp)™^m.

when r and n

are large* while

small

is

p) remains bounded, the Legendre function

(i.e.

should approximate to a Bessel function; in other words,

we must expect

Bessel functions to be expressible as limits of Legendre functions.

The

actual formulae by which Bessel functions are so expressed are, in

effect, special cases of

Hansen's

limit.

The most important formula

of this type

is

limPn (cos^)=/ (4

(1)

This

result,

which seems to have been known to Neumannt in 1862, has been

investi-

gated by Mehler, Journal fiir Math, lxviii. (1868), p. 140; Math. Ann. v. (1872), pp. 136, 141 144; Heine, Journal fur Math. lxix. (1869), p. 130; RayJeigh, Proc. London Math.

—

61—64 Proc. Royal Soc. xcii. A, (1916), pp. 433—437 [Scientific Papers, 338—341 vi. (1920), pp. 393—397]; and Giuliani, Giorn. di Mat. xxn. (1884), 239. The result has been extended to generalised Legendre functions by Heine

Soc. ix. (1878), pp. i.

(1899), pp.

—

;

;

pp. 236 and Rayleigh.

It has usually been

assumed that n tends

values in proving (1); but

it is

easier to prove

it

to infinity through integral

when n tends

to infinity as

a continuous real variable. * If

?i

were not large, the approximate formula for

t Cf. Journal fUr Math. lxh. (1863), pp.

36—49.

Pnm (cos 0)

would be

(sin

m 0)/m

!.


156

We

[CHAP.

V

take Murphy's formula

Pn (cos z/ri) =

sjFx

(—

n,

n

+1

and the reasoning of the preceding section modification that

we use the

1

;

sin2 \z\n)\

;

applicable with the slight

is

inequality

|rin(f»/»)|*f|(*/n)|,

when

\z

$ 2 n

|

,

|

|

& (-

n,

|

and then we can compare the two

n+

1

1

;

;

sin2 \z\n\

,F, (1/S

1/S6

,

series

+1

1

;

^$

;

2

z

1

2 !

),

an arbitrary positive number less than § z _1 atid the comparison made when \n\>lj8 The details of the proof may now be left to the reader.

where B

is

|

|

is

.

When

n

is restricted to

terminates, and

the proof.

be a positive integer, the series

for

Pn (coaz/n)

convenient to appeal to Tannery's theorem * to complete This fact was first noticed by Giuliani the earlier writers took for it is

;

granted the permissibility of the passage to the

limit.

In the case of generalised Legendre functions (of unrestricted order m), the definition depends on whether the argument of the functions

+1

and

—

1 or not

;

x (between

for real values of

Pw-™ (cos -J =

v

r

'

'

^_

& (-

n,

and

n+ 1 m + 1;

is

between

we have

ir)

sin" **/»).

;

so that

lim n m

(2)

Pn~m (cos -) = Jm

(as),

but otherwise, we have

m

z\ tanh (iz/n) ( Pn~m (cosh = F (~ n,n + l; m + 1; ~ sinh,„,\z\n), r m î) -J r>

_,

i

*

,

_

,

.

,

2

s

*

(

so that

lim n m

(3)

The corresponding formula

Pn-m (cosh -) = Im (z). for functions of the

from the equation which expressesf ,,,

,.

hm

(4)

m sinri7r rn -^—

w _».ooLsm

(

m+

Qnm

second kind

in terms of

z\ r— Qnm ( cosh n)7r nj \ ~

,

may be deduced

Pnm and P„~ m

Km

;

it is

(z).

This formula has been given (with a different notation) by Heine +; it is most easily proved by substituting the integral of Laplace's type for the Legendre function, proceeding to the limit

and using formula

* Cf. Bromwioh, Theory of Infinite Series, § 49. t Cf. Barnes,' Quarterly Journal, xxxix. (1908), p. 109

2r(-m-n)

nnir

BinVlir

t

the equation

;

P«~" Q n»= r(l-m +

which is adopted in this work. + Journal fur Math, i/xix. (1868), p. 131.

in Barnes' notation,

(5) of §

62 2.

is

P-"*

1

?j)

r(l + i« + n)

p


5*72]

Another formula,

slightly different from those just discussed, is

K

(5)

.

this

is

157

"/»(i£l*)" /,(

due to Laurent*, and

it

*

may be proved by

);

using the second of Murphy's

formulae, namely

Pn (cos 6) = cos" %6 F .

2

(-

X

-n

n,

1

;

- tan

;

2

\6).

[Note. The existence of the formulae of this section must be emphasized because it used to be generally believed that there was no connexion between Legendre functions and

Thus

Bessel functions.

was stated by Todhunter

in his Elementary Treatise on Laplace's BesseVs Functions (London, 1875), p. vi, that "these Bessel functions] are not connected with the main subject of this book."] it

Functions, Lame's Functions [i.e.

5*72.

A

and

Integrals associated with Mehler's formula.

completely different method of establishing the formulae of the last by Mehler and also, later, by Rayleigh this method depends

section was given

;

on a use of Laplace's integral, thus

Pn (cos $) — —1

f*

(cos

+

i sin

I TT.'O

V cos $) n d

en Iok (cos* + < sin • cos $)

fifo

7T T.'o '

(

Since

n log uniformly as n

-»- oo

{cos (z/n)

when lim

+ i sin

(z/n) cos

<£} -*-

iz cos

$ ^ ir, we have at once

Pn (cos z/n) - - f «*«»

d

=J

(z).

Heine f and de Ball J have made similar passages to the limit with integrals of Laplace's type for Legendre functions. In this way Heine has defined Bessel functions of the second and third kinds results in § 6*22

when we

;

reference will be

deal with integral representations of F„

made

to his

(z).

Mehler has also given a proof of his formula by using the Mehler- Dirichlet integral

Pn (cos6) = «2If n=yjr, it

may

§

[

; o

™*<££&4*L V{2 (cos

be shewn that

+

P

- I' zln\ i n (cos (cos*/*)*-J o

but the passage to the limit presents some proper integral.

C0S

W

-^—

.

,

}

little difficulty

because the integral

is

an im-

Various formulae have been given recently which exhibit the way in which * Journal de Math. (3) i. (1875), pp. 384—385; the formula actually given by Laurent is trioneouB on account of an arithmetical error. t Journal fur Math. lxix. (1868), p. 131. See also Sharpe, Quarterly Journal, xxiv (1890)

pp.

383—386. X Attr. Nach. cxxvm. (1891), col. 1

—

1.


158

the Legendre function approaches its limit as

Thus, a formal expansion due to Macdonald*

[CHAP. V

degree tends to

its

infinity.

is

Pn -™(cos0)

(1)

= (n + £)"'» (cos £0)-™ [Jm (x) + sin where x = (In

+

2

\0 {^x

J^ (x) - Jw (x) + ^x' Jm+l 1

(x)}

+

.

.],

1) sin \0.

Other formulae, which exhibit an upper limit for the error due to replacing a Legendre function of large degree by a Bessel function, aref

Pn (cos

(2)

rj)

±

= ^(sec rj)

iir- 1 •

Qn (cos rj)

e ± <* + i>«,»- t*n '>> [Jo {(»

+ h) ^n v}

±

iY

{(»

+ £) tan rj\] 4î-\/(sec v)

+ 7r{R(n)

Qn (cosh f) = e -(»+i)(f-tanh« ^( S ech f)

(4)

.

K

{(n

+ \) tanh f

ffl3

+

+ ^'

V(sechg).e-(»+*>*

R{n)+$

and (4), f > the numbers lt 2 < t) < where, in (2), 3 are n may be complex provided that less than unity in absolute magnitude, and its real part is positive. But the proof of these results is too lengthy to be %tt, and, in (3)

;

,

given here. 5*73.

The formulae of Olbricht.

by Hansen's formula as a limit of a hypergeometric function has led Olbricht j to investigate methods by which Bessel's equation is expressible as a confluent form of equations associated with Biemann's P-functions.

The

If

fact that

a Bessel function

is

expressible

we take the equation dz*

+

dz

z

+

V

of which a fundamental system of solutions

z~*Jv (z),

)

z* is

u

y

'

the pair of functions

z-»Y

v (z),

and compare the equation with the equation defined by the scheme fa,

b,

c,

Pio,

£,

y,

ff,

7.

1

l«',

z\, -I

—

London Math. Soc. (2) xm. (1914), pp. 220 221 eome associated results had been obtained previously by the same writer, Proc. London Math. Soc. xxxi. (1899), p. 269. t Watson, Trans. Camb. Phil. Soc. xxn. (1918), pp. 277—308 Messenger, xlvii. (1918), * Proc.

;

;

pp. 151—160.

X Nova Acta Caes.-Leop.-Acad. (Halle), 1888, pp. 1—48.


5*73]

159

namely d* y

Q- a -a

dz*

\

'

\-fi-fi'

l-y-yl

dy

z-b

z-c

dz

z-a

(

aa'(a-b)(a-c)

\

z-a

+

+ fifi'

)

-c)(b-a)

(b

yy' (°

-a)(c- &) )

z-c

z-b

)

y

-a)(z -

(z

we

b) {z

— c)

= 0,

see that the latter reduces to the former if

a while

finite (their

We

a

= v — fi,

a

= — v — fi, way

to infinity in such a

y tend

b, c, fi, fi', 7,

remain

= 0,

sum being

2/*

+ 1)

that

+ fi' and 7 + and b + c = 0.

fi

while

— yy - $b fifi'

2

7'

thus obtain the scheme 2ifi,

-2ifi,

v-p,

fi,

y,

z\,

l-v-fi,

-fi,

7,

J

0,

(

limPJ

\

~*°°

where

7,

7

- M + i ± V{0* + £) + tf 2

Another similar scheme

2 }-

is

"*"

ifi,

v-fi,

fi,

y,

z\

i-v-fi,

-fi,

y,

)

J

with the same values of 7 and y

A

scheme

for

J

v

(z)

qo,

0,

-4a&

,

a-\v,

0,

[-% v>

fi-\v,

v+l-a-fi.

pj

ilZl)

as before.

derived directly from Hansen's formula (

lim

oo,

0,

(

limP<

\v

is

\

zA. J

Olbricht has given other schemes but they are of no great importance and those which have Note.

It

now been

constructed will be sufficient examples.

has been observed by Haentzschel, Zeitschrift

fiir

Math, und Pkyt. xxxi.

(1886), p. 31, that the equation

whose solution

when the zero.

(§ 4*3) is xS'Wv (hit),

invariants

g2 and g3

may be

derived by confluence from Lamp's equation

of the Weierstrassian elliptic function are

made

to tend to

CHAPTER VI INTEGRAL REPRESENTATIONS OF BESSEL FUNCTIONS 6*1.

Generalisations of Poisson's integral.

In this chapter we

shall study various contour integrals associated with

Poisson's integral (§§ 2*3, 3"3)

and Bessel's integral (§ 2-2). By suitable choices numbers of elegant formulae can be obtained

of the contour of integration, large

which express Bessel functions as definite integrals. The contour integrals will Chapters vu and VIII to obtain approximate formulae and

also be applied in

asymptotic expansions for

Jv (z) when

z or v is large.

It happens that the applications of Poisson's integral are of a more elementary character than the applications of Bessel's integral, and accordingly

we

now study

shall

integrals of Poisson's type, deferring the study of integrals

The investigation of generalisations of now give is due in substance to Hankel*.

of Bessel's type to § 62.

which we

integral

The simplest

shall

of the formulae of §

33

is § 3'3 (4),

Poisson's

since this formula contains

a single exponential under the integral sign, while the other formulae contain

which are expressible in terms of two exponentials. We therefore examine the circumstances in which contour integrals of the type

circular functions, shall

r

e izt

Tdt

J a

are solutions of Bessel's equation;

not of of

it is

supposed that T is a function of t but b, are complex numbers independent

and that the end-points, a and

z,

z.

The

result of operating

V,, defined

in § 3*1, is

on the integral with Bessel's

differential operator

as follows:

b

L»

V,

[

e

izt

Tdtl

= z v+i

I

e

lzt

T{\-

2

)

dt

l

e

izt

T (t* -

1)1

—

(2i/

+

izt +1 1) iz» j e

b

h

= iz"+

+

+ iz»+*

f

e izt \{2v

+

1) Tt

Ttdt

- ^ {T (i 2

1)}1 dt,

Math. Ann. i. (1869), pp. 473 485. The discussion of the corresponding integrals for Iv(z) is due to Schlafli, Ann. di Mat. (2) i. (1868), pp. 232—242, though Schlafli's results are expressed in the notation explained in § 4-15. The integrals have also been examined in great detail by Gubler, Zurich Vierteljahrsschrift, xxxm. (1888), pp. 147 —172, and, from the aspect of the theory of the linear differential equations which they satisfy, by Graf, Math. Ann. xlv. (1894), pp. 235 2C2 lvi. (1903), pp. 432 444. See also de la Vallee Poussin, Ann. de la Soc. Sci. de *

and

K v (z)

—

;

Bruxelles, xxix. (1905), pp.

—

140—143.

INTEGRAL REPRESENTATIONS

6-1]

by a partial if T, a, b

integration.

161

Accordingly we obtain a solution of Bessel's equation

are so chosen that %- {T(t>-l)}

= (2v + l)Tt,

e

izt

= 0.

T(t2 -l)

t

The former of these equations shews that (P

— I)" - *,

and the

either so that

it is

value after

initial

t

T

is

a constant multiple of

shews that we may choose the path of integration, a closed circuit such that e izt (t2 — l) v+i returns to its

latter

has described the circuit, or so that

e

izt

l) v + i vanishes

(&-

at each limit.

A t

=1

contour of the

first

type

is

temporarily that the real part of z

one which starts from

—

1,

a figure-of-eight passing round the point

counter-clockwise and round

+

+ coi and

t

is

=—

clockwise.

1

And,

if

we suppose

positive, a contour of the second type is

returns there after encircling both the points

1 counter-clockwise (Fig. 1

and Fig.

2).

If

we take

a, 6

=±

1, it is

Fig. 1.

Fig. 2.

necessary to suppose that

R(v +

£)

> 0, and we

merely obtain Poisson's

integral.

To make the many- valued of

t

—1

and

t

+

1 to

axis on the right of

We

z"

at

(t

vanish at the point t

—

2

— 1)""* definite*, we take

A

where the contours

the phases

cross the real

1.

therefore proceed to /•(1+,

function

examine the contour integrals

-1-)

/-(-i+.i+j

e™ («a - 1)"-* dt,

eizt

z"

* It is supposed that v has not one of the values \,\,\\ ±1, and both integrals vanish, by Cauchy's theorem.

...;

(t*

- 1)--* dt.

for then the integrands are analytic


162

[CHAP. VI

It is to be observed that, when R (2) > 0, both integrals are convergent, and differentiations under the integral sign are permissible. Also, both

integrals are analytic functions of v for all values of

In order to express the

v.

integral in terms of Bessel functions,

first

we

expand the integrand in powers of z, the resulting series being uniformly convergent with respect to t on the contour. It follows that /"(1+.-1-)

zu

Ja

m=o

Now tm (t — I)" - * is an even

—^-r-jm+, -1-) V*(P-l)—*dt.

;m z Vrm,

00

J*(P-l)'-*dt= 2

ml

Ja

an odd function of t according as m is even or odd; and so, taking the contour to be symmetrical with respect to the origin, we see that the alternate terms of the series on the right vanish, and we are 2

then

with the equation

left

J* (t* -

t" J

or

1)"-* dt = 2

v

2

;a

= 2

K

'

on writing

t

=
To evaluate the

R(v + %)>0; from

to

1

«•*-* (u

,

,

-

1)"-*

du

=

u and u

—1

vanish

when

1.

on the right, we assume temporarily that into the straight line

first part,

{«-(»-«

™ - e<"- ««'}

n /

to

1,

1 to 0,

we have we have

zero.

um -l (1 - «)"-£ du

Jo

=

Now

u"-i(u-l)-*du, Jo

going from and on the second part, returning from where, in each case, the phase of 1 — u is

Jo

1)"-* dt

,-(1+)

, !

-

Jo

may then be deformed

taken twice; on the

fd+> I

integrals

the contour

u— 1— (1 — u)e~ vi u — 1 = (1 — ii) e +ni We thus get

t*» (t2

,

in the last integral the phases of

on the real axis on the right of u =

is

x

(2m)

w -o

u

x

(2m)! m-o w (—\m z v+2m

a

„. 2t COS

Z>7T

r(m + £)I> + £) Z \,/ — -—— r(m + /;+l) '

.

both sides of the equation "

J

(1+

V-» (» -iy-iau- u cos „ ri» + »r(»+» r (w + v +

i)

are analytic functions of v for all values of v) and so, by the general theory of analytic continuation*, this result, which has been proved when (v + £) > 0,

R

persists for all values of * Ii ("

Modern Analyst*,

+ £)~-0,

§ 5-5.

v.

The

reader will also find

Jo which

is

it

possible to obtain the result,

when

by repeatedly using the recurrence formula

»+«+J Jo

obtained by integrating the formula

l{ u »'+i( U -l)-'+'«+i. =(Ml + v + n + l )Mw-i (u _ ir + M +l + the integral

is

(>,

+ n + i)um-*

{M

_

1)

v+ M -i.

then expressed in terms of an integral of the same type in which the exponent of

u - 1 has a positive real part.


6-1]

Hence,

for all* values of v,

r(l+, -1-)

(—\m z v + *m Y( m

oo

= 2" +i i r (i) r (i» + $> cos i/tt Therefore,

if v

(i)

and

-f-

£

j.

is

.

1\

a.

m-o (2m)!r(j/+m +

a

J

163

l)

not a positive integer, ,

w - ^"ty /r

" ** ((> - i)"

i

*

this is Hankel's generalisation of Poisson's integral.

Next let us consider the second type of contour. Take the contour to lie wholly outside the circle 1 = 1, and then (t 2 — l)" -i is expansible in a series of descending powers of t, uniformly convergent on the contour thus we have |

1

;

m

m=o

and

in the series the phase of

Assuming^ the

t

!

I

(|

between

lies

— §it and + ^tt. i')

we have

permissibility of integrating term-by-term,

/•(-i+,i+)

<*>

TVA —

zy

v

A-m\ r(-i+.i+)

m -Q Wil (£-

J<*>i

V)

J

J

exp ia

oot

But

J

oo i

.

where a

aa

the phase of z (between + \tr)\ and, by a well-known formula*, the

is

last integral equals

— 1iri\Y (2m — 2v + 1).

Hence

r

,J

+

'

1+)

*- <* - 1)-'

*><

s * - «,-o

2

"

(

when we use the duplication formula § T(%-v + m) and T (- + m + 1).

r(

->7;%7r " (£ - r (2m — i~; 2^

»*

v)

!

to express

-h

T (2m —

2i>

+

r

)

1 1)

terms of

1) in

1/

* If v

-J

is

a negative integer, the simplest

residue of the integrand at

t

To

justify the

way

of evaluating the integral is to calculate the

u= 1.

term-by-term integration, observe that

-^

'

/

|

e

tzt

dt

l

is

convergent

J oot converges uniformly,

;

let

- l)" it follows that, wheu we independent of t, such that the remainder an integer We then terms of the expansion does not exceed e/K in absolute value when M ^ after have at once its

value be

A'.

Since the expansion of

are given a positive

number

e,

we cau

(t

2

M

find

M

M

I

J

cot

.

jnir{±-i>) J aci f(-l+.l-+ > ,izt '\e tzt dt\=e,
'

aud

J cot the required result follows from the definition of the

X Cf. Modern Analysis, § 12-22.

sum § Cf.

of

an

infinite series.

Modern Analysis,

§ 12-15.

.


164

R (z) >

when

Thus,

and v + %

is

[CHAP. VI

not a positive integer,

This equation was also obtained by Hankel.

Next consider (-1+.1+) eizt

J oo

i

{t*-iy-*dt,

exp ( - 1*>)

where co is an acute angle, positive or negative. This integral defines a function of z which

is

when

analytic

— \ir +

a>< arg z <

fyrr

+ co;

arg 2 < £77-, the contour can be deformed into the second of the two contours just considered. Hence the analytic continuation of J- V (z) can be defined by the new integral over an extended range of values of arg z; so that we have and,

if

z

is

subject to the further condition that

|

J

r^-«\,>" »n7V/'(-i+,i+) f

J-*(')~

(3)

\j?rm £.1Tl

where arg z has any value between

By for

giving

co

a suitable value*,

**V-iY-*dt,

Jf ociexp(-Ju))

X \-%)

— \ir + to

we can

any assigned value of arg z between

When R (z) > and ^(» + J)>0we may

and

\ir

+

co.

obtain a representation of J_„ (z)

— tr

and

it.

take the contour to be that shewn in Fig.

3,

\/ /\

Fig. 3.

in

it is supposed that the radii of the circles are ultimately taking each straight line in the contour separately, we get

which

By

J-> {z)=

¥Y

T{

K^n\)

ei

"

[/I +

e

' sni(v ~

Je * e-3»r«C4

made

indefinitely small.

h)

(1

-^" *

J)

(1

- #)»-\

€ izt e -«Hi>-l)

i

dt

(\-t*y-l dt

1

e*t

+ * If I

w be I

sentation of

valid for

e* «**("- i)

(1

- t*y~i dt~]

an appropriate value (greater than any preassigned value of arg z may be obtained.

increased in a series of stages to

J- v (z)

f°*

e^C-V (l-Py-h dt

£*-),

a repre-


6-11]

On

bisecting the third path of integration

-t, ±t,

t,

it

respectively,

we

sponds to Poisson's integral (4)

and,

if

J_ v (s)=

for

2{

\fW 7Y-J r (v+#) "($) |_ ,

and replacing

obtain a formula for

Jv (z) S

;

mvn

the formula " f jo

e-'til

in the various integrals

r

'V-Tb$&l»[fl'*<*< 1

by

it,

to Gubler* which corre-

J- V (z), due

is

+ tty-hdt+f'cosizt + v^.il-fiy-idt], J

Jo

this be combined with Poisson's integral,

w

t

165

it is

found that

- f>rl

rm,:l+ *r ~ >

*-K'

a formula which was also discovered by Gubler, though "Weber t in the case of integral values of p.

it

*l•

had been previously stated by

After what has gone before the reader should have no difficulty in obtaining a formula closely connected with (1),

*w- r-*k^/>-ir.~«>.*

(6)

in which

namely

it is

supposed that the phase of

t2

—1

vanishes

when

is

t

on the

right of t—\.

6*11.

real axis

on the

i

Modifications of Hankel's contour integrals.

Taking R (z) > 0, let us modify the two contours of § shewn in Figs. 4 and 5 respectively.

Fig. 5.

Fig. 4.

By making

6*1 into the contours

those portions of the contours which are parallel to the real

* Ziirich Vierteljahrsschrift,

xxxm.

(1888), p. 159.

See also Graf, Zeitschrift fur Math, und

Phys. xxxvin. (1893), p. 115.

t Journal fUr Math, lxxvi. (1873), p. 9. Cf. Hayashi, Nyt Tidsskriftfor Math. xxni. b, (1912), The formula was examined in the case v=0 by Escherich, Monatshefie fiir Math,

pp. 86—90.

und Phys. m.

(1892), pp. 142, 234.


166 axis

move

off to infinity (so that

[CHAP. VI

we

the integrals along them tend to zero),

obtain the two following formulae:

0)

J.{z)

2ttiT(i) d+) e

izt

(t

-

2

l)"-i dt

+

_J l+»i

J _„{*)-

(2)

e .'

first

phase of

in 1

t

t-

t

—1

2

eizt (t2

- 1)*-* dt

iy-i dt

rzt

(t

-

2

,

I)""* dt

f(-i+)

+

l+ooi

J

1

-1+ooj

many-valued functions are to be interpreted by taking at A and to be + ir at B, while in the second the

result the

the phase of

-

2 7nT(£) r(i+)

In the

e izt (t2 -1+ooi

J

—1

to

be

is

at

A

and

is

— it

at B.

To avoid confusion it is desirable to have the phase of t2 — 1 interpreted the same way in both formulae; and when it is supposed that the phase of

—1

is

+ 7r at

B, the formula (1)

J-.(*)-

(3)

is

of course unaltered, while (2) is replaced

2™r(i) /(i+) d+) 1

e

izt

(t

2

-

,

.

1)"-* dt

f(-i-)

+ e~ vin

-i

e

izt

(t

2

-

1)"-* dt

J -1+ooi

J 1+ooi

In the

last of these integrals,

J

the direction of the contour has been reversed

and the alteration in the convention determining the phase of necessitated the insertion of the factor e~ 2{y ~^ )H

On

by

comparing equations (1) and (3) with

see that

T(\-v).(\z

y[^

2

t

—1

has

.

§ 3'61

equations (1) and

(2),

we

.

(4)

(5)

unless v

We

is

an integer, in which case equations

can, however, obtain (4)

and

(1)

(5) in the case

from a consideration of the fact that continuous functions of v near v — n. Thus value

(n),

HJ»

(z)

-

lim

HJV

and

all

(3) are not independent.

when

v has

(z)

dt

and similarly

for

Hn ®

(z).

an integral

the functions involved are

~


6-12]

(5)

*

167

As in the corresponding analysis of § 6*1, the ranges of validity of (4) and may he extended hy swinging round the contours and using the theory of

analytic continuation.

Thus,

if

— \nr < a> < f ir, we J5Tr

(6)

while, if

«

(z)

=

have

T(

*""

Z)V

*~rn?

^

f

2

(*

~

W*

dt '

— f 7r < a>< \ir, we have

gw(#)B E fl-^-y):

(7)

7r * * <.£)

provided that, in both (6) and

(

f

" 1_)

^(^-i)"-*^ »*exp(-t«)

the phase of z

(7),

Representations are thus obtained of

between — ir and — 2tt and jr.

2tt,

H

and of

v

{z>

(z)

lies

between

H^ (z) when

when arg

s has

— \ir +
and

arg z has any value

any value between

If co be increased beyond the limits stated, it is necessary to make the contours coil round the singular points of the integrand, and numerical errors are liable to occur in the interpretation of the integrals unless great care is taken. Weber, however, has adopted this procedure, Math. Ann. xxxvn. (1890), pp. 411 412, to determine the for-

mulae of § 362 connecting Note.

The formula

HM V

2i*F„

(z)

(

- z),

HW V

(

- z)

with

HJ» (z),

= ffvW(z) — Hv V)(z) makes

— H

it

V V) (*).

possible to express

Yv (z) in

terms of loop integrals, and in this manner Hankel obtained the series of § 3'52 for Y»(«); this investigation will not be reproduced in view of the greater simplicity of Hankel's other method which has been described in § 3-52.

6*12.

Integral representations offunctions of the third kind.

In the formula § 6'11 (6) suppose that the phase of z has any given value between — tr and 2tt, and define by the equation arg z so that

— \ir
Then we

so that the phase of

(1)

;

a)

+ @,

shall write

= e-l™ z- (- u), — u increases from — tt + /3 to t

contour

=

\tr.

and

it

-

1

I

ir

+ fi

as

t

describes the

follows immediately that

*,"(*)-

where the phase of

iTilt

— vSe* I'-l**

t»>

f o+i

^x*) -.Ur^-K

1

+ \iujz

has

in \ _ i

/

its principal value.

1

^)

Again,

if /3

du

-

be a given

acute angle (positive or negative), this formula affords a representation of

H w (z) valid over the sector of the s-plane in which ¥

-

\ir

+

<

arg z

<

f ir

+ 0.


168

Similarly*, from §

where

$

is

611

[CHAP. VI

(7),

any acute angle (positive or negative) and

Sincef, by

by restricting

§ 3-61 (7),

v so that

— f 7r 4- /S < arg z <%ti- + /3. #_„W (*) = «•"* #„« (*), it follows that we lose nothing R (v + |) > and it is then permissible to deform ;

the contours into the line joining the origin to

exp i/3, taken twice

oo

;

for the

integrals taken round a small circle (with centre at the origin) tend to zero with the radius of the circle J.

On

deforming the contour of

^-(i)

(3)

where

/3

may

(1) in the specified

manner, we find that

r-^(l + £)

't—f/.

be any acute angle (positive or negative) and

R (v + \) > 0, - |tt + j3 < arg z < f v + 0. In like manner, from (4)

where

.

(2),

•

ff.«(,).(l.)

# may

*,.

T?

_r

^^ (!_»)

Jt

*,,

be any acute angle (positive or negative) and

R(v + $)>0,

+ £
-fir

The results (3) and (4) have not yet been proved when 2v is an odd positive integer. But in view of the continuity near p = n+$ of the functions involved (where n = 0, 1, 2, ...) it follows, as in the somewhat similar work of § 6'H, that (3) and (4) are true when v — \, f f .... The results may also be obtained for such values of v by expanding the integrands ,

,

in terminating series of descending powers of

z,

and integrating term-by-term; the formulae

so obtained are easily reconciled with the equations of § 3*4.

The general formulae

(3)

and

(4) ar_e of

discussion of asymptotic expansions of

fundamental importance in the

J±v (z)

for large values of \z\.

These

applications of the formulae will be dealt with in Chapter vn.

A useful

due to Schafheitlin§. If we take restricted to be an acute angle), and then write

modification of the formulae

arg z = /3 (so that arg z is u — 1z cot 0, it follows that <5 >

W ( 6) *

To

^*)~

,

^

r(y+1)r(|)

is

r«"eW.

s^h,,

!

o

eoe-lg.ar**-*™ ^ sin »T(ir+J)r(i)Jo

W=

H<*(,) "

2 " +1

e

dtf >

obtain this formula, write t

+ 1 = 6"^

z" 1 (

- u),

t

- l = 2e

w (1

- ±iujz).

f There seems to be no simple direct proof that

T{h - v) is

an even function of v. % Cf. Modern Analysis,

{

\

§ 12-22.

°2

e

~ U( ~

vi, §

u)V

~^ i+i^

v

~

hdu

Journal fur Math. cxrv. (1894), pp. 31—44.

V

1


6*13]

169

and hence that (7)

J "W-r(

^

F

-*-

2

to

-

cos-*

.

cos («

f**

**'

6

sin»+^

y +|)T(i)Jo

- vO + \6)

These formulae, which are of course valid only when

^ cntede

R (v + 1) > 0,

were

applied by Schafheitlin to obtain properties of the zeros of Bessel functions (§§ 15'32-*-15*35).

They were obtained by him from the consideration

that the

expressions on the right are solutions of Bessel's equation which behave in the

appropriate manner near the origin.

/» e-'" uv ~i (l + tt)'1- * du, which occurring in (3) and (4) lix. (1904), pp.

The

when

fi

= v, has

reducible to integrals of the types

been studied in some detail by Nielsen, Math. Ann.

89—102. from the aspect of the theory of asympby Brajtzew, Warschau Polyt. Inst. Nach. 1902,

integrals of this section are also discussed

otic solutions of differential equations

nos.

is

2 [Jahrbuch iiber die Fortschritte der Math. 1903, pp. 575—677].

1,

6*13.

The generalised Mehler-Sonine

integrals.

Some elegant definite integrals maybe obtained to represent Bessel functions To construct them, observe that, when z is positive (= x) and the real part of v is less than \, it is permissible to take a = £?r in § 6*11 (6) and to take to = - \ir in § 611 (7), so that the contours are those shewn in Fig. 6. When, in addition, the real part

of a positive variable of a suitably restricted order.

of v

is

greater than

manner to

—

+

of §

612)

—

\

t

it is

permissible to deform the contours, (after the

so that the first contour consists of the real axis from

+

taken twice while the second contour consists of the real axis from 1 to — oo taken twice. oo

r,^< l'^

Â-'J Fig. 6.

We

thus obtain the formulae

HP (w) = r #„<*> (as)

=-

(

^

r

-.{?7fr^- (1

^~$;$^

the second being derived from

§

611

" e-^-^) f>< (* 2

1)"-* dt,

¥

(1

(7)

- e2 <"-i>«)

f

te < («•

- l)r~* dt,

by replacing thy —t.


170

—v

In these formulae replace v by given by § 361

bj* (x) - -

(2)

iT{ i-

and use the transformation formulae and —\
when x >

It follows that,

(7).

[CHAP. VI

t

r(i).($xy

v)

11

«'-ir*

so that

w

r

,o\

"

Y

(4)

v

w " r(i-.or(*).(i«y /

2

v

(x)

=-

f"

ain (xt).dt 2

J i

(*

- i)- + *

f <*»<*>•*

2

r<*-iOr(*).tt*) F

.'i

(*-ir**

Of these results, (3) was given by Mehler, J/a*A. Ann. v. (1872), p. 142, in the special case v==0, while Sonine, Math. Ann. xvi. (1880), p. 39, gave both (3) and (4) in the same special case. Other generalisations of the Mehler-Sonine integrals will be given in & 621. 6 "14.

Symbolic formulae due

to

Hargreave and Macdonald.

When R (*) > and R (v +£) > 0,

where the phase of If

1

— t2

lies

it is

evident from formula §6-11

that

— \n.

and

between

(6)

D denotes (d/dz) and /is any polynomial, then f{it).e<«=f(D)e",

and

so,

when

v+i

is

a positive

we have

integer,

r(v+i)r(i)

When

v

+\

is

(1+

^

i

•

not a positive integer, the last expression may be regarded as a symbolic (z), on the understanding that/ (D) (e^'jz) is to be interpreted as

representation of HJV>

Afci i

I

e

M /(it)dt.

J ±i+»i

Consequently (1)

^

(1)

(?)=

r(v+i)r(i)

and similarly (2)

#„<»

^

W - r(»+J)r(j) m ,

(l+^) (i

*

;

+i^-» 5Elti22L'

so that <

3>

<

4>

jr

'»-f5l¥rIB (1+1 ^-..

'

r'V—fi$fmO +Br**T-

,


6-14, 6'15]

The unless

series obtained

it

from

(4)

171

by expanding in ascending powers of

D

does not converge

terminates; the series obtained in a similar manner from (3) converges only

when R(y)>\. The expressions on the right of (3) and (4), with constant factors omitted, were given by Hargreave, Phil. Trans, of the Royal Soc. 1848, p. 38 as solutions of Bessel's equation. The exact formulae are due to Macdonald, Proc. London Math. Soc. xxix. (1898), p. 114.

An

associated formula, valid for all values of

we

positive integer,

v, is

derivable from § 6'11

v) H W(z) = T ^~ 7rtr ;^

z)

"

U+)

f

v

C?/

J

l

+ °o

p

so that

Bv „,_ r(.+t-.).») F (1 +i)J).

{ti

^^ = r(y+rV+i) (K

This

result,

proved when

analytic continuation

= « r and

v

case

when

;

it

R (z) > 0,

is

(1+i)2)W

m^ H^

is

any

= n+%

^

',

"y

)

w)

kind, and so

^-"

(z)} -

easily extended to all values of z

was discovered by Sonine, Math. Ann.

used by Steinthal, Quarterly Journal, v

n

_ iy - n . i€iztdtf

A similar equation holds for the other function of the third (6)

If

(<2-l)"-»-l(-)».(l + Z)2)»e*«
= (-)"r(W)(^ (5)

(4).

see from the equation in question that

xvm.

by the theory of

xvi. (1880), p. 66,

(1882), p. 338

when

v

= n+%;

when in the

the result was given slightly earlier (without the use of the notation proof by Glaisher, Proc. Camb. Phil. Soc. ill. (1880), pp. 269—271.

A

of Bessel functions)

based on arguments of a physical character has been given by Havelock, Proc. London Math. Soc. (2) II. (1904), pp. 124—125.

6*15.

If i

arg z

Schlafii's* integrals of Poissons type for

we take |

<

\tt in § 6*1 (3)

Iv (z) and

K

v (z).

find that,

when

crosses the negative real axis is

— 2tt.

and then replace z by

iz,

we

\tt,

and the phase of

2

t

—1

at the point where

t

4

»

-1

Fig. 7.

If

we take

R (v + £) >

to secure convergence, the path of integration

may be taken to be the contour of Fig. 7, in which the radii may be made to tend to zero. We thus find the formulaf #)r r( (* 2 T-" {2) =

*2^r
(1

[

" <^> i?~* ( ' ~ +

i (e

- ""

*

r

+ e-3vni )

*

of the circles

dt

[

e~zt (1

- <2)"-* dt

,

* Ann. dt Mat. (2) i. (1868), pp. 239—241. Schlafli obtained the results (1) and (2) directly by the method of § 6-1. t Cf. Serret, Journal de Math. ix. (1844), p. 204.


172

in which the phases of

t

2

—1

and of

—t

1

2

[CHAP. VI

Now, from

are both zero.

§ 3'7l (9),

we have

™-T
(2)

and so (

3)

that

/_, (.)

-*r-

1

i

W = r <*-*?*"•<»* .c*"

_ /.

«

((a

- irt

*

to say*

is

K

(4)

=

(,)

v

- I)""* dt,

2

^j^ffi J"*-*

(*

whence we obtain the formula

K

(5)

v

(z)

T = ^) îT [V* c08h< sinnÂ, v >

a result set by Hobson as a problem in the Mathematical Tripos, 1898. The and args < \ir. The reader will formulae are all valid when R (v + \) > find it instructive to obtain (4) directly from § 6*11 (6). |

j

6*16.

When x arg * ^xze H v

Now, when R(v)^ —

\,

r(» + *)-(i*»>-

^T(^)

—

written

(^-l)-+i-

,

may be opened

theorem, the path of integration

R (zt) = 0.

may be

the integral, taken round arcs of a circle from p to by Jordan's lemma. Hence, by Cauchy's

j"-t-*arg Z) tends to zero as p p e±

on which

(6)

«—*

(1+)

f J +00

611

we write

If then

zt

= iu,

out until

it

the phase of

becomes the

- (u /z ) 2

2

line

1 is —fl-

at the origin in the it-plane. It then follows from §

K

v

37

(8) that

(xz) = \Ttie-l™ H% (xze^)

«-*" r (v + £ ) _ = 2TQ)

.

r (V 4-

1)

(%xz)~ v

f

(i)

'

)«i,z

00 (2z) v f*

.

r xi z

e~ xzt dt * (* -l)" + 2

_e e-™" du

J-«>(^

and so we have Basset's formula

^

Kl)

valid

when

Basset f,

v

\xz)-

R (p + |) ^ 0,

a;

X«

>

0,

|

V (\)

arg 2

1

] o

< \tt

(u2

+z

v+ i'

2

)

The formula was obtained by

for integral values of v only, by regarding

K (x)

as the limit of

was examined in the case i»=0 by Biemann, Ann. der Physik und 130—139. (Cambridge, 1888), p. 19. t Proc. Camb. Phil. Soc. vi. (1889), p. 11 ; Hydrodynamics, n.

*

The

Chemie,

integral on the right

(2) xcv. (1855), pp.

.


6*16, 6-17]

;

173

a Legendre function of the second kind and expressing it by the corresponding limit of the integral of Laplace's type {Modern Analysis, § 1533). The formula for

j

Kn (xz)

obtainable by repeated applications of the operator

is

Basset also investigated a similar formula for /„

{xz),

— zdz

but there

j-

is

an error

in his result.

The

on the right in (1) was studied by numerous mathematicians before Basset. these investigators were Poisson (see § 6'32), Journal de V&cole Polytechnique, IX. (1813), pp. 239—241; Catalan, Journal de Math. v. (1840), pp. 110—114 (reprinted with integral

Among some

Mem. de la Soc. R. des Set. de Liege, (2) xn. (1885), pp. 26 31); and Journal de Math. vm. (1843), pp. 20, 21; ix. (1844), pp. 193—210; Sohlonrilch, Analytischen Studien, n. (Leipzig, 1848), pp. 96 97. These writers evaluated the integral in finite terms when v+$ is a positive integer. corrections,

Serret,

—

Other writers who must be mentioned are Malmsten, K. Svenska V. Alcad. Handl. lxii. -74 (see § 7'23) Svanberg, Nova Acta Reg. Soc. Sci. Upsala, x. (1832), p. 232 Leslie Ellis, Trans. Camb. Phil. Soc. vm. (1849), pp. 213—215 Enneper, Math. Ann. xi.

(1841), pp. 65

;

;

360—365

Glaisher, Phil. Trans, of the Royal Soc. clxxii. (1881), pp. 792— 815 J. J. Thomson, Quarterly Journal, xvnr. (1882), pp. 377-381 ; Coates, Quarterly Journal, xx. (1885), pp. 250—260; and Oltramare, Comptes Rendus de V Assoc. Francaise ' xxiv. (1895), part n. pp. 167—171. (1873), pp.

;

;

'

The

last

named

writer proved

_ (-) n-1 jr r d"' ~ 2**»-». (n-1)! [dp^

/°° cos xu du .

o

(u*+z*)*

by contour integration that 1

of these results

C°°

Jo and the

latter is

6' 17.

~ iff

\~^fr~)] p=1 ~

r d*~ e xtp i ~^-i.(n-l)! U/>B - 1 (l+jo)»J p =r may be obtained by differentiating the

—

The former

(~)w

/e~ xs ^P\~\

xu du u*+z2 p ~

cos

.

i

equation

ire~ xz ^P

2zjp

'

then obtainable by using Lagrange's expansion.

Whittakers* generalisations of Hankel's integrals.

Formulae of the type contained in

§ 3'32

suggest that solutions of Bessel's

equation should be constructed in the form z*

It

may be shewn by

Jf a

eia Tdt.

the methods of § 6'1 that

so the integral is a solution if T is a solution of Legendre's equation for functions of order v - \ and the values of the integrated part are the same at each end of the contour.

and

* Proc.

London Math. Soc. xxiv.

(1903), pp.

198—206.

.


174

[CHAP. VI

T be

taken to be the Legendre function Q„_j (t), the contour may start is an acute angle (positive or negative) provided that z satisfies the inequalities If

and end at

+ ootexp(— ia>), where w

— ^7r + o>< arg Z<^TT + CD. If

T

be taken to be P„_j(£)> the same contour

logarithmic singularity of P„_j

makes

{t)

at

t

=— 1

impossible to take the line joining

it

(when

—1

v

is

—\

possible;

but the

not an integer)

is

to 1 as a contour except in

the special case considered in §3*32; for a detailed discussion of the integral in the general case, see § 10'5.

We now proceed

to take various contours in detail.

First consider

J

where the phase of

t is

iexp (—tui)

oo

zero at the point on the right of

contour crosses the real axis. Take the contour to |

£

|

=

1

and expand

Q v -±{t)

the similar analysis of §

J

(1)

v

(*)

in descending powers of

K

*

t

=1

at which the

wholly outside the circle

t.

It is thus found, as in

that

6*1,

=

lie

}

u

7rlP Kt) ,

#* Q.-i (0 dt,

xiexp(-iw)

and therefore

J_ r (,) = "*>

(2)

*

S* Q_„_j (I)

vr 1

If

J o&iexp(-i
^f^

we combine these formulae and use the we find that

relation* connecting the two

kinds of Legendre functions, #,<*> (*)

(3)

^f

=

4* P,_i (0

c

(ft.

Again, consider

this is a solution of Bessel's equation, and, if the contour

right of the line z

+ oo Hence -* — oo i, we find

_».

z

R (t) = a,

i.

it is

the integral

cf.

The

relation, discovered

Hobson,

Phil.

is

a multiple of

H

v

(l i

(z).

that

J *

clear that the integral is

Tram, of

by

the

oo *

exp ( — i»)

Schlafli, is

Royal Soc. clxxxvii.

(1896), p. 461.

be taken to lie on the [z* exp (—a\z ))} as Similarly by

making


6-2] is

a multiple of Zf„(2) (4)

^

(5)

H»

a,

9

From a consideration

(z).

WJrm - W ra

<*)-

J5Tr

The of

—1

**Q.-Mdt,

I

e'*

v

« (#) -

this is also obvious

from

^^ (l*\i

&-» (*> dt

>

Schlafli's relation,

™

**<" + £>

r(-l+)

ffl (I) cos vir J »,-« P

(-M

4* P,_ x* (0
(3).

integral which differs from (6) only

is

then clear that

it is

'

(z)

and hence, by § 361 combined with (6)

of (1)

175

zero since the integrand

is

by encircling the point

+1

In (5) and (6), arg (t+ 1) vanishes where the contour crosses the on the right of — 1, and, in (5), arg (t — 1) is — ir at that point. 6*2.

instead

analytic inside such a contour. real axis

Generalisations of Bessel's integral.

We shall next examine various representations of Bessel functions by a system of definite integrals and contour integrals due to Sonine* and Schlaflif. The fundamental formula which will be obtained is easily reduced to Bessel's integral in the case of functions whose order is an integer. We

take Hankel's well-known generalisation J of the second Eulerian

integral

1 (v

in

which the phase of

t

/•(°

1

1

+m+

1)

+> '

Ittx) _„,

increases from

- ir

to

ir

as

t

describes the contour, and

then

{-r(\ z y+™ _ m= om\r(v + m+l)

|

(|,)» 2iri

no + ) _)m

—

„,t,J

(

(

^

)2wt >t

_

m!

* at"

Consider the function obtained by interchanging the signs of summation

and integration on the

This

is

right; it is

an analytic function of z

for all values of

z,

and,

when expanded

in

ascending powers of z by Maclaurin's theorem, the coefficients may be obtained by differentiating with regard to z under the integral sign and making z zero after the differentiations^

Hence

* Mathematical Collection, v.

(Moscow, 1870) ; Math. Ann. xvi. (1880), pp. 9—29. t Ann. di Mat. (2) v. (1873), p. 204. His memoir, Math. Ann. m. (1871), pp. 184—149, should also be consulted. In addition, see Graf, Math. Ann. lvi. (1903), pp. 423—432, and Cbessin, John*

Hopkins University Circulars, xiv. X Cf. Modern Analysis, § 12-22.

(1895), pp. 20

—

21. § Cf.

Modern Analysis,

§§

532,

4-44.

.


176

[CHAP. VI

and so we have at once

*W-^n «— '-pI'-sI* ,

(i)

This result, which was discovered by Schlafli, was rediscovered by Sonine;

and the

When it

was the

latter writer |

arg z

first

to point out its importance.

< \ir, we may swing round the contour about the origin

\

passes to infinity in a direction

until

making an angle arg z with the negative

real axis.

On

writing

t

=

we then

\zu,

1

= e w we

j

valid

when arg^ < |

|

This

\ir.

r oo

+ B-t

= =-.

/., (z)

(p.

-

335).

have

,

(3)

\tr,

f

(

This form was given in Sonine's earlier paper Again, writing u

\

|

f(°+) 1\) .L """ e*P |** [U ~ i)\ du

J' « = 2S

< 2>

when arg z <

find that,

is

e

the

first

z8inh «'-

™ dw,

of the results obtained

by

Schlafli.

In this formula take the contour to consist of three sides of a rectangle, as in Fig. 8, with vertices at oo — tri, — tri, vi and x> + iri.

\

ni

4

-ni

Fig. 8.

we

If

on the

write

+

t

iri for

lines joining

to

+

w on iri,

we

l

Jv (z) = - J'cos {yd -

(4)

valid

when arg z <

If if

the sides parallel to the real axis and ± id for

|

\

we make arg z true

The

when

z

z sin 0) d$

-

^^ J"

e

-»t-z**t

dt>

\tr.

~-

R (y) > 0, so also is

is still

w

get Schlafli's generalisation ofBessel's integral

is

±

\tt,

the

first

the second, and


J

v

a pure imaginary

integrals just discussed were

{z) is if

known

R{v)

is

continuous and,

to be continuous.

So

(4)

is positive.

examined methodically by Sonine in his

second memoir; in that memoir he obtained numerous de««. te integrals by be an acute angle appropriate modifications of the contour. For example, if :

^

(positive or negative)

and

if

- §tt + ^ < arg z < \ir - yjr,


6'21]

177

may be replaced by one which goes from oo — (w — yfr) i to + (V + yjr) i. By taking the contour to be three sides of a rectangle with corners at oo — {ir — ^) i, —(ir — ^jr) i, (ir + ^r) i, and oo + (tt + yfr) i, we obtain,

the contour in (3) oo

as a modification of (4),

J„(z) =

(5)

efc-^^cos^tf-scos^sintf)^

W

Jo e -"*sin>7r r~ g

T

_, ginh ,,^_,

if

JO

Again, if we take ifr to be an angle between and ir, the contour in (3) may be replaced by one which passes from oo — {\tr 4 ^r)i to oo + {\tr -4- >fr) t, and

we

so

find that

J,

(6)

(*)

*

=1

f cos(i/0-*sin0)d0 ""Jo 1 f° g-zrinht *«.*-,,(

+ provided that

When

J?

|

arg z

(i»)

>

I

gin (s^gjj

is less

|

and z

than both

is

and

(= x),

positive

£

C08

— ^. we may

^ - $ 1/W - 1^)

(ft,

tt

take

^=

in the last

formula, and get*

J¥ (#)=-!

(7)

cos

(i/0

I? J

- x sin 0) d0 +

-

*->* sin

I

TT J

(a?

cosh

*

- Ai/tt) dt.

|)

Another important formula, derived from (1), is obtained by spreading out the contour until it is parallel to the imaginary axis on the right of the origin; by Jordan's lemma this is permissible if R(v) > - 1, and we then obtain the formula

in which c

may have any

positive value

;

this integral is the basis of

many

of

Sonine's investigations. Integrals which resemble those given in this section are of importance in the investigation of the diffraction of light

by a prism see Carslaw, Proc. London Math. Soc. xxx. 121—161 W. H. Jackson, Proc. London Math. Soc. (2) I. (1904), pp. 393—414; Whipple, Proc. London Math. Soc. (2) xvi. (1917), pp. 94—111.

(1899), pp.

6*21.

If

;

;

Integrals which represent functions of the second and third kinds.

we substitute

Schlafli's integral §

62 (4)

for

both of the Bessel functions

on the right of the equation

Y

¥

we ir

(z)

= J¥ (z) cot vir — J_ v (z) cosec vir

t

find that

Y

¥

(z)

= cot vir

cos (vO

I

— zamd)dd- cosec vir

cos (v$

I

+ z sin 8) dO

Jo

J o

/"oo

— cosv7r/

e- ¥t

-ZBinht

/«>

dt-l e**-""** dt.

Jo *

Cf. Oubler,

Math. Ann. xlii. (1897), pp. 583—584.

Jo


178

by

Replace

—

it

in the second integral

[CHAP. VI

on the right, and

it is

found on re-

duction that °°

Y

(1 )

v

(z)

=-

0-v0)d0--

vt ( (e TTJo

! "sin (z sin

7TJo

+ e~

yt

cos vir) e"* 8inh *

dt,

a formula, practically discovered by Schlafli (who actually gave the corresponding formula for Neumann's function), which is valid when arg z < \ir. |

By means

we can

of this result

1

—

r

+iri

tjzsinhw-i-w

I

.

\

evaluate

dw>

TTIJ _oo

when arg z < |

\

%ir

;

we take

for

the contour to be rectilinear, as in Fig.

9,

and

ni

Fig. 9.

write

—

t,

+

i0, t

the expression

is

for

w

on the three parts of the contour

fir

1

+_

yt-zsmhtfa

.

eHzsiiiO-ye)

/

£0 +

g— vni _

'"Vo

7rl.'o

and this is equal to Jv (z) + 1 F„ (s) from formula Hence, when arg z < \tr, we have |

;

we then

see that

equal to

Too

J

±_

7rt

foo

e -v«-«sinh«

7TI

Jo

(1)

combined with

§

62 (4).

|

=

A 1

(2)

#„

(3)

#„<*>(*)=-

{1)

(*)

feo+iri

e*

—

Mnh W-' M dw, '

"V *""""^. 811

.

f°°

Formulae equivalent to these were discovered by Sommerfeld, Math. Ann. xlvii. (1896), 357. The only difference between these formulae and Sommerfeld's is a rotation

pp. 327

—

of the contours through a right angle, with a corresponding change in the parametric variable; see also

pp.

Hopf and Sommerfeld, Archiv der Math, und Phys.

(3)

xvm.

'{1911),

1-16.

By an obvious change (4)

J5T.M (z)

of variable

=A

HJ* (*)

—

write (2) and (3) in the forms

u--> exp ^z (u |

(5)

we may

- -^ du,

o

W-

-. I

o

1

<*P \\z u -u)\ du {

J


6-21]

179

the contours are those shewn in Fig. 10, emerging from the origin and then bending round to the left and right respectively results equivalent to these were discovered by Schlafli. ;

Fig. 10.

There

[Note.

is

no

difficulty in

proving these results for integral values of

of the continuity of the functions involved

;

cf.

v,

in view

§ 6*11.]

We

proceed to modify the contours involved in (4) and (5) to obtain the analytic continuations of the functions on the left. If

o> is

an angle between

HW

(6)

r

(*)

— tr

and

= -.

rr

such that

|

to

— arg z < %ir, we have |

»—» exp hz ( u -£ )

du,

[

and 1

/•«>

exp (-«—«)*

^"W-sL,,.

(7)

(

/

is")

^«p{*.(-J]

•

du,

the contours being those shewn in Fig. 11 and Fig. 12; and these formulae give the analytic continuations of the functions on the

Fig. 11.

values of z for which

between* — v and I

over the range of

Fig. 12. to

— \tr< args< m + ^rr;

and

o>

may have any

value

ir.

* If u were increased beyond these limits, the phase of u. |

left

difficulties

would

arise in the interpretation of


180

by replacing

Modifications of (2) and (3) are obtained

thus found that*

H^

(8)

—

i*cosh»-vu>

re TTl J _ao-Jiri

eizcosh

.

J

p\vni

Wl

|

Formulae of

and take

+

ie l ,

(«— Jtt* e

^h vW

-izcosh w

#

^ Wj

< \tt.

A

1.

dw

Jo

special interest arise

— 1 < R (*>) <

large

\

cogh vw dW)

e-izcoshu>-*w

:-

7rt

provided that arg z

v>

J -ao+tni

Qghrxi

=

fa

— îri

j'ao

= -—^\

(z)

it is

Jo

TTl

H®

;

/"oo+Jiri

»

(9)

why w ± \iri

g—^vwi foo+Jirt

=

(Z)

v

[CHAP. VI

by taking z

positive

double application of Jordan's

(=#)

lemma

in (6)

and (7)

(to circles of

and small radius respectively) shews that, in such circumstances, we may = %-rr in (6) and to = — \tt in (7). It is thus clear, if u be replaced by

a>

that HJti (x) =

(10)

p—\vai

foo

—^-

Og—Jnrt

H ® (x) = -—r\ Wl

- ixcoah -» f

e

v

l

Tt
=

a;

>

—

(12)

Jv (x) =

(13)

Y

(x)

v

2 r® - j sin (x cosh 2 r®

=

I

^-r-

1,

cos (# cosh

£

^

#

/

itf

.

we have

— £ i>7r)

<

cogh

7o

"TTl

— 1 < R (v) <

and

e«cosh t

JO

J -oo

and hence, when

Too /

.

J_oo

7T?

(11)

=

eixcoaht-H fa

.

cosh

— £ vtt)

.

i/tf .

dt,

cosh vt.dt;

TTJo

and, in particular

§ 6*13),

(cf.

/ta\ (14) /, -.

(15)

when we The

r

/

\

TT

,

.

2 f°°sinarf.
*<*>-; j, v^T)' 2

f " cos #£

.

^•'"iJ.WÛ' replace cosh

t

by

£.

two formulae are due to Mehler, Math. Ann. v. (1872), p. 142, and Sonine, Math. Ann. xvi. (1880), p. 39, respectively ; and they have also been discussed by Basset, Proc. Camb. Phil. Soc. "in. (1895), pp. 122—128.

A

last

slightly different

(1901), pp.

369—384;

(16)

Note.

form of (14) has been given by Hardy, Quarterly Journal, xxxn.

if

in (14)

f" The reader

we

write

x=2

«n (««+£)

y/(ab),

xt=au+b/u, we

^= "A {2 J{ab)}.

will find it instructive to obtain (14)

2

n i cos0)=as (

i>„

combined with the formula § 5*71

(1).

/"

find that

^^

sin (»

+ £)*

from the formula ,.

*

rf

)} j v{2(co This was Mehler's original method.

* Cf. Coates, Quarterly Journal, xxi. (1886), pp.

183—192.


6-22] 6*22.

The

Iv {£) and

K

t)

analysis of §

«

62

(z).

be given fully; they are mainly in terms of the

his results

easily modified so as to prove that

is

/-<«>- (

&L'-«*(' + s)*

and hence, when arg z < |

<2

v

(z) are of sufficient interest to

v

though he expressed of §415.

Schlafli*,

function F(a,

The

K

modifications of the previous analysis which are involved in the dis-

cussion of

due to

Iv (z) and

Integrals representing

181

\ir,

|

''W- s?/.T u"" exp

>

+

1)}

du-

foo -Hri

1

(3)

*•(* f

/„ (z)=er-.

^coshw-wr fa

and

when arg 2 = ± $-rr

ITTIJ aa-ni

The formulae

(2)

(3) are valid

if

R(v) > 0.

If in (3) the contour is taken to be three sides of a rectangle with corners

at 00

— in, — tri,

in, 00

/,(*)*

(4)

+ iri,

found that

it is

ifV^'coBi/tfcW-^^f t

ttJo

«-*«**«--«cft,

jo

so that

/_.(*)-/,(*)

and hence, when arg .z < |

\tr,

|

(5)

= ^52^ f V^'coshitf.cfc, T Jo

jBT.,

(z)

°°

= I

e-"osh * C osh

i><

.

a formula obtained by Schlafli f by means of somewhat elaborate transformations.

From

we can evaluate

the results just obtained, 1

foo+irt

ezcoshw-vw

when arg z < j

\ir.

For

it is

fa

easily seen that

I

J

/*oo+irt

1

ATTIJ -OO-Trt

/•»— iri

f

escoshto-âfo^

I

^"n"*

roo+irtl

+

J

(../

-0O-1T*

J

l

rao—iri

I

= 5—

ezeoshw-vw

ATTIJ -a>- w i

Math. lxix. (1868),

(2) v. (1873),

pp.

(2) v. (1873),

^w + /^ (^

Too

gviri

= 5—.

* 4nn. di Jlfot. t Ann. di Mat.

w

gzcoshw-rv, J

JoO-lrt)

e-*cosh*-„« dt

+ jv (^

199—205.

pp. 199

—201

p. 131, as the definition to

this formula was used by Heine, Journal which reference was made in § 5*72. ;

fiir


182

[CHAP. VI

and hence (6)

2i sin vtt

2ttc

Again,

we may

write (5) in the form

K

(7)

v

(z)

= ^r

and hence, by the processes used in

K, <*)

(8)

when — 7r < &><

1 j

=

*

621,

yooexp(-i*>)

f

^

exp-J

/I Oexpt'u

— \ir + a> < arg z <

and

7r

§

e -*cosht-,.t dt}

\tt

— \z (u +

-\\ du,

+ a>.

Similarly (9)

this

The contours

T

m

- " -1

J Oexp (-»+«)»

/

f

exp -Us ( u V

(

< a>< 2ir and — $tt + &> < arg z <

when

valid

is

sinv7T /""expOr-ult

- e~ vwi Iv (z)

e"" /_„ (*)

^7r

Jx")

+ - )\ du

;

tt/J

+ eo.

the formulae (8) and (9) are shewn in Figs. 13 and 14

for

respectively.

Fig. 14

Fig. 13.

Further, tion in (8)

when

may

imaginary axis

;

z

is

(= x) and

positive

be swung round until it is

— 1 < R (v) < it

1, the path of integrabecomes the positive half of the

thus found that

K* 0*0 = i« r_ *"**

I

v-"-1 exp

|

- \ix

(v

so that

(10)

K

v

(x)

= le-fc"* f

e~ i* 9inht -' t dt,

J -ao

and, on changing the sign of (11)

JSrr

v,

<

(*)=*J«*'«

j'*

(fc i

J

[

dv,

.

.


6*23]

From

we

these results

see that

2 cos | mr

(12)

183

.

K

/•oo

v

(x)

=

^

e-**sinh t cos h vt

/

so that

K

(13)

v

=

(x)

^

cos

I

(a;

sinh

t)

COS|l/7Tj

and these formulae are In particular

all

valid

when x >

cosh

j/£

.

dt,

- 1 < R (v) <

and

1.

jr.<.)-(*«.<«™ho*-["!=|2fc* .'o Jo v(f + -U

(14)

a result obtained by Mehler* in 1870. It

may

be observed that

if,

in (7),

we make the substitution

^ze t

=T

,

we

find that

jr.W-K^êxpj-r-^}^,

(15)

R (z2 > 0.

The integral on the right has been studied by numerous mathe) among whom may be mentioned Poisson, Journal de VEcole Polytechnique, ix. (cahier 16), 1813, p. 237; Glaisher, British Association Report, 1872, pp 15—17; Proc. Camb. Phil. Soc. in. (1880), pp. 5—12; and Kapteyn, Bvll. des Sci. Math. (2) xvi. (1892), PP 41—44. The integrals in which v has the special values \ and f were discussed by Euler, Inst. Calc. Int. iv. (Petersburg, 1794), p. 415; and, when v is half of an odd provided that maticians,

integer, the integral

has been evaluated by Legendre, Exercices de Calcul Integral,

1811), p. 366; Cauchy, Exercices des Math. (Paris, 1826), pp.

I.

(Paris,

54—56; and Schlomilch,

Journal fur Math. xxxm. (1846), pp. 268—280. The integral in which the limits of integration are arbitrary has been examined by Binet, Comptes Rendus, xn. (1841), pp. 958— 962.

Hardy's formulae for integrals of Du Bois Reymond's

6'23.

The

integrals

f" I

a?

sin

t

sin

in

— P"

1

f*

— f-

Jq

.

f(t)*

dt,

which f(t)

oscillates rapidly as

m t.t»~*dt,

cos

t

* 0.

By constructing a differential equation

Hardy succeeded in expressing them in terms of Bessel but a simpler way of evaluating them is to make use of the results

of the fourth order, functions

of

1

t

which x > 0, — 1 < R(v) < 1, have been examined by Hardy f as examples Du Bois Reymond's integrals

/,o

in

a?

cost. cos

dt,

t

Ja

of

type.

^621,

;

6-22. *

Math. Ann. xvm.

(1881), p. 182.

t Messenger, xt. (1911), pp. 44

—51.


184 If f"

we

sin

t

JO

by x&,

t

2 a?

.

.

sin

I

replace

— f'

1

.

= x"

dt

t

__

clear that

it is

f°°

sin

I

J

(a?e*)

_»

i»n

[CHAP. VI

sin (xe~ l ) e vt dt .

fgitzcosht

1

i

g— 2&ccosht

g—2tassinhtJ

girfzsinht

gi»t

fa

—00

./

= - \af [Trie-*™!!™ (2a?) - Trie*™ H^\ (2x) - 2er** Kv (2a?) - 2^"'* #_„ (2a:)], and hence we have (1)

sin

/

%

sin

*

JO

«--»

.

t

?°\ A = 4sin|i/7r

(2a?) [«/„ L

.

- «/_„ (2a?) + /_„ (2a?) - /„ (2a?)],

and similarly "K2

lob

(2)

«

cos

f

When

^

.

r-

1

"ITT"

=

eft

(X„ (2a?) -

,

,

v has the special value zero, these formulae

sin*sin^.^ = £7ry

(3)

it

Jo

(2a?)

r Cos
(4)

Jo 6*24.

Jv (2a?) + /_„ (2a?) - /„ (2a?)].

t

+ if

(2a?)

+

become (2a?),

Jfir

(2a?).

t

Theisinger''s .extension of BesseFs integral.

curious extension of Jacobi's formulae of § 2*2 has been obtained in the case of J {x) and Jx (x) by Theisinger, Monatshefte fur Math, und Phys. xxiv. (1913), pp. 337 341 we shall now give a generalisation of Theisinger's formula which is valid for functions of order

A

v

—

-\
where If

a

;

is

any positive number*,

j9

Now

<

X) =

2

+

2

it is

obvious from Poisson's integral that

flf)"

(

(y

f

" «>-a«ii. C os (x cos 0) sin2 " 6dd ** (l

- e-«enn#) cos (# cos 0) sin 2 " 0tf0.

(l-e _ «!,in *)cos(;FCos0)sin8 "0cW

2

1

ir

_ g— oxsinfl

-/.o sinh (x sin ff) sinh (a? sin

- ix cos 0)

sin 2 " 0cW

'

1-expJicm^-l/i)} -i~~5nh{i« (*-V«)}

ft '

J

sml1

"^(*jzl!*f*

^

2f

*'

J

where the contour passes above the origin. Take the contour to be the real axis with an indentation at the origin, and write z= ±tan £<£ on the two parts of the contour; we thus find that the last expression is equal to

y ^ —exp(-cmcot<6) ,

+ .•[** J

=4 ,

J

- eXl (a

1

sin

o

r** I

,.

sin(.rcot<£)

Jo

•

sm

/i

;

(.r

^

cot j.

.

,

,_..

,

sin (x tan id>) x "^

)

sin

<*tan v

<£)

j\

/i

i.

cot (iax cot <6) vs f/ cos (Max Vi

j

ml cot2"
.

,

d
-^-r ^sin
W^ .e-»«cot»±\ ^sin0 2

f — vir) .

/

* In Theisinger's analysis, a is

sin

(.r

tan

i
.„„

.

rf<£

2 \ r-^V cot " ) .

an even

integer.

.


6'24-6'3l]

185

and therefore

r(y+ (1)

£^ J

v

(

A.)« [

/**"

+ «2

•

**

e -««.in« C08

x

/i

cot sin (iax 2

/

Jo

j\

d>)

(-p

cos 6) sin*"

0d0

vi

.sin (a? tan Ad>) ,,„, /i AJ cot8 " d> cos (4
^~

sm (j;cotd))

dd>

^-^

•

^sind)

The transformation fails when v >£, because the integral round the indentation does not tend to zero with the radius of the indentation. The form given by Theisinger iu the case v = \ differs from (1) because he works with § 3-3 (7) which gives r(»-j)r(j) (2)

f**

a

+4 provided that \

6'3.

^

®Z-r -

2

/

{*)

( ** e-°"infl sin (x cos

=

2 6) sin

,1 j x jx /i sm (lax cot <6) cos (*a# cot
.

i.

""2

6 cos 6d6

—

sin 2 (Aar tan Ad>) r-±f

,

cW>

,,„_.,,

2 ?Z cot "

'

d>

-r-~

,

2%e equivalence of

the integral representations

Three different types of integrals which represent §§ 6'15 (4), 622 (5) and 616 (1), namely

of

K

K v

v

(z).

(z)

have now been

obtained in

K

'

(,,

'i|frr =

f

e zcoaht

e~*

{t *~

iy ' h dt

cosh vt.dt,

Jo

_ !> + £). (25)" f « cos as* cfo A 'W" (u> + «/+* ,

.

x

^T(J)

The

equality of the

J

and second was

first

directly demonstrated in 1871

by

Schlafli*; but Poisson proved the equivalence of the second and third as early as 1813, while Malmsten gave a less direct proof of the equivalence of the

second and third in 1841.

We

proceed to describe the three transformations

in question.

6*31.

Scklq/li's transformation.

We pp.

first give an abstract of the analysis used by 199—201, to prove the relation

r

Schlafli,

Ann. di Mat.

(2) v. (1873),

t ™ e-'^'coahvddB ffl*y I" tr*(fi-l)"-idtK r ' Jo ("+i) J i

which

may

arises

from a comparison of two of the integral representations of

We have, of course, at

first

to taket

v (2),

and which

to suppose that

R(z)>0 to secure

convergence, and

it is

convenient

-§
* An earlier proof is due to Kummer, Journal filr Math. xvn. much more elaborate than Sclilafli's investigation.

t

K

be established by analysis resembling that of § 2*323.

The

result is established for larger values of

R {»)

either

(1837), pp.

228

—242, but

by the theory of analytic continua-

tion or by the use of recurrence formulae.

W.

B. P.

it is

7


186

Now define

by the equation

1

and then,

;

on expanding the Replacing

if

t=x - (x -

last factor of the

x by

cosh

6,

we

(*-0 r

'

Ji

where x >

[CHAP. VI

1) «,

'

we have

integrand in powers of u and integrating term-by- term.

see that

(coah 6

Ji

-ty

'

v r($)

so that, by a partial integration,

2"

€-•«*• ooBh^cM-?!^^ ("c-'^flsinhdsinh^dd w"il^ f" r (l) jo »r(J) J

—

= —7^

tf-'w'MtfsuihdcM

/

r(l-i'), Jo

dtdx

-w=i)l'!'"^r*J/

r(i-r)Ji

(

1}

'

*(*-<)* 1}

('

r(i-«0 J, J.*

*

*

the inversion of the order of the integrations presents no great theoretical difficulty, and the transformation is established.

6*32.

The

Poisson's transformation.

direct proof that

f

I

c

-~^-„

ft_

I>+*)-(2*)"

f

r cos(*M

)rftt

due to Poisson* Journal de VEcole Polytechnique, ix. (1813), pp. 239—241. The equation true when |argz|<£»r, x>0 and R(v)> -£, but it is convenient to assume in the course of the proof that R (v) > | and arg z < ±w, and to derive the result for other values of z and v by an appeal to recurrence formulae and the theory of analytic continuation. is is

|

If

we

replace

sufficient to

by ajnew variable defined by the equation v=a**'«~ w, we see that

prove that

„.

*

t

|

,.

f coa(xu)du

See also Paoli, Mem. di Mat.

4r(A)

f

e di Fit. della Soe.

.

,

,

,,

,

,,

,

,

Italiana delle Sci. xx. (1828), p. 172.

it is


6-32, 6-33]

Now r

the expression on the left

/

I

Jo Jo

(«

s 2

)* + i +*2<»v,..n .

equal to

is

dtdu=

I

s"-i exp

I

/"OO

{exp (

I

- ««2) cos #m dv)

t»s (uP+z*) and change the order

Now

.

.

*"- J

exp ( — ss2 )

ds,

Jo

o

write

{-sfvP+z2)} cos xit.dsdu

Jo Jo /OO

when we

187

of the integrations*.

exp(— *»*) cos #u.efa=£r(£)*-i exp ( — £#*/*),

/

y o

and so we have r(v + ^ )

/o"^Spi ==ir(*

'"

,

)

IeXp{

/o"'

-^- i

^

v(2z])"J

which establishes the [Note.

result.

It is evident that

8=\xe~ tlz=\v1 lvlz. i

The only reason

for modifying

r

by taking v as a parametric variable

to obtain an integral which is ostensibly of the

is

same form as the integral actually investigated by Poisson exp

— jf - a2 .*~ B

(

)

with his notation the integral

;

is

dx.]

/;

6*33.

Malmsten's transformation.

The method employed by Malmstent

in proving that,

when R(z)>0 and

R {?)>

-i,

then

m coB(xu)du r(i)(^)- m r(»+*) - v*, * _ [ [ c {l t (**)T(J);. («*+^)" + i~ ro+*) Ji

1Y X)

- idt

ah

not so direct as the analysis of §§ 6*31, 6*32, inasmuch as it involves an appeal to the theory of linear differential equations. It is first shewn by Malmsten that the three is

expressions

qua functions of x, are annihilated J by the operator <*2

/«

^ d

*d^-V v - l) dx- XZ and that as x-*~ that

R (v) > 0.

+ oo

,

the third

is

(«•**)

9 '

first and second are bounded, provided and third expressions form a fundamental system

while the

It follows that the second

of solutions of the equation

«S- <*->£-«•*'-* * Cf.

Bromwich, Theory of Infinite Series, § 177. Svenska V. Akad. Handl. lxii. (1841), pp. 65—74. X The reader should have no difficulty in supplying a proof of

t

A'.

this.

THEORY OF BBSSBL FUNCTIONS

188

[CHAP. VI

and the first is consequently a linear combination of the second and third. In view of the unboundedness of the third as a?-^+oo, it is obvious that the first must be a constant multiple of the second so that (cos xu)

du

/; where

V is

independent of x.

To determine

2z*T(v+$)

and the required transformation for the

^cryv)

r(„)r(i)

so that

make x-*-0 and then

C,

follows,

Z**

'

when R(v)>0,

if

we use the

duplication formula

Gamma function.

An immediate

consequence of Malmsten's transformation

is

that

/

7o expressible in finite terms

;

c a ?

gtt

'

.'

-

is

(«*+82 r

for it is equal to

n-i

(2^)»»(2n-m-

(2s)«"-*(»-l)! m=0

1)

m.'(ra-ro-l)!

*

This method of evaluating the integral is simpler than a method given by Catalan, Journal de Math. v. (1840), pp. 110 114; and his investigation is not rigorous in all its stages. The transformation is discussed by Serret, Journal de Math. vnr. (1843), pp. 20,

—

21;

ix. (1844),

pp.

[Collected Papers,

6*4 Airy'8

The

i.

193—216;

see also Cayley, Journal de Math. xn. (1847), p. 236

(1889), p. 313.]

integral.

integral

cos

±

(<*

Joo

act)

dt

which appeared in the researches of Airy* "On the Intensity of Light in the neighbourhood of a Caustic" is a member of a class of integrals which are

The integral was tabulated by Airy by quadratures, but the process was excessively laborious. Later, De Morgan f obtained a series in ascending powers of ar by a process which needs justification either by Stokes' transformation (which will be explained immediately) or by expressible in terms of Bessel functions.

the use of Hardy's theory of generalised integrals^. *

Tram. Camb.

Phil. Soc. vi. (1838), pp. 379

—

402.

Airy used the form

cos |t (tc* - mw) du>,

/: but this

is easily reduced to the integral- given above. t The result was communicated to Airy on March 11, 1818

(1849), pp.

;

see Trans.

Camb. Phil. Soc. vni.

595—599.

X Quarterly Journal, xxxv. (1904), pp. 22—66

;

Trans. Camb. Phil. Soe. xxi. (1912), pp. 1

—

48.


6'4]

Stokes observed* that the integral

as!

and he

189


satisfies

± **-<>,

also obtained the asymptotic expansions of the integral for large

values of

As was

x,

both positive and negative.

observed by Stokes

to Bessel's equation

;

(foe. cit. p. 187),

§ 4*3 (5) with 2g

cf.

Bessel functions of orderst Berichte dea natur.-med.

±

this differential equation can be reduced

= 3. The expression

£ was published

first in

Vereins in dnruibruck,

Nicholson, Phil. Mag. (6) xvn. (1909), pp.

of Airy's integral in terms of a little-known paper by Wirtinger,

xxm.

(1897), pp. 7

—

15,

and

later

by

6—17.

—

Subsequently Hardy, Quarterly Journal, xli. (1910), pp. 226 240, pointed out the connexion between Airy's integral and the integrals discussed in §§ 6*21, 6*22, and he then

examined various generalisations of Airy's integral

To evaluate

we observe

Airy's integral J,

\r.

Now

exp (»*** ±

that

— 10*22).

may be written in the form

it

ixt) dt.

consider this integrand taken along two arcs of a circle of radius p with

centre at the origin, the arcs terminating at

The

(§§.10*2

1

pe** and pe**1 pe*% respectively.

p,

,

integrals along these arcs tend to zero as p

hence,

-*-

oo

by Jordan's lemma, and

,

by Cauchy's theorem, we obtain Stokes' transformation

(<*>

cos

(t*

±

xt) dt

=

~

/•

co

expert

exp (»•? ±

o

ixt) dt

Zjooexpfiri

Jd

=\

1

I* {^

Ti

exp (- t» ± «**'«T) + «-** i exp(- t 3 ± e-^xr)} dr;

Jo

the contour of the second integral consists of two rays emerging from the origin

and the third integral

is

obtained by writing re*", re**1 for

t

on these

rays.

Now,

since the resulting series are convergent, it

exp(-T» +

« ± â!T)dT=

* Trans.

2 w=0

/,

Camb. Phil. Soc.

ix. (1856), pp.

^

m

may be shewn

T*exp(-T»)<*T,

; -

166—187.

that§

.'o

[Math, and Phys. Papers, h. (1883),

pp. 329—349.] See also Stokes' letter of May 12, 1848, to Airy, Sir G. O. Stokes, 160. Scientific Correspondence, n. (Cambridge, 1907), pp. 159

Memoir and

—

+ For other occurrences of these functions, see Rayleigh, Phil. Mag. (6) xxvm. (1914), 609—619; xxx. (1915), pp. 329—338 [Scientific Papers, vi. (1920), pp. 266—275; 841—349] on stability of motion of a viscous fluid; also Weyl, Math. Ann. Lxvm. (1910), p. 267, and, for pp.

approximate formulae, § 8*43 infra. X The integral

is

convergent. Cf. Hardy,

Soc. Sci. de Bruxelles, xvi. (1892), pp. §

Bromwiob, Theory of Infinite

toe. cit. p.

150—180.

Series, § 176.

228, or de la Vallee Poussin, Ann. de la


190

[CHAP. VI

and so cos

|

(<*

Jo

+ ~

art) eft

= 2

——

'?

ml

m =o

rm exp r\(- t*) dr /

Jo

= i 5 (±x)™sin§(m + l)7r.r(£m + £)/m! o m =o

"* This

is

U-omir(m + |)

the result obtained by

De

a;

is

to be taken to be positive)

(i)

(2)

we

the series on the right

obtain the formulae (in which

due to Wirtinger and Nicholson

/;oos ( , - **> « = j, v(w

.

I/-,

d$) + a $%)]

£«.(,+*)*- w<*.>. [/., (f$H(|$)] V# v

6*5.

mt.iti!r(m + |)J

When

Morgan.

are expressed in terms of Bessel functions,

+ **

(1x\lx\

Barnes' integral representations of Bessel functions.

By using integrals of a type introduced by Pincherle* and

Mellinf, Barnes J

has obtained representations of Bessel functions which render possible an easy proof of

Kummer's formula

of § 442.

Let us consider the residue of

- T (2m - s) at s

- 2m + r, where

of the residues

is

r

= 0,

1, 2,

(— ) m zim e~ iz

.

{izy

This residue

. . . .

(—

is

r

(iar)

lm+r/r

!,

so the

sum

.

Hence, by Cauchy's theorem,

J ^ Z)e if

-

27ri\ntoJoo

the contour encloses the points 0,

2»».m\r(p + m + 1, 2,

—

It

aS l)

may be

'

verified,

by using

Stirling's formula that the integrals are convergent.

Now suppose that R (v) > — \, and choose the R (y + s) > — $. When this last condition is satisfied r ( 2 m - g> 5 .m!r(i/ + w + l) =o2 m

contour so that, on

it,

the series

2r'l

is

convergent and equal to

r <-*)

wt l.l

l..

.

1

n

* Rend, del R. Istituto Lombardo, (2) xix. (1886), pp.

Lincei, ser. 4, Rendieonti, iv. (1888), pp.

r(-«)l> + j + t) 559—562

;

Atti delta R. Accad. dei

694—700, 792—799.

—

t Mellin has given a summary of his researches, Math. Ann. lxviii. (1910), pp. 305 337. X Gamb. Phil. Tram. xx. (1908), pp. 270 279. For a bibliography of researches on inteRraU of this type, see Barnes, Proc. London Math. Soe. (2) v. (1907), pp. 59 65.

—

—

'


6*5]

by the well-known formula due to Gauss. of summation and integration* we have

If therefore

r(-«)r( y + « +

l0+)

-_<*££

t (,\ *-*

f

191

we change the

order

|).(irr
The only poles of the integrand inside the contour are we calculate the sum of the residues at these poles, we

at 0,

When

1, 2,

find that

so that

^

(1)

which

is

-« =

( ^ )e

Rummer's

_(ML- F i

In

relation.

like

+

,

(I

l

j

i; -2iz),

manner, we find that

M*) *= r £*+ x yK (» + {

(2)

+

2„

;

2"

*-,

+

l;

2iz).

These formulae, proved when B(v) > — £, are relations connecting functions of v which are analytic for all values of v, and so, by the theory of analytic continuation, they are universally true.

by

It is also possible to represent Bessel functions

exponential factor

To do

involved.

is

we

this,

integrals in which no

consider the function

r(-*-«>r (-«)(!»)«'+« qua function of s. It has poles at the points

= 0,

5

The

residue at s

—m

....

is iri

(~)m (^Y +Sm

v

sin vnr

while the residue at s

- v ,-v+l,-v + 2,

1, 2, ...;

= — v + in

'

ml r

+m+

(v

1)

is

(-)w (|s)- y+OTt

iri-"

sin V7r

'

ml T

(v

+m+

'

1)

so that

H® (z) = - 2^.

7re-4<" +1 >'rf

(3)

j

T (-

v

- s) T (- s) (îz)"^ ds,

and, in like manner,

ire»w H

(4)

v

where the contours

w ( z)

=

start from

-~lr(-v-s)r (- *) (- \izy*" ds, and return

the integrand counter-clockwise. in (4) the contours

—

oo

i.

(5)

If

we

may

When

|

to

+ oo

arg iz

\

<

after encircling the poles of \ir in (3) or arg (— iz)

reverse the directions of the contours

7re-*<" +1 "*

f" H® (z) = -U I'm J

* Cf

.

|

be opened out, so as to start from c+ °° l

we

oo i

find that

r ( _ „ _ ,) r (- *) (ii>)' +M ds,

-c-aoi

Bromwich, Theory of

Infinite Series, § 176.

\

<

\tt

and end at

;;,


192 provided that argiz

I

|

vJW H

(6)

p

< \ir and ;

o)

( z)

= -L r frm]

e+x>%

r (_ „ _ S ) r (- s) (- \%zy*~ ds,

-c-oot

provided that arg (— iz) \<\nr; and, in each integral, c |

exceeding jR

and the path of integration

(v)

[CHAP. VI

is parallel

is

any positive number

to the imaginary axis.

There is an integral resembling these which represents the function of the and the argument first kind of order v, but it converges only when It (v) > of the function

is positive.

The

integral in question is

'.»-K£,¥$g&*> and

it is

will notice that,

6*51.

By

way

obtained in the same

when

j

s

|

is

large

as the preceding integrals

on the contour, the integrand

the reader

;

is

0(\s |~* -1 )-

Barnes' representations offimctions of the third kind.

using the duplication formula for the

Gamma

function

we may

write

the results just obtained in the form

-M+j).(±2uy* ^- r r^r(+ ^r^ + s+^sin**-*

m UJ

y

(z) ** j «>rW« 2ty*rJ.

Consider

now

the integral 1

_ §*l. Ziyir

r(-8)r(-2p-8)r(p + 8 + i).(2igyd8,

I" J _aot

which the integrand differs from the integrand in (1) by a factor which is It is to be supposed temporarily that 2v is not an integer and s. that the path of integration is so drawn that the sequences of poles 0, 1, 2, ... — 2v, 1 — 2v, 2 — 2i/, ... lie on the right of the contour while the sequence of poles — v — \,— v~\,— v — \, ... lies on the left of the contour. In the first place, we shall shew that, if arg iz < f ir, the integral taken round a semi-

in

periodic in

|

\

p on the right of the imaginary axis tends to zero as p •* we have

circle of radius for, if s

v

and,

'

by 1

= pe**,

*

v

v

'

r (s) T (2v + * -f l)sin*7rsin(2i' + *)7r

Stirling's formula,

T(v + s+i).(2izy g r(s + l)JT(2i/ + s+l)

~ pe

i9

log (2iz)

- {v + pe") (log p + id) + pe ie -

^ log (2tt)

;

and the real part of this tends to — co when — ^7r<6<\ir, because the dominant term is — p cos 6 log p. When is nearly equal to ± |7r, sin sir is comparable with \ exp [pir sin 6 and the dominant term in the real part of the logarithm |

of s times the integrand

p cos 6 log 2z |

and

this tends

|

\

j)

\

is

— p sin $ arg 2iz — pcosd log p + pd sin 6 + p cos 6 - 2p7r|sin &\ arg iz < \w. to -
if*

j

j


6*51]

Hence

s times the integrand tends to zero all along the semicircle,

the integral round the semicircle tends to zero to pass

193

if

the semicircle

is

and so drawn so as

between (and not through) the poles of the integrand.

when arg iz < \tr and 2v

It follows from Cauchy's theorem that,

j

is

not an

J

integer, then

Pr

- ^rJLi^/tr j may be

(-

r (-

«)

2*

- «) r (* + « + j)

.

-oof

^y

ds

calculated by evaluating the residues at the poles on the right of the

contour.

The

residues of

r(-s)r(-2v-s)r(v+8 + i).(2izy at s

—m

and

7T

ein2inr

s

= — 2v + m

are respectively

r(v+m + $).(2iz)m m\T{2v + m + l)

_ '

v + m + \) (2iz)-»+m mlV(—2p + m + l)

tt_ r (sin2*"7r

.

and hence

_(2z£

p r(-s)r(-2v-s)T(v +

= ( 2,M r(, + i) sin 2vir I (2j/ + 1)

l

sin 2v7r

It follows that, (2)

s

+ l).(2iz)°ds

„

+

'

when

j.argiz

< f ir,

|

rn~' mir>-Mr

B.*to-

rrrt

x ["* r(-«)r(-2v-«)r(i» +

«

+

f).(2w)»
>

J -oe»

and

similarly,

when arg (— iz) < f tt, |

|

e

(3)

i*->n)

*,«(*)--

cog

(„,„.)

^2^)"

7T*

*r

r(-«)r(-2v-*)r(i/+s + ^).(-2wyrf*.

The restriction that v is not to be an integer may be removed in the usual manner by a limiting process, but the restriction that 2v must not be an odd integer cannot be removed, since then poles which must be on the right of the contour would have to coincide with poles which must be on the left.

CHAPTER

VII

ASYMPTOTIC EXPANSIONS OF BESSEL FUNCTIONS Approximate formulae for

7"1.

In Chapter

Jv {z).

various representations of Bessel functions were obtained

ill

argument z, multiplied in some by log z. These series are well adapted for numerical computation when since the series not large compared with 4 (v + 1), 4 (v + 2), 4 (v + 3),

in the form of series of ascending powers of the

cases

z1

is

.

converge

fairly rapidly for

such values of

But,

z.

converge slowly, and an inspection of their

initial

when

|

z

.

.

,

is large,

I

the series

terms affords no clue to the

approximate values of Jv (z) and Yv (z). There is one exception to this statement when v + \ is an integer which is not large, the expressions for J± v (z) in finite terms (§ 3*4) enable the functions to be calculated without difficulty. ;

The


is

the determination of formulae which render

possible the calculation of the values of a fundamental system of solutions of Bessel's equation

There are

when

really

z

is large.

two aspects of the problem to be considered

the investi-

;

when v is large is very different from the investigation when v is not The former investigation is, in every respect, of a more recondite

gation large.

character than the latter, and It must, however,

it is

postponed until Chapter

be mentioned that the

the more recondite problem was

first

step towards the solution of

made by Carlini* some years

investigation of the behaviour of

J

(x),

vm. before Poisson 's -J"

for large positive values of x,

was

published.

The formal expansion obtained by Poisson was r

z

N

/o( *)

=

/ 2

\»T

U)

,

COB [

<

g

,

,

L

1

12

.3 2

"^>-t -2l(8^

+ (

+ sin(*-i7r).j when x

is

large and positive.

.3 8 .5 2 .7 a

l2

4!(8*)' l2

)

-"j 3 2 52

^-3^ l2

IT

+

..

But, since the series on the right are not con-

vergent, and since Poisson gave no investigation of the remainders in the series, his analysis (apart

from his method of obtaining the dominant term) and ingenious rather than convincing.

is

to be regarded as suggestive

* Iticerche sulla convergenza delta serie che serva alia soluzione del problema di Keplero (Milan,

1817).

An

account of these investigations has already been given in

§ 1*4.

—

t Journal de VEcole Polytechnique, xn. (cahier 19), (1823), pp. 350 352 ; see § l 6. An investigation of Jv (x) similar to Poisson's investigation of J (x) has been constructed by Gray and

—38.

Mathews, A Treatise on Bessel Functions (London, 1895), pp. 34

-

1

ASYMPTOTIC EXPANSIONS

7-1]

195 -

be seen in the course of this chapter that Poisson's series are asymptotic has been proved by Lipschitz, Hankel, Schlafli, Weber, Stieltjes and Barnes. It will

;

this

It must be mentioned that Poisson merely indicated the law of formation of successive terms of the series without giving an explicit expression for the general term such an ;

expression was actually obtained by

W.

R. Hamilton*

The analogous formal expansion

for

J

x

(cf.

(x) is

§ 1*6).

due to Hansenf and a few ;

years later, JacobiJ obtained the more general formula which

is

now

usually

written in the form

Jn (x)~(

— |W7T — \tr)

(4n2 - I s) (4rt2

(

X

COS(#

j

,

x

4n2 -

2

2UMf

f .

-3)

,

)

(4n*

- 3 ) (4>n* - 5 2

2

)

-

(4n 2

7 2)

_

4!(8*)<

(4»2 -l 2

-sin^-^Tr-i^.j-yj^ ,

8

+ (~

(4n 2

-l 2)(4n2 -3

2

)(4n2 -5 2)

gy^^

+L

Y\ .

...JJ

These expansions for J (x) and Jx (x) were used by Hansen for purposes of numerical computation, and a comparison of the results so obtained for isolated values of x with the results obtained from the ascending series led Hansen to infer that the expansions, although not convergent, could safely be used for purposes of computation §.

Two years before the publication of Jacobi's expansion, Plana had discovered a method of transforming Parse val's integral which placed the expansion of «/„ (x) on a much more satisfactory basis!". His work was followed by the ||

researches of Lipschitz**,

tion

;

who gave the

first

rigorous investigation of the

with the aid of the theory of contour integraLipschitz also briefly indicated how his results could be applied to Jn (z).

asymptotic expansion of Jo

The general formulae

(z)

for

Jv (z)

and

Y

v (z),

where v has any assigned (commemoir by

large and complex, were obtained in the great

and z is Hankelft, written in 1868.

plex) value

* Some information concerning W. R. Hamilton's researches will be found in Sir George Gabriel Stokes, Memoir and Scientific Correspondence, i. (Cambridge, 1907), pp. 130—135. + Ermittelung der absoluten Storungen [Schriften der Stermoarte Seeburg], (Gotha, 1843),

pp. 119—123.

is

t Astr. Nach. xxvm. (1849), col. 94. [Get. Math. Werke, vn. (1891), p. 174.] obtained by making the substitutions

Jacobi's result

n+1>cosx+(- l^-^sinx, J2 cos(*- i«ir-iir) = - l)*»< (

.

^/a.8ln(*-iii«-J»)=:(-l)*" (

" +1) Bin*-(-l)*" (, '- 1, cos*,

in the form quoted.

a note by Niemoller, Zeitsehrift fiir Math, und Phys. xxv. (1880), pp. 44—48. della R. Accad. delle Sci. di Torino, (2) x. (1849), pp. 275—292. expansions of Jv («) and IV (z) by If Analysis of Plana's type was used to obtain the asymptotic JVlcMalion, Annals of Math. vin. (1894), pp. 57—61. ** Journal fur Math. lvi. (1859), pp. 189—196. § See II

Mem.

+t Math. Ann.

I.

(1869), pp.

467—501.


196

The general character of the formula

Yn (z)

for

[CHAP.

had been indicated by Lommel, Studien

Functionen (Leipzig, 1868), just before the publication of Hankel's and the researches of Weber, Math. Ann. vi. (1873), pp. 146 149 must also be


memoir

VH

;

—

mentioned.

The asymptotic expansion

K

was investigated (and proved to be this result was reproduced, with the addition of the corresponding formula for /„ (z), by Kirchhoff f and a littleknown paper by MalmstenJ also contains an investigation of the asymptotic of

asymptotic) at an early date by

¥

(z)

Kummer*

;

;

expansion of

A

K

¥ (z).

J

close study of the remainders in the asymptotic expansions of

(x),

T

(x),

I

(x)

and Kq{x) has been made by Stieltjes, Ann. Sci. de VEcole norm. sup. (3) in. (1886), pp. 233 252, and parts of his analysis have been extended by Callandreau, Bull, des Sci. Math. (2) xiv. (1890), pp. 110 114, to include functions of any integral order; while results concerning the remainders when the variables are complex have been obtained by Weber, Math. Ann. xxxvu. (1890), pp. 404—416.

—

—

The expansions have 1906, pp. 239

also been investigated

by Adamoft§, Petersburg Ann.

Inst, polyt.

—265, and by Valewink|| in a Haarlem dissertation, 1905.

«/„ (z) and Yv (z), when seem to be most simply carried out with the aid of integrals of Poisson's type. But Schlafli1[ has shewn that a large number of results are obtainable by a peculiar treatment of integrals of Bessel's type, while, more recently, Barnes** has discussed the asymptotic expansions by means of the Pincherle-Mellin integrals, involving gamma-functions, which were examined in §§ 65, 6*51.

Investigations concerning asymptotic expansions of

j

z

is

large while v

is fixed,

J

7'2.

We

Asymptotic expansions of shall

now

H^ (z) and i/„

modifications, will follow that given

Take the formula

§ 6*12 (3),

Hankel.

of the factor (1

(v-j)iu "*"

2z

|

z

; |

the analysis, apart from some slight

by Hankel ff.

namely

when -\ir
The expansion

(z) after

obtain the asymptotic expansions of the functions of the

third kind, valid for large values of

valid

(2)

— |7r + j8<

args <

|7r

+ jS,

+ $iu/z) v -l in descending

+ (,,-fr)(„-f).(^ +

270

t

provided that

powers of

z is

*"'

s

* Journal fUr Math. xvn. (1837), pp. 228—242. f Ibid, xlviii. (1854), pp. 348—376. t K. Svenska V. Akad. Handl. lxh. (1841), pp. 65—74. § See the Jahrbuch Uber die Fortschritte der Math. 1907, p. 492. Ibid. 1905, p. 828. IT Ann. di Mat. (2) ** Trans. Camb. Phil. Soc. xx. (1908), pp. 270—279. II

tt Math. Ann.

i.

(1869), pp.

491—495.

vi. (1875),

pp.

1—20.

.


7-2]

197

but since this expansion is not convergent all along the path of integration, we shall replace it by a finite number of terms plus a remainder.

For 1

\

all

+ 2z)

+ (p-l)\{2iz)

-J -^TWz) =0

It is convenient to take

any

positive angle 8

p

which

satisfies

tf

{1

;

dt

1

I

J

R {v — p — \) ^

so large that

2iz)

and we then choose

the inequalities

|arg*_(|7r + £)| $ 7r-$.

|£|^|7T-S, The

we have*

positive integral values of p,

effect of this choice is that,

when

8 is given, z is restricted so that

— 7r + 28 ^ arg z % 2?r — 28. When

the choice has been made, then ut

|arg(l-^)

>sin8,

1

<7T,

2iz

and u under consideration, and so P~i < e'l 'wi (sin 8)*<"-p-*> (i _

for the values of

t

~Y~

I

= Ap

,

j

say,

where

On

of

z.

substituting its expansion for (1

we

term,

A p is independent

+ liu/*)*"*

and integrating term-by-

find that

- (AY e**-*—*> Pi H *'w (z) w-UJ

1

(*-*>»- r ( ,> + , »+*>

U-o «ir(v + i).(Kf)-

1L»1 + jKp J'

where

where

2?p is

Hence,

p and

when R {v - p - f) <

8 which

and

is

independent of z.

R (v + J) > 0, we have

**-»— 1" Pi' ^""hi^^ H « (*) - (--Y + m!(2t«)m \irzj m=0

(1) v 7

v '

when angle

a function of v,

^ ;

is

such that

[

— it + 28 ^ arg z ^ 2tt — 28,

,

'J

8 being any positive acute

Bachmann-Landau symbol which denotes a of the order of magnitude^ of z~* as z -* oo

and the symbol

function

(#->)]

The formula

is

the

|

(1) is also

valid

when

\

R (v - p — |) >

;

this

may be

seen by

* Cf. Modem Analysis, § 5-41. The use of this form of the binomial expansion seems to be due to Graf and Gabler, Einleitung in die Theorie der BesseFschen Funktionen, i. (Bern, 1898) pp. 86—87. Cf. Whittaker, Modem Analysis (Cambridge, 1902), §161; Gibson, Proe. Edinburgh Math. Soc. xxxvm. (1920), pp. 6 9 ; and MaoBobert, ibid. pp. 10 19. •

—

t Cf.

Modem Analysis,

§ 2-1.

—

:


198

supposing that

R {v — p — J) >

R(? — q — \) <0;

and then taking an integer q so large that

the expression which

if

—p + 1

terms

it

may be

R (v + |) >

provided that

then

terms

p

sum

throughout the previous

i

§ 6*12 (4) that

Hi* (.)-£)'.-«-«--«

(2)

is

(z~ p ) or o (z~ p ) ; and the

In a similar manner (by changing the sign of work) we can deduce from

in (1)

[ ]

expressed as

(z~ p).

therefore

is

contained in

is

rewritten with q in place of p throughout, followed by q —p + 1 terms each of which is of these q

[CHAP. VII

g;
(*-,)]

and that the domain of values of z

is

,

now given

by the inequalities

If,

following Hankel,

=

2tt

we

{4>v*

+ 28 % arg z ^

tr

-

25.

write

-

I s } {4* 2

-3

2 }

.

..

{tv*

- (2m -

l)2 }

2 2m .w!

these expansions become

H,«

(4)

W = (A)*e-<«-^g |^ +

<)(*-*)]

.

o

For brevity we write these equations thus

HP (z) ~ f A) V^-w

(6)

jJ«(,)^,(A e-»- £ fc^}. y ' ' WzJy m . (2iz)m

Since §

m>

(5)

361

(7),«

(

5

-=£-.
t

(j>, m) is an even function of v, it follows from the formulae of which connect functions of the third kind of order v with the corre-

sponding functions of order — v, that the restriction that the real part of v exceeds - \ is unnecessary. So the formulae (1)— (6) are valid for all values of

v,

when z

confined to one or other of two sectors of angle just less than

is

Sir.

In the notation of generalised hypergeometric functions, the expansions are

~ (A)* ^*~-w

(7)

HP

(8)

HP (z) ~ (A)* e-uz-ir-iu)

of which (7)

is

(z)

valid

when

— tt < arg z <

.

.f, (i

.

2

f

2-tt,

o

+ „,

i

_„

(^ + v>i

and

(8)

;

_^

t

_ v _ _1_ .

when —

2-ir

y

<

arg z

< ir.


7*21]

Y

Asymptotic expansions of Jv {z), J- V (z) and

7*21.

we combine

If

199

the formulae of § 7

2,

v {z).

we deduce from the formulae of § 3*61

(which express Bessel functions of the

and second kinds

first

in terms of

functions of the third kind) that

2

/ i Tt\ / VT[coe^-W-iir).^ i s *(*)-[-)

/-n (1)

^

(-)w -(">2m)

{2gym

/ i \ v i -sin^-Ji/w-iir).^ •

n W ~(iy

(2)

sin(z — \i vtt •

/

x

.

.

(-)'*(»'.

v

" (-)m .(^,

^

'

- Sin (* + |„7T - iTT)

2w)

.(f,

(2zfm

•

m-0 i

.

+ iy

J2zjî

(-)w

i x v — $ir). 2.

/i

(

2w+l)"

2m +

1)" '

^y w+1

^.(ij^^w-wyt^-)

( 4)

and

(-)w -(^,2m

(in the case of functions of integral order

T.(.)~(

(5)

T)

n

only),

L«n(*-W-i»).jS o

(2 , )m v

+ cos(^-^7r-|7r).^ o

These formulae are

all valid for

(

^ )2m+1

J.

large values of z provided that arg z |

|

|

|

<

tt;

and the error due to stopping at any term is obviously of the order of magnitude of that term multiplied by \\z. Actually, however, this factor \\z may be replaced by ljz2 this may be seen by taking the expansions of HJ$ (z) and {z) to two terms further than the last term required in the particular v combination with which we have to deal. ;

H

(2)

As has been seen

in § 7'2, the integrals

R (v) > - \ represent H

{l)

v

(z)

and

H®

(z),

but,

which are dealt with when

when R(p)% — ^

t

the integrals

from which the asymptotic expansions are derived are those which represent {2) .ff (,) _„ (z) and -y (z). This difference in the mode of treatment of J, (z)

H

and

Y

v

{z) for

formula (1)

is

such values of v seems to have led some writers to think* that not valid unless jR

(v)

> — \.

Sheppard, Quarterly Journal, xxm. (1889), p. 223 Searle, Quarterly Journal, xxxix. The error appears to have originated from Todhunter, An Elementary Treatise on Laplace's Functions, Lamfs Functions and Bessel's Functions (London, 1875), pp. 312 313. * Cf.

(1908), p. 60.

;

—

1l 1

1

:


200

[CHAP. VII

of J (z) was obtained by Lipschitz* by interound a rectangle (indented at ± 1) -with corners at + 1 Cauchy's theorem gives at once

The asymptotic expansion grating

e***

and ±

+ oc

1

f J

(1

-

8 )-*

£

i.

eizt (1

- < )-* dt + e** 2

f °V-">*

-1

+ tu)-* du

«-* (2

.'o rct>

- «l«

e-(f+«)z

M -i (2

- tu)-* du = 0,

o

and the analysis then proceeds on the

lines already given;

but in order to

obtain asymptotic expansions of a pair of solutions of Bessel's equation

it

seems necessary to use a method which involves at some stage the loop integrals discussed in Chapter vi. It

may be

convenient to note explicitly the

involved in equations

m v (-)

.

(v,

(1)— (4); they

2m)

2

(4i/

-

(2*)*

m-o

2

2

« (-)™

mt

.

(y,

(2^)

) (4i/

2!(8*)

+

-

terms in the expansions

2

)

2

-3

s

)

2

(4y»

- 32) (4y2 - 5«) (4»/a - 7 8 ) 4!(8s)

4^ -

2m + 1) 2»»+ 1

The reader should

(4i/

initial

are as follows

8

(4g»

-

2

)

(4i^

4>

- 32) (4»» - 5 8 ) +....

3!(8s)T

l!8s

notice that

a formula given by Lommel, Studien,

p. 67.

The method by which Lommel endeavoured to obtain the asymptotic expansion of Yn (z) in his Studien, pp. 93 97, was by differentiating the expansions of J± v (z) with respect iX> v but of course it is now known that the term-by-term differentiation of an Note.

—

;

asymptotic expansion with respect to a parameter raises various theoretical

difficulties.

be noticed that Lommel's later work, Math. Ann. iv. (1871), p. 103, is free from the algebraical errors which occur in his earlier work. These errors have been enumerated by Julius, Archives Nderlandaises, xxviit. (1895), pp. 221 225. The asymptotic expansions of Jn (z) and Yn (z) have also been studied by McMahon, Ann. of Math. vni. (1894), pp. 57 61, and Kapteyn, Monatsheftefiir Math, und Phys. xiv. (1903), pp. 275—282. It should

—

—

A novel

application of these asymptotic expansions has been discovered

in recent years

:

they are of some importance in the analytic theory of the

In such investigations the dominant terms of the exThis fact combined with the the of Bessel functions that theory forms only a trivial part of consideration has made it seem merely to mention in question desirable the investigations and more by Hardy §. Wigert the recent papers the work of Voronoif and J divisors of numbers.

pansions are adequate for the purpose in view.

* Journal far Math. lvi. (1859), pp. 189—196. t Ann. Sci. de Vticale norm. sup. (3) xxi. (1904), pp. 207'—86b. Math. Kongresses in Heidelberg, 1904, pp. 241 245.

459—534

;

Verh. des Int.

—

% Acta Mathematica, xxxvn. (1914), pp. 113 § Quarterly

pp. 1—25.

Journal, xlvi.

(1915), pp.

—140.

263—283

;

Proc.

London Math.

Soc. (2) xv. (1916),

w ASYMPTOTIC EXPANSIONS

7*22] 7*22.

Stoked phenomenon.

The formula § 7-21 (1) for Jy (z) was established and arg z < 7r. If we took arg z to lie between between — tr and tt) we should consequently have |

201

for values of z

2ir (so that

lies,

|

Jv {z) = e vwi Jv (ze~ oo e v

1ci

V* T

2

/

)

-i

/

m v (-)

n

i

,-i

when

<

arg z

<

[cos <,

+ i^)j

+

v2 ;r

;

(1).

We

shall

The expansions

is

2m +

1)"

)

/.

from the expansion of

superficially quite different

now make a

close

}

2 , Vj^h V/^

JH-0

this expansion

(v,

\&ze

-sin(* + i.wr + iw) 2

721

2m )

(*.

27r,

*w~ **<(-) and

•

(-)"*•

5

x

,

m-o so that,

§

such that

arg ze~ ni

examination of this change.

of § 7*21 are derived from the formula

and throughout the sector

in

— ir < arg z <

which

2rr,

the function

HP (z) has

the asymptotic expansion

W

\irtj

The corresponding expansion

HP

for

HM(z)co(--)

(1)

{z),

namely

i

e-i«-l<>«-i*

,irz)

however, valid for the sector

is,

valid for the sector

and

-

27r

< arg z < 2tr we

HP (z) = 2 cos vtt

(2iz)m

TO -o

X m.

< arg z <

it.

("'

w)

(2iz)m

'

To obtain an expansion

use the formula of

namely

§ 3'62 (6),

HP (zer*) + e HP (ze-™), VTd

.

this gives

( 2)

iw«>~£)'.^^.sj$ VtTS/

TO _o

(2t^) TO

The expansions (1) and (2) are both valid when < &rg z < them has the asymptotic expansion

tt

]

now the

difference between

2COBV7T.

(—)'*<#*"+*« 5 Kir* J

and, on account of the factor e

iz

m-o

<7^£J2>, m (2t*)

which multiplies the series, this expression -is z is large) than the error due to stopping

of lower order of magnitude (when

j

|

„

:


202

any

at

we

,

definite

term of the expansion (1);

stop at the pth term.

occurs

when 0< arg*<

Generally

for this error is

Hence the discrepancy between

tt,

used in conjunction with

[CHAP. VII

is

its

when

(e~ iz z-P-l) (1)

and

(2),

which

only apparent, since the series in (1) has to be

remainder.

we have

where the constants c,, c2 have values which depend on the domain of values assigned to arg z. And, if arg z is continually increased (or decreased) while \z\'\% unaltered, the values of c x and c have to be changed abruptly at various 2 stages, the change in either constant being made when the function which multiplies it is negligible compared with the function multiplying the other constant. That is to say, changes in c, occur when I (z) is positive, while changes in c2 occur when I(z) is negative. It is not difficult to prove that the values to be assigned to the constants Ci

and

g2 are as follows

= £ e*p(*+iW +1) +V"\
{v

where p

is

any

= $a**"4)« c, = i**»+iM

C3

[(2p

-

1)

7T

< arg z < (2p +

[2P

7r

<

arg 2

1)

tt],

< (2p + 2) tt],

integer, positive or negative.

This phenomenon of the discontinuity of the constants was discovered by Stokes and was discussed by him in a series of papers. It is a phenomenon which is not confined to Bessel functions, and it is characteristic of integral functions which possess asymptotic expansions of a simple type*.

The fact that the constants involved

in the asymptotic expansion of the analytic function

Jv (z) are discontinuous

was discovered by Stokes in (March?) 1857, and the discovery was apparently one of those which are made at three o'clock in the morning. See Sir George Gabriel Stokes, Memoir and Scientific Correspondence, I. (Cambridge, 1907), p. 62. The papers in which Stokes published his discovery are the folio wing t: Trans. Cavib. Phil. Soc. x. (1864), pp. 106—128; xi. (1871), pp. 412 425; Acta Math. xxvi. (1902), pp. 393— 397. [Math, and Phys. Papers, iv. (1904), pp. 77— 109 ; 283—298 v. (1905), pp. 283-287.]

—

;

The

third of these seems to have been the last paper written

7*23.

Asymptotic expansions of Iv

The formula

K

v

(z)

and if

(l)

and

K

v

{z).

combined with equation

§ 7*2 (5)

(iz),

(z)

by Stokes.

§ 3*7 (8),

which connects

shews at once that

-(2z)

e

L

~n«r +

2i(8^

+

-J'

Bromwich, Theory of Infinite Series, §133. t Stokes illustrated the change witb the aid of Bessel functions whose orders are * Cf.

the latter being those associated with Airy's integral

(§ 6-4).

and ±

^


7*23, 7*24]

when arg z < |

§ tt.

\

r

the formula I„ (z)

*

\

v

— \ir < arg s <

provided that

On

,

And

(-) w ( y

»

= e* vni J

m)

r'+(,,+ *"''

=

er*"*

* V (-) ™) t t \ ^~(2,r*)i m%> (2*)» + (»>,

provided that

— |7r < arg z <

lies

— \tt and

between

(",

"0

Jv (e* ni z)

e-'-<"+*>"

|

,

(2tt^

shews that (y,

m)

to(2^r'

TO

\ir.

The apparent discrepancy between which arg z

5

<

TO

W

-*™ z) shews that

f 7r.

the other hand, the formula /„ (z)

<*\

(e

v

203

phenomenon which has just been

|7r

and

(2)

when

(3)

z has a value for

of course, an example of Stokes'

is,

investigated.

section were stated explicitly by Kummer, Journal far Math. xvn. 228—242, and Kirchhoff, Journal fur Math, xlviii. (1854), pp. 348—376, except that, in (2) and (3), the negligible second series is omitted. The object of the retention of the negligible series is to make (1) and (3) formally consistent with § 37 (6).

The formulae of this

(1837), pp.

The formulae are also given by Riemann, Ann. der Pkysik unci Chemie, (2) xcv. when v = 0. Proofs are to be found on pp. 496 498 of Hankel's memoir.

(1855),

—

p. 135,

A number of extremely interesting symbolic investigations of the formulae are to be found in Heaviside's* papers, but it is difficult to decide how valuable such researches are to be considered

A x f

remarkable memoir

cos

jo (1

when modern standards

ax dx .

+ a*) M+1 ~2»+

obtained

1

.M!

+ nC

°°

cos

Jo (1 [ ]

+ l).(2a)n- + n C. 1

i

is

.

+x

n+1

2 )

is

f

_

\2a

2 :W+1 .?i!(

to be replaced

_1

|>]

j

,

this, in

Malmsten's

means

be observed that this notation

7*24.

1

+

by (n)_ TO and

l/{(w+ l)(»+2)...(n It will

(n-rl)(n+2).(2ay-*-r...]

written symbolically in the form

ire~ a

ax dx

denoting that \n\~ m

notation,

(n

1

This formula

(cf. § 6*3). f

the

due to Malmste'n f, in which the formula

is

tre~ a

x[(2a)n is

of rigour are adopted.

is

+

m)}.

different from the notation of § 4*4.

The asymptotic expansions of her (z) and

bei(z).

From the formulae obtained in §§ 721, 7*23, the asymptotic expansions of Thomson's functions ber (z) and bei (z), and of their generalisations, may be written down without difficulty. The formulae for functions of any order have been given by Whitehead^:, but, on account of their complexity, they will not * Proc.

Royal Soc.

lii. (1893),

pp.

504—529; Electromagnetic Theory,

ii.

(London, 1899).

My

thanks are due to Dr Bromwich for bringing to my notice the results contained in the latter work. t K. Svenska V. Akad. Handl. lxii. (1841), pp. 65-74. X Quarterly Journal, xlii. (1911), pp.

329—338.

:

THEORY OF BESSBL FUNCTIONS

204 be quoted here.

The

Russell*; he found

functions of zero order had been examined previously by

it

convenient to deal with the logarithms! of the functions

of the third kind which are involved, and his formulae

ber <*) - exP a (*) cos r (,\ \bei(z)~ V(27r^)sin^

n ^ K

W

)

'

(9\ V

[CHAP. VII

J

'

_ ex P«(-^) c OS o

ker (*)

(kei(s)

may be written as follows

'

v

,

sin^

V(2*/tt)

;'

where .

.

a (z)

z ~ -r= + '

-v/2

R(

^

\

_

\

25

13

8^V2

384^ V2

128*4

7r

_

25

*

*

'

+ ""

^^~V2~8~~8772"T6V~384*V2 The ranges |

arg z

|

of validity of the formulae are

< |tt

arg^

|

|

< \ir

in the case of (1)

and

in the case of (2).

These results have been expressed in a modified form by Savidge, Phil. Mag.

(6) xix.

(1910), p. 51.

7*25.

A

Hadamard's

result

which

modification of the asymptotic expansions.

is

of considerable

Hadamard J; he has shewn

that

importance

theoretical

is

due to

possible to modify the various asymptotic

it is

expansions, so that they become convergent series together with a negligible re-

mainder term. The formulae will be stated for real values of the variables, but the reader should have no difficulty in making the modifications appropriate to complex variables.

We u,

take

first

when

the case of /„ (x)

v

> — \. When we

replace sin

\B by

we have I' ( * )=

uMml? 2

(2a?)" e*

a"' ,An'' em

exp (- 2u*x) uiv (1 - u*y~i du .

J

i>+i)r
2

(2a?)" e*

S

-I> + 1)1^)^0

{\

- v)m m!

f

1 ,

the last result being valid because the series of integrals

We may C\\ {)

„ ,~ , 2Ma?)rfW '

Wa „+2m eXp(

io is

convergent.

write this equation in the form

—

T (t\— K{a!)

S

(J

v )m

rzx

[

-r( V + l)V(i)^2x)J: ml(2xr] e*

fv+m-hf-tfif l dt e

§ (l-v)m .y(v + m + ^,2x) I> + i).ro!(2aOm

V(27r«) TOto

'

where 7 denotes the "incomplete Gamma-function" of Legendre§. * Phil. Mag. (6) xvn. (1909), pp. 531, 537. t Gf. the similar procedure due to Meissel, which will be explained in § 8-11. § Of. Modern Analysis, § 16-2. t Bull, tit la Soc. Math, de France, xxxvi. (1908), pp. 77—85.


7*25, 7*3]

For large values of

the difference between

x,

y

+n + \,

(v

2x)

r( „ + $) is

0(x v + n + i.e~ Sx ) which

205

is

ana

tt

+v >»

o(l) for each integral value of n.

In the case of the ordinary Bessel functions, we take the expression

for

the function of the third kind

*' w

H, 1+ <«>-U)T>+tfJ.' "^( s)

WxJ r(v+t)

OT

m!(2#) w

_

d»

j

so that

and

similarly

^w

(3 )

From

-( i)» r

these results

of the first

^- M

;

>

easy to derive convergent series for the functions

it is

Hadamard gave the formulae

In

)+o

and second kinds.

functions of any order exceeding

Formulae for

7*3.

i^K^ ^

§ 7*2

the

we gave an

for functions of order zero only

—\

is

;

but the extension to

obvious.

remainders in the asymptotic expansions.

investigation which shewed that the remainders in

Hw

H

® (z) are of the same order of and v magnitude as the first terms neglected. In the case of functions of the first and second kinds, it is easy to obtain a more exact and rather remarkable theorem to the effect that when v is real* and x is positive the remainders the asymptotic expansions of

v

(z)

after a certain stage in the asymptotic expansions of

numerically

less

than the

first

J± v (x) and Y± v (x)

condite investigation (§ 7*32), it same sign as the first terms neglected.

Let us write

w

2

!

i).f;--{(îr-(-r }*=^->'

+

*

We may

are

terms neglected, and, by a slightly more recan be proved that the remainders are of the

take

v^Q

without losing generality.

-


206

[CHAP. VII

so that \*

(2

[cos (x

—J

(2)

Y±v (x) = (^) yrrx/

Now I(v) —

T \vtt - {-n) P (x,

[sin (x*\vTr-±Tr)P (x, v)

+ cos (x + \mr

so large that 2j»>i/

V

+

-2 ™Y~* "

j.

2x) |

1

**)

\

^1

and, since

;

there exists a

O

is

"v

(WW

»to

(2m)!

1

obviously real,

/

we have*

JLY" 4.

UW

'

WM*

2*» •(§-*>» / (2p)I

'

- 1 ^ $ $ 1.

on integration that

It follows

P( *'* )=

— J,

V

±2^

™ Y~* + fi

where 6

v)\

B to be \ir since

<*-">» f ±!?Y" + *'•<*-">» f ±2!\* ™Y~* _ ~ TOto m! V2tW "^ (2^)! ^2**/

x

and, on adding the results combined in this formula, _,_

Q (x,

not exceeding unity in absolute value, such that (i1 I

/i

±tt)

1.

p be taken

It follows that, if 0,

=

- sin (x *\vir-\ir)Q (x, v)\

we may take

and, in the analysis of § 7*2,

the variables are real, and so A^,

number

v)

+

(2,n)!(2*)-

mto

(2p)!(2^r(, + |)J

^

*

du

>

and since " f

9

e~

u

W +2p-* cZw U

|

e~tt u»+w-* dw

= T (»/ + 2p +

P

(x, v) does not we see that the remainder after p terms in the expansion of exceed the (p + l)th term in absolute value, provided that 2p>v — \.

From

the formula (i

j.

*U

Y~*

V-2W we

manner that the remainder after p terms

find in a similar

Q (x, v) does 2p>v-§.

of

V"* _ t (*-">» f ±«Y" 4. ?iftr y W' f ±* ",* 2**/ (2p+~l)» ml \2**/ \

not exceed the (p

+

1 )th

1 '

in the expansion

term in absolute value, provided that

—

These results were given by Hankel, Math. Ann. I. (1869), pp. 491 494, and were reproduced by Gray and Mathews in their Treatise on Bessel Functions (London, 1895), p. 70, but small inaccuracies have been pointed out in both investigations by Orr, Trans. Camb. Phil.

Soc.

xvn. (1899), pp. 172—180.

In the case of

* This result (1859), pp.

K

v

(x)

was obtained 189—196.

we have the formula

in a rather different

manner by

Lipschitz, Journal filr Math. lvi.


7-31]

207

and p

+ iX' = * (-¥lJLz2!k ( u y h

(i V

p>v — \,

when

and,

mZ

2x)

ml

\2x)

may be

the last term

written

(p-D! where

<

6

l

<$

1,

and

on integration,

so,

*.(«)-

v -\(

(£)V

where This

$ #2 is

<$

when p >

1

v

(2a )"pJ'

(2ar)»

,t

f

- \.

a more exact result than those obtained for

by the same methods

why

the reason

P (x, v)

the greater exactness

and Q (x,

v)

secured

is, of course, the fact that (1 + %ut/x)"-P-l is positive and does not oscillate in sign after the manner of (1 + \iutjx) v -P-* liut/asy-P-*. ± (1 ;

is

-

7*31.

The researches of Stieltjes on

J

(x),

Y

(x)

and

K

(x).

The results of § 7 3 were put into a more precise form by Stieltjes*, who proved not only that the remainders in the asymptotic expansions associated with J (x),Y (x) and (x) are numerically less than the first terms neglected, -

K

but also that the remainders have the same sign as those terms. Stieltjes also

examined I

(x), but his result is complicated and we shall not reproduce be expected that I is intractable because in the dominant expansion have the same sign whereas in the other three asymptotic expansions the

It is only to

it+.

the terms

all

(.».•)

terms alternate in

sign.

It is evident from the definitions of § 7 '3 that

P (*' 0) = Q (X

0) '

2^r

C

e

=

2J^So

~ UX

e

In these formulae replace (1 +

U~k K 1

+ ** M) "* +

^

{(1

+ ¥url " (1 ~

£j'w) -i

by

~ UX

2

/'*»

"".'o *

Ann.

f

The function Iv (x) has

Sci. de

VEcole norm. sup.

in Taylor's

± |iw

-

** M >"*J

du

~'

du

hiu)

]

>

-

_

sin 2

'

pp. 233—252. examined by Schafheitlin, Jahresbericht der Deutschen pp. 120—129, but he appears to use Lagrange's form for the (3) in. (1886),

also been

Math.-Vereinigung, xix. (1910),

remainder

d

1

(1

theorem when

it is

inapplicable.


208

[CHAP. VII

It is then evident that

*

«,

[*"{!

-} w*sin«tf> + ... +(-)*-

1

($u*Bm*)r-1

+ (-)p ($«» sin* 4>¥/{l + \u* sin4 0)} (ty, where

p

is

Now,

any positive integer (zero included).

obviously,

l+Ksin

where 8

lies

If

and

between

\ (a + \iu)-i + (i

9

2

Jo

1

K%

Jo

^

and hence

;

- \wr*} - 1 -

~f ({uY + ^jf^- «»)• -

M the positive function e" *

we multiply by

*'

w_i and

• •

•

integrate, it is evident

that (1)

P(*.0)-l-g^ s ^,+

.

..+(-)

(2p-2)!(8a>)*-«

+ ^ rUi and jj is any positive integer result which had to be proved for P (a?, 0). where

<

0,

<

1,

i».y.y...(4p-iy (2p)!(8
(zero included);

and

this

is

the

Similarly, from the formula -

4 ,„ n ii + iiu) 4 _ (i _ itU )-»i - - j o ,

i ((1

we

find that

m (2)

n/ n\ c^,o)_-

v

.

—

i

,

la

,

+

i'-

fi"

•

3*- 5'

8! (8*)

.

r

,

"

1

K

)

f

6 *^&± ittsin9

dd>

,

l'.ff.5«...(4p-8y (2p-l)!(8«)«- 1

and p is any positive integer (zero included) and result which had to be proved for Q {x, 0). where

< 6t <

1,

In the case of

;

K

(ai),

K and replaced

(1

this is the

Stieltjes took the formula

(a)

* C/°° e-™ u-i (1 +!«)"* du,

+ $w)~ J by -

i

.

i,

•

a

method just explained, and gives again the

a

»

t îe

Proced ure then

follows the

result of § 7*3.

By an ingenious device, Callandreau* succeeded in applying the result of Stieltjes to obtain the corresponding results for functions of any integral order ; but we shall now explain a method which is effective in obtaining the precise results for functions of any real order. * Bull, det Sci.

Math.

(2) xiv. (1890),

pp. 110—114.

"

f


7-32] 7*32.

J

with

v

the remainders in the asymptotic expansions associated

The signs of

(x)

and

Y

v

209

(x).

been seen that Jv (x) and Yv (x) are expressible in terms of two functions P(x, v) and Q{x, v) which have asymptotic expansions of a simpler type. We shall now extend the result of Stieltjes (§ 731) so as to shew that for any real value* of the order v, the remainder after p terms of the expansion of P (x, v) is of the same sign as (in addition to being numerically It has already

p

that 2p

>

- \:

v

a corresponding result

which these conditions Q enable the theorem to be stated in the following manner:

holds for

on

+ l)th term provided The v) when 2p > v - f

thanf) the (p

less

In

(x,

.

the oscillatory parts

restrictions

P {x, v) and

of the series for

Q {x,

v),

the

lay

remainders

are of the same sign as, and numerically less than, the first terms neglected.

By

a slight modification of the formulae of

p {x

v)

>

=

Q{x, p)=

2T^ry)/

" e~ ux " v-i

2iT^Y) \1

and, exactly as in § 7 3

e

~ ux

"""' {{l

we may shew

;

IRi + nay-* + (i - iiuy-i]

Ji

l

ur~7

)P(i

(2p~2)?

(1

* iu)v

~h

+ ** M) ""*

we have

+(1 "

i

iu)v

~ i]

du

'

~ * iuy ~ i] du

(1

'

that

trAtzîtr \&ni).

wi=0 (

+

{{1

§ 7 3,

- typ

"

o

2

¥

{(1

+

^y^ - & - wr**)

dt.

The reader will see that we can establish the theorem if we can prove that, when 2p > v — \. the last term on the right is of fixed sign and its sign is that of

It is clearly sufficient to 1

[

shew that

\\ _ t?P-* U

— v — $] positive. Now this

+ \%vty-** - (1 - liuty-v+l)

{(1

dt

2p

is

expression

is

equal to J

1 f (1 - 1)**-* ii \w-*-i \e~ K {i+i * Jo (2p-v-$)T(2p-v-$)] K ' -

.

=

=r7K-— 1

^

I "( 1

f {2p-v + $)JoJ 9

l

iut)

- e~ x

As

in § 7-3

we may take

vÔ

2

'

.

e~x d\dt

sin ($\ut)dtd\.

without loss of generality,

t This has already been proved in § 7*3. t Since isin^Xwt) |^$Xuf, the condition infinite integral.

- i M° 1 d\ dt

- m-2 \v->-* sin ($ Xut)

= --_—^ ,- f"\*--»^ P(l -t)*1 {2p — v + $)Jo Jo *

<1

2p>»-b

secures the absolute convergence of the


210

Now

(1

— t)w~*

a monotonic decreasing function of

is

second mean- value theorem, a number 2?-*

|(1-

t)

between

f,

sin ($\ut) dt

=

Jo is positive,

1 (1 C ~ O^"*** K

2F-T^Ti

VH

t;

and hence, by the

1,

exists such that

( sin (l\ut) dt

> 0.

Jo

T (2p — v + £)

Since

and

[CHAP.

we have succeeded

W-*

+

+

*

in transforming

- (1 " ImO'-*^} dt

into an infinite integral in which the integrand is positive, and so the expression

under consideration

That

is positive.

is to say,

i{(i

+ 4»«y-*+(i-iwy-»} " mto

where

(2wi)!

0^0 when 2p> v — \. And

these circumstances

6 ^

|

(2p)!

has already been seen

it

^

Consequently

1.

|

the last equation by e~ux u v ~* and integrating, stated for

that in

obtain the property

P (x, v).

The corresponding property for Q (x, \.

(§ 7*3)

and then, on multiplying

^1 we at once ;

«i + 1 »r. - (i

_ }»)-*) =

-g

v) follows

from the equation

<->"-^ -;^ft">""

o

(

the details of the analysis will easily be supplied by the reader. Note.

(l±$m) r -*

The lies

when

analysis fails

same sign as unity, and, in \{v-\)u, and hence P{x,

like v)

if

we take p=0, but then the phase \

{{l+\iu) v -\ + (\ -iiw)"

*}

of

has the

manner, £ {(1 + %iu) v ~h — (1 - \iu) v ~ i}/t has the same sign as $(j-, v) have the same sign as the first terms in their

and

expansions, so the conclusions are

7*33.

—\
and ±\{v-\)ir, and so

between

still

true

;

and the conclusion

is

true for

Q (x,

v)

when

Weber's formulae for the remainders in the expansions of functions

of the third kind.

Some

inequalities which are satisfied

expansions of H,

and

H

by the remainders

in the asymptotic

have been given by Weber*. These inequalities owe their importance to the fact that they are true whether z and v are real or complex. In the investigations which we shall give it will be supposed for ll)

simplicity that v

(z)

is real,

only are adequate to

¥

(2

>

(z)

though

it will

make the mode

pe obvious that modifications of detail

of analysis applicable to complex values

» Math. Ann. xxxvn. (l|890), pp.

404—416.

+

,,


7-33]

of

There

v.

We

:

no further

is

= r,

shall write \z\

sideration,

If v

we

assuming that vÔ, R(z)^0. are primarily under con-

and, since large values of z j

suppose that

shall

- \ > 0, we

loss of generality in

have*, by §

|

1r^v — \.

612 (3),

j(iy y- -'-'.r e

«y-'

,

2\i e'G-W-1")

w r(- +

\TTz)

;iy If

^ v<

£,

we use

-.,

M

+

>->f 1

/ *oxp {- »

d„

- ^*)} «-• A*

(l

-

i)

_^p

(l j.


//« (z) = (2/*) („ +

1)

and apply the inequality just obtained It is thus

211

^

(*)

- f£>, (z)

to each of the functions

on the right.

found that

HM (z)^G\ (%Trz)-i

(1)

e*(»-W-

W

|

and similarly

H® (z)^0\ ($ir*)-*e-*fr-*"«-W

(2)

|

where

(3)

("
The

may be called Weber's crude inequalities satisfied by Hv (z) By an elegant piece of analysis, Weber succeeded in deducing

results

H

{1)

and v ® (z). more refined inequalities from them in the following manner

Take the first p terms of the series involved in Hankel's two expansions and denote them by the symbols 2„ (z p), 2„ 2 (z p), so that '

(1)

(

'

;

2

u>

(z

•

v)

V

_

(-)m -(v>™)

;

2 m

(z

.

o)

_

V (W)

m=0 It is easy to verify that

Yd dz*

We

+

Z%

dz

+ z>

z " {z,p)

-z*{-2izy-i

regard this as an equation to be solved by the method of variation of para-

meters; we thus find that

V

(z;p) = {\ttz)* e -^-w-H |4 (^ jyr

* In the third line of analysis the inequality e x

t When !»<£ we take 2r>y + $.

,i,

^+B ^^

^ 1 + x (xÔ)

(

(2)

^

has been used,

o


212

A (z)

where

B (z) are functions

and

(A' (z) H v v

Vn

of z so chosen that

+ B' (z) H® (z) « 0,

(z)

|4' (z) ±H."

[CHAP.

+ B (z)± H.»

(z)

(z)

= - (*x#)-* e^—M

It follows that

A' (z)

£^

= *» <*«)r* *<-»— W

Jfr «

/^^

.

(,),

and so il (#)

- 4 - 1 Tr/^ {£tT (5 + t)}-> 6«**-*~-W

A is a constant. We obtain a similar expression

P

H » (Z + 9

h 2 :^'f

t) dt,

t)]p

where

2, m

(*

;

p)

- {-4#„

(1

(*)

>

+ ##,

for

B (z), and hence

-**P-toP)J. \^t)

By considering is

not

that

A=l

Hu y

(z)

H,»

(z)

Sinee

{V

'

P)

R (z) ^ 0,

inequalities,

we

(

—

= (~\

Rp w may j,

+ oo

,

it

in the forms

e -«*- W-i*> {2„<*>

+ Rp

(z;p) +

<

Rp %

be defined by the equation

\JTV

we have

-*-

B = 0.

and

J*-*"-**) {2 r « (z; p)

)

\ttz)

where the remainder '

=

*

F*F+0|5

Hence we may write Hankel's formulae

" ***

follows that

the behaviour of both sides of this equation as z

difficult to see

^

it

-'(*-*--i') (*)} (£**)* e

,2)

\z

\-2i(z

+ t\^ ^/(r2 + t'), and

+ t)}* so,

dt

by using the crude

see that. the modulus of the last integrand does not exceed 2*-p 7T"1

Q3

.

(r3

+ t*)-* lp+1)

.

Hence |

Rp w

|

^ 2 1-* G*p

(v, |

p)

|

f°

(t*

+ «»)-*
.'

and

so,

when p >

(4)

1,

we have

IVI<»l(y)l^f|^r

and similarly

W

I-«p

K^w

K

,p;i

r(Jp + J).|(2*)»r and it will be observed that in Weber These are the results obtained by the relative values of v concerning made the analysis no hypothesis has been results obtained by from the differ and p\ in this respect Weber's results ;

other writers.


7'34] 7*34.

Approximations

When

the argument of a Bessel function

expansion

term in

remainders in

to

213

the asymptotic expansions.

is not very large*, the asymptotic not well adapted for numerical computation because the smallest (with the remainder after the smallest term) is not particularly

is

it

may

small; at the same time the argument

be sufficiently large

for

the

ascending series to converge very slowly.

An by

ingenious method for meeting these numerical difficulties was devised

we

Stieltjesf;

shall explain the

method

in detail as applied to the function

K (x) and state the results which were obtained by Stieltjes by applying the method

We

J

to

Y

and

(x)

(x).

apply the transformation indicated in

§ 7*31 to

the formula § 6"15 (4),

so that

v

\- e

i

~x

~mi du

r

e

e~ xu dOdu _ ejV2 f * J** ~ it J Jo m*(! + |wsin2 0)

V

-

*-?-(lu V

2 m = oJo

7T

M*

JO

*

Bin* 0)' m

n"^"(-i-^^ds*."

JO

That

is

to say,

at f"

f*

D Rp=~i\

t'.'o

when x

the value of is

large

;

m for which

is

,

,

m)j(2x) m

(0,

,

K

.

a

0)

is

(x),

we choose p

x=

hp +

numerically less than unity

Rp=-i

tt*Jo Jo

least is nearly equal to

_1

2x

;

so that

and then

2 (1 ««-*" sin 0)p

.

..,

m*(1

-

+\$usm :

-

2

m d0dn.

0)

Now, as u increases from to qo \ue~* H increases from (when u = 2) and then decreases to zero so we write ,

e

0)

r~-^rd0du. 2

+ £usm

w*(l

2

accordingly, in order to consider the remainder after the

smallest term of the series for

where a

—

e~ xu (hi sin8 0)*

w,

.'o

+ |usin

«*(1

.'( 'o

where

Now

d0dn

$««-*«

where £ increases from

up

to a

maximum

;

—x

to oo

,

= e~

l

~S\

and, for similar reasons,

sin 2

we

write

(x),

Y

= e—f.

* The range of values of .r under contemplation for the functions from about 4 to about 10. t Ann. Sci. de VEcole norm. sup. (3) in. (1886), pp. 241—252.

J

(x)

and

A' (x) is


214

The domain

of integration becomes the whole of the

[CHAP. VII

(f, rj)

plane

;

and

it is

found that

2f^r

R

7T*

f-

e

pM)

-

J -oo J -oo

{

l

a r>s

? v>}dtdV

lr%»=0

,

)

where a o,o

= i»

«2,o

=

by some rather tedious arithmetic. the asymptotic expansion of

2o- 2

-

ao,2

^j,

It follows* that the

Rp for large values

of

p

= i>

dominant terms of

are

so that

It is easy to verify

by

Stirling's

theorem that

(2xy

K

r

'

W

p

'

so that the error due to stopping at one of the smallest terms

is

roughly half of the

first

term omitted.

In

like

manner

Stieltjes

proved that,

if

P (x, 0) and Q (x, 0) are defined as

in § 7-3, then

{>

(3)

}

~Jlo

(2*)™

K

r

p

'

where (5)

(6)

provided that

be x

p

is

chosen so as to be nearly equal to

and t

x,

is

defined to

— p.

Results of this character are useful for tabulating Bessel functions in the critical range;

some similar formulae have been actually used for that purpose by Airey, Archiv der Math, und Phys. (3) xx. (1913), pp. 240— 244-; (3) XXII. (1914), pp. 30 43 and British Association

—

;

Reports, 1913, 1914. It would be of some interest to extend the results, which Stieltjes has established for Bessel functions of zero order (as well as for the logarithmic integral and some other

functions), to Bessel functions of arbitrary order. * Cf.

Bromwich, Theory of

proved in §

8*3.

Infinite Series, §§ 133, 137,

and 174, or the lemma which

will

be

.


7*35, 7'4]

Deductions from Schafheitlin's integrals.

7'35. If

we

replace a by 2 tan

p/

.

n,

in the formulae of §

(2x) v+ i

+

(v

1

(v

sin-* 6 cos (v

f*»

i)

+

o

sin"-*

[**

f).

- V) 6

t

sin

(i;

- 4)

.

„

,

,„

cos 2 " +1

o

which resemble Schafheitlin's integrals of It is obvious

732, we deduce that

cos 2 " +1 6

]

(2x) v+ l

,

A

An

215

612.

§

from these results that

P (X,

V)>0,

Q (*,

>

(_£<„<

3)

o,

(!<"
Q(ar,i')<0.

(~\
*)

interesting consequence of these results

is

that

we can prove

that

Q(x,v)/P(x,v) is

-

an increasing function of x when

function of #

when

|

<

£ < y < £ and that

it is

a decreasing

i>< f

For we have

Q'(x,v)P(x,v)-P'(x,p)Q(x,v) (2x) v+ l

(

2 )

ft* ft*

where

F (e >

*>

=

£l^£^ (cos0cos<£)

:

<

tan *

- tan *> c ° s <* -

e sin ("

*>

-

*)

<*>>

so that

!(», *)

+ *<*

0)

=

g=|^|^

(tan «

-

tan

sin (i

-,,)(«-

*).

If we interchange the parametric variables 0, $ in the double integral and add the results so obtained we see that, when — | < v < f the double integral has the same sign as \ - v and this proves the result. ,

;

7"4.

Schlafli's investigation

of th e asymptotic expansions of Bessel functions.

In a memoir which seems hardly to have received the recognition which importance deserves, Schlafli* has given a very elegant but somewhat

its

elaborate investigation of the asymptotic expansions of the functions of the third kind.

The

integral

formulae from

which he derived these expansions are although Bessel's integral is not so well

generalisations of Bessel's integral

adapted as Poisson's integral *

Ann. di Mat.

(2) vi. (1875), pp.

the importance of this

memoir

is

for

;

constructing the asymptotic expansion of

1—20. The only standard work on Bessel functions is the treatise by Graf and Gubler.

recognised

in

which


216

J

v

(z)

when

z

and

large

is

v is fixed, yet Schlafli's

in obtaining the expansion, but also

[OHAP.

VH

method not only succeeds

expresses the remainders in a neat and

it

compact form. procedure consisted in taking integrals of the type

Schlafli's

and selecting the contour of integration in such a way Ire* (u is constant.

is

first

positive

negative,

phase* of

+ l/«)

and

;

it is

supposed that r

take the phase to be

and 6

is real,

=

1

and

is

positive

is

ir

and a

is real.

write

;

+ pe a

,

and then

reia p*e2i0 l(l is

the

took two contours, the constants for the respective contours

u

where p

it,

it

Let us

(I)

2

on

He

and

being

-

that,

consequently equal to

+ pe *) 1

conjugate complex.

its

Hence we have sin(a

(1) {L)

P

+ 2fl)

sinfl

U

sm(a + 6)'

Next choose a new parametric

sin(a

variable

+ 0)

e

such that

= 20 + a - 7T,

and then

/^

cos \ (a T7 cos |(a

w=

' ( v2)

-
.

,,

re

TT e *.
(u

- l) u

- r sin JX cos i/ a ~ 9) f( + 2

a

~=

T7(a cos |

,

.x

-

9)

— (ir — a) to (ir — a), u traces out a contour emerging from the origin at an angle — {ir — a) with the positive real axis and passing to < a < 2v. infinity at an angle (ir — a) with the positive real axis, provided that Now,

as

varies from

on a the contour passes to

If this restriction is not laid

infinity

more

than once.

We

shall

now lay

this restriction

on a

;

and then the contour is of the type we give a> and arg z the same

specified for formula § 6*22 (9), provided that

value

a,

as is permissible.

It follows that

2i sin vrr

where u *

is

The reader

"method Chapter

^bC'-'M-DI-J*

defined in terms of

will find it interesting to

of steepest descents" which

viii.

by equation

(2).

compare the general methods of

is

this section with the

applied to obtain various asymptotic expansions in


7*4]

217

Changing the sign of is equivalent to replacing u by 1/u, and so, replacing the expression on the left by its value as a function of the third kind, we have

e*"«"

(8)

From as

H® (re'>-**>) = — TV"' + .

(2) it follows that

i*")

t is

when u

positive

du

is

iy/u

=

+ 1)1

^~-

d.

to

+

oo

on the contour, we have dt

- reu (w - 1/u) ~ e -t(T-«)i ( wi + u-*) *J{rt)

~u

.

t,

dt

=

\reu (u

;

- re u (u so that

j

— re* (u — Yf/u increases steadily* from to it — a and, if we write

varies monotonically from

<£

exp

the range of values of arg w being less than

'

ir.

Next, by Cauchy's theorem,

it is

convenient to take the point £ = 1 inside the contour, but f = is a branch-point.

must be

outside the contour because the origin It follows that

n. +u-. = <*+j£

( 4)

--

1

f

™ ,jr?(.t-}W„

Hence

W Now

'"

^

'

it is

27r»irM"

Jo

(f-iy +

J

W^)

'

evident that p- 1

1

(-)»»

g>»

$»

(-)p

ft>

tP

" w = (C- l)""* (r+r + (£- l)v (re-)P {(?- 1)' + #/(re*)} (f - 1)' + #/(re*) where

p

is

any

positive integer (zero included).

quently to suppose that

On making

j_

,

1+

^

*

(with the notation of § 7

H.»

and

(«•<-.-.)

-

2),

a*

;

we deduce

R (— v — $). and observing that

I> + m + i)

,

2-Tri]

(6)

exceeds both

It will be convenient subse-

this substitution in the last integrand

h«+ w«+ ,

p

R(p — £)

I>-m + £).(2m!)

= m\(v,m) (2m)!

*

that

= (^)*exp {nf -

I

™) [£ <=£^> + V>]

where

"*

2mV(2V); 27rtV(27r)io d

, „.

dtp cos

W.

B. F.

-* £"+p-*

/•»/•(«+, l/w+.n-)

(_)p

(£- V)^ (re¥{(r1

J i sin 2

^

a + cos ft

_ ~

sin

ft

(1

d£dt

1)'

+ ?*7(™ia )}

+ 2 coa a cob ft + cos" ft) (cos a + cos ft) 2 *

,


218

[CHAP.Vn

First consider £" +*>-*dZ

r(«+,i/«+,i+)

I

(c-iy^-M^-iy+w^M is so large that it exceeds both R (v — £) and R{—v — ^), we take the ton]

When p

contour to be as shewn in Fig. 15; and

tend to

circles

If

now we

oo

and

when the

radii of the large

respectively the integrals along

and small

them tend

to zero.

write

£=e ±,ri (l-a;)Joo on the two rays (which are 1

r («+, l/t*+,

i+)

ton)

all

that survives of the contour),

£»+p-i

we

find that

d£

(£-l)*p->{(f_l)*+#/(re*)} (-)* cos

vir

1

f

aP—-* (1

- xY+*-*dot;

tx(l-x)/(reia )

'

Fig. 15.

Now

the numerator of the integrand is positive (when v is real), and the modulus of the denominator is never less than 1 when \nr< a< § tr\ for other values of a

never less than

it is

|

sin

a

]

Therefore

where is

|

O

|

complex, (7)

is 1

or

it is

|

cosec a

|

according as cos a

is

negative or positive.

easy to see that

Vk

cosi/tt

cos

R

(i/tt)

I

o

\(R(v),p) |

(2r)*>

When

v

,

,


7'4]

Hence,

*.<,

(8)

where

|

0j

when

finally,

— £ir < arg z < § w,

$v~ [s fcg^>

w.

+ * fcggtfj

>

does not exceed 1 or sec (arg s)

|

219

|

negative, provided that v is real

and p + $ >

according as I(z)

|

v

|

|.

is

positive or

When v is complex, the modi-

form of the remainder given by (7) has to be used.

fied

Since

has

R{1 -te(l - x)jfrd*j\ > when ^(e^Ô, we part positive when v is real and l(z)^0.

see that, in (8),

X

its real

If z be replaced

when

and,

The

by

we

iz in (8)

when arg z\
find that,

|

v is real, (i)

i2(0,)>Oand|08 j0,

(ii)

|0,|<|cosec(args)|,ifl*(s)
modifications necessary for complex values of v are left to the reader.

(II)

We

next discuss the consequences of taking the phase of

ire<-(M-2

As

to be zero.

before,

we

+ l/tt)

write

u=

1

+ pe**,

and then reu />9 €**/(! +pei9) is positive, and complex, so that we obtain anew equation

therefore equal to its conjugate (1).

We

then diverge from the

preceding analysis by writing
= -(20 + a)

so that

tt-" *^**) sin£(a-0) 11

(10)

Now,

as

—a

rCfa

(«-iy = u

'

to

a,

u

—a

with the positive real

rsin'ft

sin£(a-«£)sin£(a +

emerging from the and passing to infinity at an provided that a lies between — ir and it.

axis,

real axis

The contour is then of the type specified for formula missible, we give a> and argz the same value a. It follows that,

K

¥

where u (11)

is

when — tr < a <

(re**)

= £cos vir

j

H

¥

<®

(/*'<-*»>)

§ 6'22 (8)

if,

as

is

tt,

u-"' 1 exp 1

defined as a function of

4>)'

traces out a contour

an angle a with the positive

origin at

angle

varies from

^

- ^re* (u +

by equation (10)

;

-) f

and therefore

= 1-^1T( m-k + u») exp |- Ire * (u + 1

jt d,

d log u 1)

,

per-


220

and hence,

if

now t

we

[CHAP. VII

= reu (u-lflu,

find that

We

have consequently expressed a second solution of Bessel's equation in a and the analysis its asymptotic expansion can be deduced (z), the final result being that, when proceeds as in the case of v form from which

;

H

{1)

- §tt < &rg z <

H® 0) = (—}

(13)

e-*

fa, pz 1

<*-*'*-*»>

(m»)

L«-o(2i*)"

-

(v,p)

,

(2**)M

argz according as I{z) ^ or I(z) > 0, and .ft (02 ) ^ when I(z) ^ 0. If v is provided that v is real and p + 1 > u complex the form of the remainder has to be modified, just as in the case of (8). where

2

|

|

does not exceed 1 or

|

sec |

j

;

|

It should be observed that, since the integrands in (3)

functions of

and (11) are even

unnecessary in this investigation to suppose that R (v) as was necessary in investigations based on integrals tof

v, it is

must exceed — |

,

Poisson's type.

Barnes' investigation * of asymptotic expansions of Bessel functions.

7'5.

The asymptotic expansions of functions of the third kind follow immediately from Barnes' formulae which were obtained in §§6*5, 651. Let us consider " t " '~p r (- s) r (- 2v - s) r ( v + s + j) (- iizy ds ,

/.

ao

i—v—p

= (- 2u)-*-*

["'

T(-s + p+p) r(-s-v +p) V(s-p + 1) (- 2iz)°ds.

J -TCi

If arg |

(— iz) ^ f it —

8,

|

we have

r(-s+v+p)r(-s-v+p)r(s-p + i)(-2izyds\ I/: 1

< j"

\r(-s + v + p)r(-s-p+p)r(s-p + $)el*- s)

'

ds\,

l

J -coi

and the

last integral is

convergent and so the

first

integral of all

is

0{(-2i*)-'-p}. But, by the arguments of

§ 6'51,

the

first

integral

is

— 2iri

times the

of the residues at the poles on the right of the contour, and so

— 7T- Hv w {z)j\ei(z~ v " s = - p — £,— p — f,

)

(-)"'

r (p + m + £) T(- p + m + $) m\(-2izy +m+ *

* Trans.

sum

equal to

— 27ri times the sum of the residues — p— p + \. The residue at — v — m — \ is

cos pit (2z) v ] plus ...,

it is

Camb. Phil. Soc. xx.

(1908), pp.

273—279.

at


7'5, 7*51]

and

221

when arg (— iz) < \ir — $

so,

j

|

M

(

- 1Fr(,

:y£S +

' ti)

and

this is equivalent to the result obtained in §7*2.

HJ®

(z)

may be

constructed by replacing

i

^«-H

The

investigation of

by - i throughout.

The reader should notice that, although the determination of the order of magnitude of the remainders by this method is transparently simple, it is not possible to obtain concrete formulae, concerning the magnitude and the sign of the remainders, which are ultimately supplied

by the methods which have

been previously considered.

7'51.

Asymptotic expansions of products of Bessel functions.

It does not seem possible to obtain asymptotic expansions of the four products J±n(z) J±,(z) in which the coefficients have simple forms, even

when ifM

<2>

fi

(z)

= v. The

H^ (z)

exists for the general

term

;

H^

H

(z) ¥ ® (z) and which no simple expression the leading terms in the two expansions are

reason for this

is

that the products

have asymptotic expansions

2 e±g**ri0»+>+i).ri trz

t

for

2/^+2^-1 ...}.

4dz

1

The productsiTM « (*)HW * (z) and J? M « (z)H^(z), however, do possess simple asymptotic expansions ; and from them

we can deduce asymptotic

expansions

for

j;(i)/P (#)+r,(i)F,(i) and

J, (z)

for

The

simplest

Y

¥

(z)

- FM (z) J¥ (z).

way of constructing the expansions

just explained in § § 541 suggests that

7*5.

A

is

by Barnes' method,

consideration of series of the type obtained in

we should examine

the integral

is to be chosen so that the poles of T (2s + 1) lie on the left of the contour and the poles of the other four Gamma functions lie on the right of the contour and it is temporarily supposed that /*, v and fi±v are not

the contour

;

integers, so that the integrand has

provided that

|

arg(t>)

|

< f 7r.

no double

poles.

The

integral is convergent

THBOBY OF BESSEL FUNCTIONS

222

First evaluate the integral

sequences of poles which is

VH

to enclose the

to the right of the original contour ; the expression

lie

sum

equal to minus the

by swinging round the contour

[CHAP.

of the residues at these poles, and the residue at

m + JO* + v) is r (fi + v + 2m + 1) (-)m (^zy+* +am ) sin (/a + v)ir' m\T + m + 1) T (v + ra + 1 ) T (ft + v + m + 1

g»gH»+ww ~~

sin fiir .sin vtr

.

(jj.

.

It follows that

^D<^ +1 r (H-H r >

ft

(

?--)

xr (^- a r (-^-^) ( )

~ sin fiir sin pit \

sin (ji

J¥ (z)

e*"-»>- J_^ (z)

sin (^

=

)s,
+ v) tr

sin

sin

(/*

- v) ir

(ji

g-»(M+")X JL (*) /_, M

— fi) w

sin (v

^

(,g) | )

+ v) ir

+ v) 7r sin (/* — i>) ir

2lnT^ ^ C

(^

)J' (^)+

^

(^

)F ' (

^

- cot | (/* - *) ir {/M (*) Fr (s) - FM (z) J¥ (*)}] "*•

.[{•/„(*)/,(*)+ r,(*)F.Cr)}

2 COS J

(/LI

+ V) IT

+ \,*&\(r-v)Tr{J {z)Y¥ {z)-Y

tl

ti

By |

writing

—i

for i

throughout the analysis we deduce that, less than $ w, i.e. if arg z\
arg iz and arg (— iz) are \

|

|

- COt J (/. " ") »

x

{z)J¥ (z))l

|

•

(J» (*)

F, (*) - 1 , (*) J. Wl]

r (£=*-•) r(-^-*)cos*7r.(|*)»
if

both


7*51]

223

and (2)

»-

M . t » +>))r [|J">WJ.W+F,(,)r.(.)» + tan i 0» - ") »

•

[J* (') Y. (*) -

F„ (») /, (*))]

-BD^êr-'-) "^-) 1

xr(»J- s) r (-^-») sin, w .(i,)-<&. These results hold for all values of p and v (whether integers or not) provided that, in the case of the former p + v and p — p are not even integers, and. in the case of the latter p + v and p — p are not odd integers.

(1)

We

now

and

(2) after the

We first

obtain the asymptotic expansions of the functions on the

take

tegrands on the

p

left

manner of §

to be

left

of

7*5.

an integer so large that the only poles of the in= -p-\ are poles of Y (2s + 1) ; and then

of the line R(s) /oof

raoi-p—l

-x*

J

-xj — p— J

(when either integrand is inserted) is equal to 2wt times the sum of the residues at the poles between the contours. Since , x.-p-J J

we deduce (8) .

[

/*

-xt-p-J

^ ^ ^ ^ ^_^

that the asymptotic expansions,

when arg z <

Jh (z) Jv (z) + F„ (z) F, (#)] - cot \{p - v) 7T

^

+»

i

j

.

ir,

are

[J, (,) F„ Cr) -

J

v

(z)

F„ (*)]

7T»

"

.«=o

ipi-rini^-Oir-^l

(2m +

+1

2

'

1) !($*)*»+»

+^ -§- +

2

1,

1

g"

2'"W

f

and

^

2

~7rzcosi(/*-v)7r"

4

At +

|/

\

2

2

+1

/A-I/

2

+ 1 I/-/A+1 2

1-/A-1/ 2

1 '

2'

_1\ ?/*

THEOEY OP BESSEL FUNCTIONS

224

In the special case when fi—v, the

last

[CHAP. VII

formula reduces to

JS(*)+7S(s)~2- 5 {1.3.5...(2m-l)}^,

(5)

and, in particular,

Jo

(6)

Formula

(5)

(,)+I.

*

(2m)l(&)-

<*)~~J& o

seems to have been discovered by Lorenz, K. Danske Vidensk, Sehkabs

Skrifter, (6) vi. (1890).

[Oeuvres scientifcpies,

I.

(1898), p. 435], while the

more general

were stated by Orr, Proc. Camb. Phil. Soc. x. (1900), p. 99. A proof of (5) which depends on transformations of repeated integrals was given by Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), pp. 245 247 the expansion (5) formulae

(3)

and

(4)

—

is,

pp.

;

however, attributed to Walter Gregory by A. Lodge, British Association Report, 1906,

494—498.

It is not easy to estimate exactly the magnitude or the sign of the remainder after any number of terms in these asymptotic expansions when this

method of J,2

(z)

is

+

used.

An

alternative

method of obtaining the asymptotic expansion § 13'75, and it will then be possible to form

F„2 (z) will be given in

such an estimate.

—

CHAPTER

VIII

BESSEL FUNCTIONS OF LARGE ORDER 8*1.

Bessel functions of large order.

The subject of this chapter is the investigation of descriptive properties, including approximate formulae, complete asymptotic expansions, and inequalities of various types connected with Bessel

functions and the probe examined are of primary importance when the orders of the functions concerned are large, though niany of the results happen to be, true

which

perties

;

will

for functions of all positive orders.

We

shall first obtain results

which are of a purely formal character,

associated with Carlini's formula (§1*4).

Next, we shall obtain certain

approximate formulae with the aid of Kelvin's* "principle of stationary phase." And finally, we shall examine the contour integrals discovered by

Debyef

;

these will be employed firstly to obtain asymptotic expansions

the variables concerned are

secondly, to obtain

real,

numerous

when

inequalities of

varying degrees of importance, and thirdly, to obtain asymptotic expansions of Bessel functions in which both the order and the argument are complex.

In dealing with the function J¥ (as), in wfiich v and x are positive, it is found that the problems under consideration have to be divided into three classes,

according as xjv

is less

than, nearly equal to, or greater than unity.

Similar sub-divisions also have to be

made

in the corresponding theorems

concerned with complex variables.

The and

trivial

z is fixed,

problem of determining the asymptotic expansion of J,

may be noticed

It is evident,

by applying

Stirling's

theorem to the expansion of

§ 3*1,

^(*)~exp{v+vlog(^)-(v+|)logv}.[co+^+^ + where

c

— 1/J(2n)',

(«),

when

v is large

here.

this result has been pointed out

that

...l,

by Horn, Math. Ann.

lii.

(1899),

p. 369.

For physical applications of approximate formulae for functions of large order, may be consulted: Macdonald, Proe. Royal Soc. lxxi. (1903), pp. 251 258; lxxii. (1904), pp. 59—68; xc.A (1914), pp. 50—61; Phil. Trans, of the Royal Soc. cxx. A 144; Debye, Ann. der Physik und Chemie, (4) xxx. (1909), pp. 57—136; (1910), pp. 113 March, Ann. der Physik und Chemie, (4) xxxvii. (1912),, pp. 29—50; Rybczynski, Ann. der [Note.

the following writers

—

* Phil. Mag. (5) xxiii. (1887), pp. 252—255. [Math, and Phys. Papers, iv. (1910), pp. 303—306.] In connexion with the principle, see Stokes, Trans. Comb. Phil. Soc. ix. (1856), p. 175 footnote.

[Math, and Phys. Papers, n. (1883), p. 341.] t Math. Ann. txvu. (1909), pp. 535 558; MUnehener Sitzuvgsberichte, xl.

—

[5], (1910).

5


226

[CHAP. VIII

Phyrii und Ckemie, (4) xli. (1913), pp. 191—208 Nicholson, Phil. Mag. (6) xix. (1910), Love, Phil. Trans, of the Royal Soc. ccxv. A (1915), pp. 105 131 Watson, pp. 516 537 Proc. Royal Soc. xcv. A (1918), pp. 83—99, 546—563. The works quoted all deal with the problem of the propagation of electric waves over the surface of the earth, and are largely concerned with attempts to reconcile theoretical with experimental results.] ;

—

8*11.

—

;

;

Meissel's first extension of Carlini's formula.

The approximation

obtained by Carlini

(§ 1*4)

the

is

asymptotic expansion of a Bessel function of large order

;

first

term of the

subsequent terms

by Meissel, Astr. Nach. cxxix. manner:

in the expansion were formally calculated (1892),

col.

281—284,

^

in the following

may

It is clear that Bessel's equation

+»^-^(i-^.W=o;

)

(1) if

we

define a function

be written

u (z) by the equation

u

^( y*>* r(iT+l) exp {/ then equation

(1)

If

+ zu (z) -

i/

now we assume

that, for large values of

v,

of descending powers of

i*o,

«i, Ma,

us>

...

{u (s)} 2 ]

+

a

- 5 ) = 0. 2

(1

u

(z) is

expansible in a series

thus

v,

u (z)

where

^

transforms into *» [it' (*)

(2)

^ dz

— vu +

it,

+ v^/v + u

+

3 /v*

...,

denote functions of z which are independent of

v,

by

substituting in (2) and equating to zero the coefficients of the various powers of v on the

left,

we

find that

«,«{V(i

Ws= iu

=

45+105* + 8(1 -i-y I62

s

Ui

64^ + 560a

7

s

8

8 (i!^r 4565 5 + 25s 7

+ i28(i-^)V

_

'

+ 3685 + 9245 + 374* + lSz

3

'

9

32(l-5 )

2 7

Hence, on integration,

J'u (z) dz -

"2== -

^=20^).

-«•)}/*,

found

it is

v {log 1

+

^

1

(cf.

_ 2, +

(

2

1

(

+ 57607

s

+

3252

+

t

16

V(l

-

*•)

- l} - i log (1 - z*)

)

+ 353 _ 24i/|(i_^ 1

§ 1*4) that

4>z

\

16r> 2

j

2

+ z*

(l-5 )3 2

-1512s 2 - 3654^-3755" (l-*8 ) 1

+

23 25s

128^(1

-5 )

2 88-g4

2 6

+

13-g8

|

3

—

i

*

'

FUNCTIONS OF LARGE ORDER

8*11, 8*12]

Hence we have

j

(3)

227

Meissel's formula

( pz)=s

&T (v + 1) (1

^- - *» *»)* {1

•

exp (- F*>

+

V(l

- *s )}'

where

+ z*

4>z*

(1-*«)t 16i^(l-^) *-£t£S-}-

<*>

s

j

1__

f

!6

- 1512.? - 3654g« - 375s« _ 8

5760^1 32**

(i_*.)*

J °J

+ 288** + 232*« + ISz* + "" 128v*(l-^)8

It will appear in § 8*4 that the expression given for

V

is

¥

the

sum

of the

four dominant terms of an asymptotic expansion which is certainly valid

z

lies

between

It is stated

and 1 and v

by Graf and Gubler* that the first approximation derivable from

j

(ve)

when

is large.

(3),

namely

*«pW(r-/»

„

(2irv)i(l-**)i{l+V(l

-«*)}•''

due to Duhamel but a search for the formula in Duhamel's writings has not been and it seems certain that, even if it had been discovered by Duhamel, his discovery would have been subsequent to Carlini's. is

;

successful,

Note.

The reader should t/, v (v

(5)

observe that (3)

seen a)' =

may

also

——

—

be written in the form

-

y/(2trv

.

.

*

' , '

tanh o)

where

^=^^(2 + 3sech»a)-^^(4sech»a+sech*a)

(6)

- gsLrJ (16 - 1512 sech8 a - 3654 sech4 a - 375 sech8 a)

-C

Ma

^

4

(32 sech* a+ 288 sech* a + 232 sech8 a + 13 sech*- a)

+ .... 8*12.

Meissel's second expansion.

The expansion obtained

in § 8*11 obviously fails to represent

and greater than unity;

Jv (vz) when

such values of z, Meisself obtained two formal solutions of Bessel's equation and, if we write z = sec /3, the reader z

is real

for

;

will see,

by making some modifications

in §

81 1

(5),

that these solutions

be written in the form

/(*-£*) «P{-

A t«M.

* Einleitung in die Tfieorie der BeueVsehen Funktionen,

f Attr. Nach. exxx. (1892),

col.

363—368.

i.

(Bern, 1898), p. 102.

may


228

[CHAP. VIII

where*

P, = ^£(4sec*,8

(1)

-

^^

+ sec
(32 sec8

+ 232 sec

288 sec4

+ 13 sec8 £)

6

+ g^T (768 sec £ + 41280 se&0+ 14884 sec«/S + 17493 sec £ s

8

+ 4242 sec Q,

(2)

= ,(tan£-/3)-^^(2 + 3sec I

-

C

^ (16

5 yg

-

3 225(30i;»

- 1512 sec £ 8

< 256

2

+ 103 sec

10

12

£)

/3)

3654

sec*

- 375 sec

6

0)

+ 7872 ° sec2 3 + 18 9!200 sec4 £ + 4744640 sec6 £ + 1914210 sec8 /3 + 67599 sec £) '

10

+

....

To determine to far,

Jv (vsec 0)

and compare the

type given in § 7'21

and we

;

in terms of these expansions,

we take

to tend

results so obtained with the expansions of Hankel's

we see that, as -*- \tt, P* — 0, Q v ~v (sec -

%tt),

infer that

(3)

JT.» (v sec 0) =

(4)

#„<"

(i/

sec

0)

J(~~)

= J(r^!~)

•

'

e-^+^-K e- p-- iQ-+ini

It follows that /

-

(5)

J, f> sec

(6)

F, (v sec 0) =

/S)

^/

(^~

)

^/(~^)

•

•

•- p" cos (Q„

- ^r\

«- p " sin

- «,

(Q,

where P„ and Q„ are defined by (1) and (2). It will appear subsequently (§ 8 41) that these formulae are actually asymptotic expansions of Jv {y sec 0) is any assigned acute angle. and F„ (y sec 0) when i> is large and #

Formulae which are valid for small values of 0, i.e. asymptotic expansions valid when z and v are both large and are nearly „ (z) and F„ (z) which are equal, cannot easily be obtained by this method but it will be seen in § 8*2 that, for such values of the variables, approximations can be obtained by rigorous methods from Schlafli's extension of Bessel's integral.

of

J

;

*

The reader

will observe that the

the corresponding analysis of § 8*11.

approximation has been carried two stages farther than in


8*2]

The dominant terms

Note. form

in the expansions (5)

*'-

Wh6re

(6),

which

may be written

(*«*-*) )*>

Qv ~ */(*? — v 2 ) - \vir + v arc sin

(v/x),

had been obtained two years before the publication of Meissel's paper by memoir on Physical Optics, K. Dantke Videnskabernes Selskabs Skrifter, [Oeuvres Scientifiques, L (1898), pp. 421 436.]

—

The procedure of Lorenz was to take for granted which has been proved in § 7*51,

-* Jf *'

in the

Yv {x)=Mv &va{Q v -±ir),

Jv (x)=Mv coa(Qv -br),

(7)

and

229

1

Til+ 8'

y2

t*L

y2

-JLTi *#!_

-*,

*«

that, as

L. Lorenz in a (6) vi. (1890).

a consequence of the result

1-3 (*>-*). (*»-f) ** 2.4

+

"I

-J

-*T* J

tf*

'

and then to use the exact equation

£SL'._L_

(8) V

dx

'

which

is easily

-nxM*'

deduced from the Wronskian formula of § 3*63

whence the approximation stated

for

(1),

to prove that

Qv follows without difficulty.

Subsequent researches on the lines laid down by Lorenz are due to Macdonald, Phil. Trans, of the Royal Soc. ccx. A (1910), pp. 131—144, and Nicholson, Phil. Mag. (6) xiv.

697—707 (6) six. (1910), pp. 228—249; 516—537; Proc. London Math. Soc. (2) 67—80; (2) XL (1913), pp. 104—126. A result concerning Qv+1 -Q„ which connected with (8), has been published by A. Lodge, British Association Report,

(1907), pp.

;

ix. (1911), pp. is closely

1906, pp. 494—498.

8*2. The principle of stationary phase. Bessel functions of equal order and argument.

The principle of stationary phase was formally enunciated by Kelvin* in connexion with a problem of Hydrodynamics, though the essence of the principle is to be found in some much earlier work by Stokes f on Airy's integral (§ 6'4) and Parseval's integral (§ 2*2), and also in a posthumous paper by Riemann %.

The problem which Kelvin propounded was

to find

an approximate expression

for the

integral 1

2irJ

cos [to {x — tf(m)}] dm,

which expresses the effect at place and time (x, t) of an impulsive disturbance at place and time (0, 0), when /(to) is the velocity of propagation of two-dimensional waves in water corresponding to a wave-length 2n/m.

The

principle of interference set forth

by Stokes

* Phil. Mag. (5) xxra. (1887), pp. 252—255. [Math, and Phys. Papers, rr. (1910), pp. 303—306.] t Comb. Phil. Trans, a. (1856), pp. 175, 183. [Math, and Phys. Papers, n. (1883), pp. 341, 851.] t Ges. Math. Werke (Leipzig, 1876), pp. 400-406.

F

.

:


230 and Rayleigh

[CHAP. VIII

and wave-velocity suggested to Kelvin x-tf(m), the parts of the integral outside the range (ft -a, /* + «) are negligible on account of interference if /x is a value (or the value) of m

in their treatment of group-velocity


m

of values of

which makes

^[»{*-*/(«i)}]=0. In the range (ji-a, /a+a), the expression m{x-tf(m)} three terms of its expansion by Taylor's theorem, namely

M {*-
it is

-/0-iW .00 + 2/'

is

0*)}

then replaced by the

first

(m-tf,

found that, if*

m

M

W2 "n/[-Mm/"(m) + 2/'(m)]' cob

then

U

{*/»»/' 0»)

+ «!*} Ar

f ~7r^/[-2t{hf"( H.) + 2f'(H.)}]

«»{»»/' Qi) + jir} V["2ir<{f*/"(M) + 2/'0*)}]-

In the last integral the limits for replaced by

-

oo

and

a,

which are large even though a be small, have been

+ oo

be seen from the foregoing analysis that Kelvin's principle is, effectively, that in of the integral of a rapidly oscillating function, the important part of the integral is that part of the range of integration near which the phase of the trigonometrical

It will the case

due

to

function involved

is

stationary \.

It has subsequently been noticed J that it is possible to give a formal mathematical proof of Kelvin's principle, for a large class of oscillating functions,

by using Bromwich's generalisation § of an

integral formula

The form of Bromwich's theorem which

will

due to

be adequate

for

Dirichlet.

the applica-

tions of the principle to Bessel functions is as follows

Let F{x)be a function of x which has limited total fluctuation when xÔ; Then, if—l
ybea function

let

i

v*

a?- 1

F (x)smvx

.

.

dx-~

(+ 0)

jo and,

if0
sines

may

/

tr~ l sin t dt

*

This

is

= F (+ 0) V (ft) sin \fitr

;

be replaced by cosines throughout.

The method which has just been explained

maximum

.

Jo

the appropriate substitution

will

now be used

when m{x-tf(m)} has a minimum

to obtain at »»=/*;

an

for a

the sign of the expression under the radical is changed.

t A persistent search reveals traces of the use of the principle in the writings of Cauohy. See equation (119) in note 16 of his TMorie de la propagation des Ondes, crowned Sept. 1815, Mem. prisent&s par divers savants, i. (1827). [Oeuvres, (1) i. (1882), p. 230.] e.g.

t Proc. Camb. Phil. Soc. xix. (1918), pp. 49—55. § Bromwich, Theory of Infinite Series, § 174.


8*2]

approximate formula for J¥ (v) when v which was discovered by Cauchy*, is

and

positive.

This formula,

r(*)

J9 (v),

(1)

large

is

231

2^.3*7T^' This formula has been investigated by means of the principle of stationary phase, comparatively recently, by Nicholson, Phil. Mag. (6) xvi. (1909), pp. 276 277, and Rayleigh,

—

Phil.

Mag.

also Watson, Proc.

From

1001—1004

(6) xx. (1910), pp.

Camb. Phil.

Papers, v. (1912), pp. 617—620]; see

[Scientific

Soc. xix. (1918), pp.

42—48.

§ 6'2 (4) it is evident that

J( v) =

1 f'cos

tJo

[v

(0-

sin 0)}

d$-^^ Jo

( V-ie-fsinh*) dt,

ff

and obviously

smw

/

"

^ {t+Anh() dt

^

1 'cos

- sin 6)}

e

["^t dt=Q î v)

1

Hence

J

v

Now

let

(„)

=

[v

f T JO

= — sin 0, and

l

_£i_

im

*-»ol

it

*/(l

— cos 0)

+

(1/v).

cosv

I'

Jo 1

But if

d$

then

- sin 0)\ d0 -

f 'cos { v (0 Jo

and hence,

(6

T

d.

-cos

2

-cos0

6*

has limited total fluctuation in the interval

(0, v),

follows from Bromurich's theorem that ['

cosi/0

2

,. -*— dq>

I

Jol-costf

It

still

this result

has to be proved that 0*/(l we observe that

- cos 0)

4>*

£.

so that

^

,.

,

* cos v
r ^

ra)C° Si7r

'


1

f

<# (1-costfJ where

...

|

=S

and then

/•*

6 r ~ -= r 6* Jo

o ^ * v '

has limited total fluctuation

_
2(1-008^ _ sin ^

^(0)=0,

;

to establish

'

3 $ _ 8m 0)

gr(,r-0)=+a>,

^ (0) « (1 - cos 0)*/(l + cos 0) > 0, and therefore, by integration, g(0)^0 when O$0
[Oeuvret,

(1)

xn. (1900), p. 663.]

A

is

proof by

THEORY OF

232

By means

of

BjSSSEL FUNCTIONS

some tedious integrations by parts*,

[CHAP. VHI

it is

possible to obtain

a second approximation, namely

and

it

may

also

be proved that

J/W-lr^ + o (*"*);

(3)

an associated formula

is

F.(,)~-?»

(4)

2*7Tl/*

The asymptotic

expansions, of which these results give the dominant

terms, will be investigated with the aid of

more powerful analytical machinery

in § 8-42.

8*21.

The

Meissel's third expansion.

been used by Cauchyf and Meissel $ to Jn (n) when n is a large integer. expansion was obtained by Cauchy and (in

integral just discussed has

obtain the formal asymptotic expansion of

now be explained how this a more complete form) by Meissel the theoretical cesses employed will be investigated in § 8*42. It will

;

justification of the pro-

Taking the formula

Jn (w) let

us write

— sin — £$*

;

=-

cos {n (0

— sin

0)} d0,

then follows that, for sufficiently small values of t,

it

m=0 \q =

and

\ = £$,

1,

^* =

\2 = TlW»

26&00>

^<

~ T7¥ llooo

>

It follows that

Jn (n) - -

When n

is large,

f( I

(2m +

\ni? is large at the

«/n(w)~- 2 (2m 7T»t=0

1)

X. *"»l cos Qnfi)

upper

limit,

+ l)*™. O .'0 \

%

d0.

and Meissel inferred that

t*" cos

(\n&)dt,

* BeeProc. Camb. Phil. Soc. xix. (1918), pp. 42—48. t Comptet Rendu*, xxxvm. (1854), pp. 990—998, 1104—1107. [Oeuvret, (1) xu. (1900), pp. 161—164, 167—170.] t Astr. Nach. cxxvn. (1891), col. 359 362; cxxviu. (1891), col. 145—154. Concerning formula (1), Meissel stated "Sohon vor dreissig Jahren war ich zu folgenden Forme! gelangt."

—


8*21, 8*22]

Q

where

is

the sign indicating a "generalised integral"

233 (§ 6*4)

;

and hence, by

integrating term-by-term and using Euler's formula, Meissel deduced that

Jn (*) ~ 7TI I A™ T (fm + i) (l \«/)

(1)

,m+} cos

in _

Meissel also gave an approximation for X m , valid

m is large;

when

tion exhibits the divergent character of the expansion

(*m +

*) ».

and

this approxima-

(1).

The approximation is obtainable by the theory developed in the memoir of Darboux, "Sur l'approximation des functions de tres grands nombres," Journal de Math. (3) iv. (1878), pp. 5—56, 377—416.

We consider the singularities of 6 qua be monogenic) are the points at which

and near*

**=

function of

d—irn and

t

;

the singularities (where 6

t=(12rir)*, where

fails

r— +1, ±2, ±3,

to

...

±(12w)» the dominant terms in the expansion of 8 are

* + -41'

±2*- + (36»r)*{l1 J

By the theory of Darboux, an approximation to X m is the sum of the coefficients of fr** 1 in the expansions of the two functions comprised in the last formula ; that is to say that

Xw ~2.(36,r)i

*-M-<»».-»> (2w + l)!(12ir)*" + ,

1

2r(2m+§) 3^r(|)r(2m + 2).(127r)*m and

by

so,

Stirling's formula,

Xm

(2)

1

~

(18)*r(|)(m+J)*(12ir)* This is Meissel's approximation by Cauchy, loc. tit., p. 1106.

8*22.

The

I

»

an approximation of the same character was obtained

to

J

v

(v sec 0).

principle of stationary phase has been applied

in § 8'2

— sec

for

J¥ (v sec 0) where

is

i.

by Rayleighf

to obtain

fixed positive acute angle,

we have

J, (v sec 0)

and

;

The application of Kelvin's principle

an approximate formula and v is large J.

As

'

sin

Write — sec creases from to

is

= - (cos [v (0 - sec

stationary (a

sin

—

sin 0))

minimum) when

— tan +

,

and then increases as

so that

d0+O (1/v), = 0.

decreases to zero as

increases from

to

in-

tr.

* These are the singularities which are nearest to the origin,

t Phil. Mag. (6) xx. (1910), p. 1004. [Scientific Papers, v. (1912), p. 620.] % See also Macdonald, Phil. Traru. of the Royal Soc. ccx. A (1910), pp. 131—144; and Proc. Royal Soc. lxxi. (1903), pp. 251—258; lxxii. (1904), pp. 59—68.


234

[CHAP. VIII

Now cos [v (0

— sec

dd

sin 0)}

Jo f»-/5

r/-0

and 0*

as

+ tan/5-]

,

0-/8.

V(2 tan 0)

Hence,

i/^

has limited total fluctuation in the range

(d0/d)

*

follows from Bromwich's theorem

cos \v (0

- sec

sin 0)}

J*

0^0 ^ tt,

it

tftat

d0 ~ 2

1°°

+ £ - tan 0)}

cos (v (<£

î/

and

dd

//

2 tan£)

tan 0,

so

n\ (1) v '

ti

«/„ "\(v

o\ sec r'/ 0)

{ir) ~ cos {y— tan 0-0)m ( 777

*

.

V(i*"rtan£J)

The formula sin{*>(tan/9-#)-£?r} v/ o\ 7 *— ^ Y (y sec #-/ 0) ~ i-v os v(£i/7rtan/8)

/0 \ (2)' is

v

derived in a similar manner from § 6 21 -

The reader expansions

(1).

dominant terms in Meissel's

will observe that these are the

§ 8*12 (5),

To complete the

(6).

rigorous proof of these formulae

we have

to

shew that 0*

(d$/d)

has

limited total fluctuation.

Now

the square of this function, namely

(dO/chf))

2 ,

is

equal to

- sec ft sin - ft + tan ft _ ... _AW cos tf)* (1 - sec .

'

But

say.

., ^

cos

'~

ft

cosecd

(1

- sec ft cos fff- 2 (fl-secftsin - ft+tanft) cosftcosecd(l-secftcos0) 3

The numerator, k (0),

of this fraction has the differential coefficient

- cos ft cos $ cosec2 0(1- sec ft cos 0) 2 as $ increases from \ir to

when

0^0
and

n

;

,

£ir, and then increases steadily since k(0)=O wben 0=ft<£»r, it follows that h! (0)^0

and so k(0) decreases steadily as $ increases from

to

A'(0) changes sign once (from negative to positive) in the range

ft^d^JT.

Hence point (a

when proved.

|

yjh

(ff)

|

is

minimum)

< 6 <«

it

monotonic (and decreasing) when

<6<

< 6<

slh (6) consequently has limited total fluctuation in the range

ft

ir

;

since

|

|

and it has one stationary bounded and continuous when ^ 6 ^ »r, as had to be ft,

is

;


8'3]

The method of steepest

8*3.

235

descents.

A development of the theory of contour integration, called

the method of by Debyef to obtain integral representaof Bessel functions of large order from which asymptotic expansions are

steepest descents*, has been applied tions

readily deduced.

If,

in general,. we consider the integral e"/<«" J,

in which

(w) dw,

v is supposed to be large, the contour is chosen so that it passes through a point w at which /'(w) vanishes and the whole of the contour is then determined by the assumption that the imaginary part of f(w) is to be constant on it, so that the equation of the contour may be written in [

|

;

the form

If(w)=If(w ). To obtain a geometrical conception of the contour, let w = u + iv, where u, v and draw the surface such that the three coordinates of any point

are real

on

it

;

are u,

If

v,

Rf(w).

Rf(w) = z, and if the z-axis be supposed to be vertical, the surface has no maxima or minima except where /(w) fails to be monogenic; for, at

absolute all

other points,

du*

The

points [u

,

v

,

R/(w )]

dv 2

are saddle points, or passes, on the surface, so that

the plan of a curve on the surface which goes through one of the passes on the surface. This curve possesses a further property derived from the equation of the contour for the rate of change of the contour of integration

is

;

f(w), at any given value of w, has a definite modulus, since /(w) is supposed to be monogenic and since If(w) does not change as w traverses the contour, ;

it

follows that

the curve it,

is

is

Rf(w) must change

characterised

so chosen that it is

as rapidly as possible

by the property that

;

that

is

to say, that

any point of the steepest curve through that point and on the its direction, at

surface.

It may happen that we have a freedom of choice in selecting a pass and then in selecting a contour through that pass our choice is to be determined from the consideration that the curve must descend on both sides of the pass for if the curve ascended, Rf(w) would tend to + oo (except in very special ;

cases) as

w

left

the pass, and then the integral would diverge

if

R (p) > 0.

* French "Methode du Col," German "Methode der Sattelpunkte." t Math. Ann. lxvii. (1909), pp. 535 558; Miltichener Sitzmigsberichte, xl. [5], (1910). The method is to be traced to a posthumous paper by Biemann, Werke, p. 405 ; and it has recently been applied to obtain asymptotic expansions of a variety of functions.

—


236

[CHAP. VIII

The contour has now been selected* so that the integrand does not on it and so we may expect that an approximate value of

oscillate rapidly

;

the integral will be determined from a consideration of the integrand in the

neighbourhood of the pass the interference effects

(cf.

:

from the physical point of view, we have evaded which occur with any other type of contour.

§ 8*2)

The mode of derivation of asymptotic expansions from the integral will be seen clearly from the special functions which will be studied in §§ 8*4 8*43,

—

convenient to enunciate at this stage a lemma f which will be useful subsequently in proving that the expansions which will be obtained

8*6, 8'61

but

;

it is

are asymptotic in the sense of Poincare.

Lemma. Let F{r)

when r ^ a

be analytic

\

|

+

8,

where a

> 0,

>

8

and

;

let

F(t)= I am ^ m^~\ when

|

t

positive

^ a,

j

r being positive ; also,

numbers independent of

t,

let

\

F (r)

when t

\

<

lTe*

r ,

is positive

K

where

and r > a.

and

b are

Then

the

asymptotic expansion

1 am T (m/r) v~ m lr m=l valid in the sense of Poincare when \v\is sufficiently large and e- VT F{r)dT~

/,

is

where

A

is

an arbitrary positive number.

It is evident that, if

M be any fixed integer, a constant K

x

can be found

such that m=l

I

whenever t

^

whether r ^ a or t j

e- vr F{r)dr=

;

2

I

m=l JO

J

where

> a and

\e~»T

l^jfl^f

\.

therefore

eâ^^^dr + RM

,

Kî^^dr

Jo

= K r (M/r)/{R (v) - b) M* = 0(v-x'r x

),

provided that

R (v) >

b,

which

is

the case

when

|

v

\

> b cosec A. The

remains valid even when b is a function of v such that compared with v. We have therefore proved that f JO

and

so the

* For

lemma

er»F (t) dr = X am T (m/r) it-**" + m-l

is

analysis

R (p) — b is not small {v~ M'r ),

established.

an account of researches in which the contour is the real axis see pp. 1343 Burkhardt's article in the Encyclopadie der Math. Wits. n. 1 (1916). t Cf. Proe. London Math. Soc (2) xvu. (L918), p. 133.

—1350 of


8*31]

237

8*31.

The construction of Debye's contours* when

It has

been seen in §§ 6*2, 6*21 that the various types of functions associated

the variables are real.

with J, (x) can be represented by integrals of the form

On

taken along suitable contours.

we

shall

the hypothesis that v and x are positive,

now examine whether any

of the contours appropriate for the

method of steepest descents are of the types investigated

in §§ 6*2, 6*21.

In accordance with the principles of the method of steepest descents, as 8*3, we have first to find the stationary points of

explained in §

x sinh w — vw, qua function of w,

i.e.

we have

(1)

to solve the equation

a;

w— v— 0;

cosh

and it is at once seen that we shall have three distinct cases to in which xjv is less than, greater than, or equal to 1, respectively.

consider,

We con-

sider these three cases in turn. (I)

When xjv < 1, we

can find a positive number a such that

x — v sech

(2)

and then the complete solution of (1)

a,

is

w = ± a + 2mri. It will

be sufficient to confine our attention to the stationary points + a; at f

these points the imaginary part of of the contour to be discussed

a?

sinh

w — vw

is zero,

and so the equation

is

/ (x sinh w — vw) — 0. Write

w = u + iv,

where

cosh u sin v so that v

= 0,

and

u, v are real,

this equation

becomes

— v cosh a = 0,

or

/ox (o)

The contour

coshw = ,

v

=

v cosh — siny

a .

gives a divergent integral.

contour given by equation

To

We

therefore consider the

values of v between

and ir, correspond pairs of values of u which are equal but opposite in sign and as v increases from to n, the positive value of u steadily increases from a to + x (3).

;

.

* The contours investigated in this section are those which were discussed in Debye's earlier paper, Math. Ann. lxyii. (1909), pp. 535 558, except that their orientation is different; cf. § 6-21. t The effect of taking stationary points other than ±a would be to translate the contour parallel to the imaginary axis.

—


238

[CHAP, vin

is unaltered by changing the sign of v and so the contour is symmetrical with regard to the axes the shape of the part of the contour

The equation

;

between v

= — tr

and v = tr

is

shewn in

Fig. 16.

TTl

> -77 Fig. 16.

t

If it is

= sinh a — a cosh a — (sinh w — w cosh a),

easy to verify that t (which

As

+ oo

w

on the curves shewn by the arrows.

is real

increases in the directions indicated

—mi

travels along the contour from oo

+ oo

and then increases to

to

and

;

since,

to oo

by

+ iri,

in the figure)

t decreases from

§ 6*2 (3),

IlTlJ ao-vi

we have obtained a curve from which we can derive information concerning Jv {x) when x and v are large and x\v< 1. The detailed discussion of the integral will be given subsequently in §§ 8*4, 85.

— oo

± iri give information concerning a second but this problem is complicated by Stokes' phenomenon, on account of the two stationary points on the contour.

The contours from

to oo

solution of Bessel's equation;

(II)

When x/v>l, we

can find a positive acute angle

the relevant stationary points, cosh

which are now roots of the equation

w — cos /8 = 0,

w=± i#.

are

When we

take the stationary point

I (sinh so that, replacing /

such that

a;=i'sec)9,

(4)

and

/3

5\

why

u

if$,

the contour which

we

w — w cos /3) = sin ft — fi cos ft,

+ iv,

cos h

the equation of the contour

u=

—+ 1^_J^ —__

sin ft £

(v

ft)

cos

ft

is

obtain

is


8*31]

Now,

for values of v

sin ft

has one

minimum

and

between

and

+ (v — ft) cos ft — sin v

and

ir,

The shape

v is

or

;

for other

equation (5) gives two real values

(equal but opposite in sign), and these coincide only

when

zero

+ (v — ft) cos ft > sin v.

Hence, for values of v between are infinite

is

ir,

sin ft

u

the function

= ft) at which the value of the function

(v

values of v between

of

tr,

239

when

v

= ft.

They

ir.

of the curves given by equation (5)

is

as

shewn

in the upper

half of Fig. 17; and if

tsj (sin ft — ft cos ft) — (sinh w — w cos ft), it

easy to verify that r (which

is

directions indicated to oo

+ iri,

is

by the arrows. As

r decreases from

+ oo

—^^

to

on the curves) increases in the

real

w

travels along the contour from

and then increases

to

-f-

oo

—«

and so we

ni

\

S^3

—

—

-"""*^

-ni

*

Fig. 17.

have obtained a curve from which [§ 6'21 (4)] we can derive information concerning (#) when x and v are large and x/v > 1. The detailed discussion v of the integral will be given in §§ 8*41, 15'8.

H

If

{1)

we had taken the

stationary point

— ift, we

should have obtained the

curves shewn in the lower half of Fig. 17, and the curve going from oo

— 7rt

gives an integral associated with

in § 8*41.

The two

integrals

H

v

<

1

—

oo

to

(x); this also will be discussed

now obtained form a fundamental system of is a marked distinction between

solutions of Bessel's equation, so that there

the case x/v

{i)

and the case xjv >

1.

•


240 (III)

from

case in which v = x

The

(I) or

may be

from (II) by taking o or

sidered are v

—

#

[CHAP.Vm

derived as a limiting case either

equal to

The curves now

0.

to be con-

and cosh u

(6)

and they are shewn in Fig.

=

v/sin

v,

18.

Fig. 18.

We

obtain information concerning

curves from

— oo

to oo

±

iri,

H,

ll)

(v)

H

{i)

v

(p)

by considering the

while information concerning

from the curve which passes from oo — mi to tion will be given in §§ 8'42, 853, 8*54.

8*32.

and

oo

+ vi. The

Jv (v)

is

obtained

detailed investiga-

Geometrical properties of Debtee's contours.

An interesting result which will be found to be important in dealing with zeros of Bessel functions (§ 15*8), and which is also used in proving certain approximate formulae which will be stated in § 8*43, is associated with the second of the three contours just discussed (Fig. 17 of § 831).

— oc

to

» + jrt

is

positive

The theorem in question and does not exceed *JZ.

is

that the slope* of the branch

from

It is evident that, for the curve in question,

du

sin(» — /3) — (v — ff)convcos3

dv

sin 8 v

smh u -T- = .

,

—

-

,

.

.

But sin(«;— /3)secv-(i>-/3)cos/3 has the positive derivative cos /3 tan 2 v, and hence

it

follows that sin

(v- /9) - (v — f3) cos

v cos

since v — /3 and u

has the same signt as v— /3. Therefore fur the curve under consideration, dv/du

is positive.

* Proc. Comb. Phil. Soc. xix. (1918), p. 105.

3=0,

the slope

is

on the

left

of the origin

better results of this type exist.

f This

is

obvious from a figure.

£

are both positive or both negative

and

Since, in the limiting case (Fig. 18) in is

^3

which

immediately on the right of the origin, no


8*32, 8*4]

Again, to prove that dv\du does not exceed

^Wr) =

we

write

8iDg + (i, ~ i3)COS

1

and then

it is sufficient

241

'

sin v

to prove that

3^'*(*>)-iP(t;)+l>0.

Now 2*'

the expression on the («)

:

(v) 3

(which vanishes when v=j8) has the derivate

+ (»)]

[«*" (»)-

»W

left

- [(*•-£)

{sin'p+3 cos2 »} cos 0+ sin 2 v sin

/o\ o (v - 0) cos 0+

v, But

,

.

8in 2

— ^— ^ nce

when

ir.

- /3)J.

--5

s

sin2 r +3 cos* v

'

positive

(i>

vsin£-3cost>sin(v-/3) £-' i

x

has the positive derivate

0- 3 cos v sin

*.

T~\%> an<*

2

Therefore, since

y

(v)

8i

it is positive

has the same sign as

v

when »=0,

- 0,

it

it is

follows that

2*' «[3*'>) -*(»)]

has the same sign as v — 0, and consequently

3^'»(»)—ip(»)+l has »=/3 for its only minimum between This proves the result stated.

8*4.

v—0 and

t>=»r

;

and therefore

it is

not negative.

The asymptotic expansion* of Jp {v sech a).

From

the results obtained in § 8'31 we shall now obtain the asymptotic first kind in which the argument is less than the order, both being large and positive.

expansion of the function of the

We retain the notation of § 8*31 (I) and it is clear that, corresponding to any positive value of t, there are two values of w, which will be called wx and Wj the values of wx and w% differ only in the sign of their imaginary part, and it will be supposed that ;

;

l(w )>0, 1

We

J(>8)<0.

then have

where x — v sech a.

Next we discuss the expansions of «/i and w% in ascending powers of t. Since t and dr/dw vanish when w = a, it follows that the expansion of t in powers of w — a begins with a term in (w — a)* by reverting this expansion, we obtain expansions of the form ;

Wl *

_«= i

Wa _ a=

_^!» Ti(«+D »=o>» + l

The asymptotic expansions contained

by Debye, Math. Ann.

in this section

—558.

lxvii. (1909), pp. 535

2

(-)'n+1

« =0 and

-^

ni+1

T*<"»+1 >,

in §§ 8*41, 8*42 were established


242

[CHAP. VIII

and, by Lagrange's theorem, these expansions are valid for sufficiently small

Moreover

values of Ti.

1

am

f(Q+.o+)/d t i;A

dr

Ut/t*<»« +1

~2w»J

>

dw Ti(m+i)

2vi]

The double

*

circuit in the T-plane is necessary in order to dispose of the

powers of t and a single circuit round a in the w-plane corresponds to a double circuit round the origin in the T-plane. From the last contour fractional

;

integral it folio ws that

m ascending

T-j(m+i)

am

is

the coefficient of l/(w

powers of

w — a; we

— o)

in the expansion of

are thus enabled to calculate the

coefficients a m .

Write

w—a= t

where

c

W and we have W — 1) — cosh a (sinh W — W)

= - sinh a (cosh

= — | sinh a,

coefficient of

W

m

(wi)

ai(w) /

\

a l(TO)

a,(m)

= — £ cosh a,

c8

in the expansion of

The coefficients and so we have fa

c,

= — ^ sinh a, Therefore a m c W + c F + ...}-* {m+l) . . . .

x

a (m),

a, (m), Oa (m), ...,

,

= Co-H-,|_^.|},

m + 1 .-Ca + (m + l)(m + 3) f _
<«,j.t>

.

T

2

.

2!

(m + l)(m + 3) 2dc,

= c -Hw|-|±|

2s 21

Co

c2

'

.

(ro+l)(i» + 3)(wi

(1)

+ 5)

2*. 3!

(m +

1)

(m +

(m+l)(m + 3)(ra + 5) 2s 3!

'

.

(to

W 3^

3) /2cj c8

c

+

we find that fa =a (0) + (-!• sinh o) *, (- | sinh a)-1 {J a, = a, (1) =

CoV

8

+ 1) (m + 3) (wt + 5) (m + 7)

substitution

coth

a},

= a, (2) = - (- J sinh a)"* {$ - ft coth* «}, 8 a, « a, (3) = - (- 1 sinh a)" {-& coth a - ft coth a}, coth'a 4 fljfr coth«a}, «4 - a* (4) - + (- i sinh a)"t T ^ a,

(2)

2

|

I

{

#

ff

c^ 'c*

2*. 4!

On

the

.

in this expansion will be called

= c -* (wt+1)

is

2

{c„ -t

c^)

'cA

.


8*4]

Now when t |

^= 2

->-i

j

small

is sufficiently

;

= cosh o — cosh w,

aw

d (u\ — w^jdr tends

follows that

T"-»

a, m

and since

-7it

243

Hence the conditions stated

lemma

in the

\dr

Jo

+ oo

to zero as r tends to

of § 8*3 are satisfied,

and

so

dr)

has the asymptotic expansion

"

when x

oÎVm + l )

is large.

Since arg

{(a/,

-o)/t*}-»- \ir as t

-*-

0, it follows that, in

has to be interpreted by the convention arg a

Jr ( y secha)^

(8)

v(2

^ tenha) J

o

= + ^7r,

($5),

the phase of a

and hence

-^^--. (||;taDha)TO

>

where

(A =l, ^^-^cotn'o, 1 ^ 2 = TlF-tVVcoth 'a + ^^coth*a,

(4)

!

The formula

J

¥ (v sech o) valid (3) gives the asymptotic expansion of number and v is large and positive.

when

a is any fixed positive

The corresponding expansion contour from

— oo

to

oo

+ irt,

v , r,(,secha)~.

,_, (5)

for the function of the second kind, obtained

,

e

K«-t«nha)

V(iffytanha)

The

by taking a

is

« r(m+i)

(~)m A m

^-T^T -(^vtanhar-

position of the singularities of

d^-w^ldr,

qua function of the

complex variable t, should be noted. These singularities correspond to the points where w fails to be a monogenic function of t, i.e. the points where dr/dw vanishes. Hence the singularities correspond to the values ± a + 2ntri of w, so they are the points where t = 2mri cosh

and n assumes

all

a,

* = 2 (sinh a - a cosh a) + 2mri cosh «,

integral values.

a formula for dw/dr in the form of a contour integral be a pair of corresponding values of (w, r), then, by Cauchy's theorem,

It is convenient to obtain (W(» To)

fdw\ = J_ /"('•+) dw dr Jl_ f («*+> dw_ t—t 2tri J dr t-t \drJo 2irt J where the contour includes no point (except w ) at which t has the value t '

.

;

if


244 8*41.

[CHAP. VIII

The asymptotic expansions of Jv (v sec 8) and F„ (v sec 8).

In § 8*4 we obtained the asymptotic expansion of a Bessel function in which the argument was less than the order, both being large ; we shall now obtain the asymptotic expansions of a fundamental system of solutions of Bessel's equation

We

when the argument is greater than the order, both being large.

retain the notation of § 8*31 (II)

passes from — oo to be supposed that

+ tri

oo

;

it is

;

clear that, corresponding to

w

two values of

positive value of t, there are

any

lying on the contour which

these values will be called Wj and

w

and

it

it will

R(w )>Q, R(w )<0. 2

l

We

then have

HJm (v sec 8) = where x =

The

v sec 8.

except that a

is

e

:

wi

XT \

~-

-r- \ dr,

[dr

Jo

dr)

now proceeds

analysis

by

replaced throughout

exactly on the lines of § 8*4 %8, and the Bessel function is of the

third kind. It is thus found that

\dr

Jo

To determine the phase arg {(Wi

— i8)/r} -*• + \w

dr) of o

,

as t -* 0, a„

xm+*

m=0 that

is

of (— £i sin £)""*,

we

obseive that

and so

= el»7V(£sin/8).

Consequently

H

(1)

In

¥

like

{1)

(v sec

8)

~

—

_

1,

L-*l

T (£)

* . '

*

(| pi tan £)*

manner, by taking as contour the reflexion of the preceding contour

in the real axis of the w-plane, (2)

X

*JQvrr tan 8) M =

we

find that

HjHv*e*e)~^T*~™;: 5 T tt h) V(Wtan/8) '(-i^tan/S)** .-

w=0

T(i)

In these formulae, which are valid when /8 is a fixed positive acute angle and v is large and positive, we have to make the substitutions :

^2 = Th

+ ih cot2 y3 + $& «*•£,

and

we

(3 > j

If (4)

we combine

(1)

(2),

find that

J¥ (v8ec8)rsj v

\pwtaa 8/ L _i_

•

/

*

^ *>

o

^ i

* n

«=o

m ? (-)

T(i)

r(2m + 8)

(±i/tan/S)»»

Atn+r

"I


8'41, 8-42]

245

F„(i/sec£)~

(5)

w r(2m l) ilw, / \*T sin(>tan£-i//3-±7r) a a x v ?'.-• 2 (-) =±-r + *-'.,-. 3/ Vi^rtan^/ L r<$) (|i»tan£J*» «=o

/ (

—

2

•

I

*

'-

'

^„

0-)"-r(2m + f)

|

-cosCi/tan^-iz/g-iTr)

The dominant terms

——

i

*.

(^tan/9)2 '» +1

r(i)

)W =o

in these expansions are those obtained

by the principle

of stationary phase in § 8*21.

8*42.

Asymptotic expansions of Bessel functions whose order and argument

are nearly equal.

The formulae which have been established in §§ 84, 8'41 obviously fail when a (or £) is small, that is when the

to give adequate approximations

argument and order of the Bessel function concerned are nearly equal. It is, however, possible to use the same method for determining asymptotic expansions in these circumstances, and it happens that no complications arise by supposing the variables

to be

complex.

Accordingly we shall discuss the functions

#.«(*),

HJ*(i)

t

where z and v are complex numbers of large modulus, such that \z — v\ not large. It will appear that it is necessary to assume that z — v *= o (z$), order that the terms of low rank in the expansions may be small.

is

in

We shall write v

and

it is

We

= z(l- €

),

convenient to suppose temporarily that

then have

H^

(1)

v

where the contour and negative.

(z)

is

= —.j

exp

\z

(sinh

w — w) + zew] dw,

that shewn in Fig. 18; on this contour sinh

w— w

is real

We write t=

w — sinh w,

and the values of w corresponding to any positive value of t will be called w and wt of which wx is a complex number with a positive real part, and w2 is a real negative number. Y

,

We (2)

then have

H& (z) = i [ V" iexp (zew,)

^-

exp (zews )

^l dr.


246

The expansion

w

of t in powers of

[CHAP. VIII

begins with a term in

its*,

and hence we

obtain expansions, of the form

-j-i

exp (zeWi)

«*»"

exp (zewt)

and these are valid when t |

To determine the

|

= t~ 3 2 m T Jm b.

^p = t-3 2 is sufficiently

we

coefficients bm 1

,

m=

e» <"*+»

« bm t*»

small.

observe that

/-(0+.0+.0+) 0+.0+) /•(<>+,

/dw

v

rf

T

<»»+!)

_j_r(o+)

~ As is

6XP (

6wt J

^w ^W) (w - sinh «/)*

<»' +1 >

*

in the analogous investigation of § 8*4, a single circuit in the r-plane

inadequate, and the triple circuit

is

necessary to dispose of the fractional

powers of t; a triple circuit round the origin in the T-plane corresponds to a single circuit in the w-plane. It follows that bm is equal to Je* ,m+1,w * multiplied

by the

coefficient of

wm

in the expansion of

exp (zew)

.

{(sinh

w — w)/w*}~l

("*+1 >.

The coefficients in this expansion will be called b (m%

It is easy -to

&i(w), b2 (m),

shew that

(b,(m) = &*"**, bx

(m) = &<*+*&,

w.).a«- {£-*£}. (3)

»w=

6

61(m+l)

o*{m)

o

tT--V-|' -»-

|

24

12Q

For brevity we write &w (™)

= 6*"»+»£m (e*),

504()0

j.

...

so that


8-42]

247

so that*

5 B

4(«)-e* B (es) = *«V - ^«,

= l, - fa % (m) - \
t

3

(4)

^5

[3,(0)

We

=

(e»)

- ^J- " ifo*

-^ ^=

3

*3

ft<0)-T7*flfanr.

(jreto,)

ttT

ir~*

2

ei<™+u« 6*
»=0

ezp(#e«^)^*-iT-< 2 (IT

\

and [exp (zew) (dw/dr)] .

It follows

and

We

— „^jb.}

satisfies

9

£TO (e*) t**

^Q^m+li Bm (e2)T^n

m+1

lemma

the conditions of the

t

of § 8*3.

of § 8'3 that

H^(z)^^î^^Bm h *{*)

e^

>

TO=0

from the lemma

similarly

(6)

4.(0)

then have

exp

(5)

+ ««*«*,

— AJ^-i(« ^ +

(ez) S

5TO(

mUni + l)^.^^,

^)sini(m + i )w .r|iî|)

deduce at once that

(7)

j^z

(8)

F,

From

^Lh

Bm{ez)

W ~-^2

^ Hm+l)v '~m^

(->.^M«in4(m + l)».5jg^). f

the Cauchy-Meissel formula §

821

(2), it is to

be inferred that, when

m is large, <»)

^m(0)~ r(§) (w + i)J(127r)Jw

,

but there seems to be no very simple approximate formula

for

Bm (ez).

The dominant terms in (7) were obtained by Meissel, in a Kiel Programing 1892; and some similar results, which seem to resemble those stated in § 8'43, were obtained by Koppe in a Berlin ProgrammX, 1899. The dominant terms in (8) as well as in (7) were also investigated by Nicholson, Phil. Mag. (6) xvi. (1908), pp. 271 279, shortly before the

—

appearance of Debye's memoir.

The values of B (0). Bt (0), ... B w (0) were given by Meissel, Astr. Naeh. czxvii. (1891), 359—362; apart from the use of the contours Meissel's analysis (of. § 8*21) is substantially the same as the analysis given in this section. The object of using the methods of contour integration is to evade the difficulties produced by using generalised integrals. The values of B (tz), Bj (cz) and JB9 (ez) will be found in a paper by Airey, Phil. Mag. (6) xxxi. *

col.

(1916), p. 524.

—

t See the Jahrbueh liber die ForUchritte der Math. 1892, pp. 476 178. I See the Jahrbueh itber die ForUchritte der Math 1899, pp. 420, 421.

:


248

We

next consider the extent to which the condition

has so far been imposed on formulae (5)

The

[CHAP. VIII

—

(8), is

|

arg z

j

<

\tr,

which

removable.

qua function of t, are the values be a monogenic function of t, so that the t 2 singularities are the values of r corresponding to those values of w for which singularities of the integrand in (2),

of t for which

w

(or

w)

fails to

= 0.

drjdw

They are

therefore the points

t

where n assumes

=

Intri,

integral values.

all

swing the contour through any angle w than a right angle (either positively or negatively), and we then

It is consequently permissible to less

obtain the

analytic

continuation of

H

(l)

(z) or

v

H® v

(z)

over the range

— £w -

V < arg z < \ir — t\. By giving v suitable values, we thus find that the expansions (5) (8) are valid over the extended region

—

— 7r <

arg z
If we confine our attention to real variables, we see that the solution of tbe problem is not quite complete ; we have determined asymptotic expansions of Jv (x) valid when x and v are large

and

(i)

xjv

<1,

(ii)

x/v>l, (iii) x- v not large compared with #*. But there (i) and (iii) and also between (ii) and (iii), and in these |

|

are transitional regions between

transitional regions xjv is nearly equal to 1 while

|

—v

a?

|

is large.

In these transitional

regions simple expansions (involving elementary functions only in each term) do not exist.

But important approximate formulae have been discovered by Nicholson, which involve Bessel functions of orders + J. Formulae of this type will now be investigated.

Approximate formulae valid in

8*43.

The

failure of the formulae of §§ 8*4

the transitional regions.

—8*42 in the

transitional regions led

Nicholson* to investigate second approximations to Bessel's integral in the following

manner

In the case of functions of integral order

Jn («) = and,

when x and n

cos (n0

n,

— x sin 0) d$,

are nearly equal (both being large),

it

follows from Kelvin's

principle of stationary phase (§ 8*2) that the important part of the path of

integration

the part on which

is

sin 6 is approximately equal to

x and n under

is

small

— \ 0*.

;

now, on this part of the path,

It is inferred that, for the values of

consideration,

Jn (*0 ~ - I'gos (n0 ffJQ

OS -

f

"cos (n0

-x0 + %x0») d0

-x0 +

\x0*) d0,

ir J a * Phil.

Mag.

xxiv. (1916), pp.

(6) xix. (1910),

239—250.

pp.

247—249

;

see also Erode, Archiv der Math,

und Phys.

(3)

:

FUNCTIONS OF LABGB ORDER

8*43]

and the

last expression is

249

one of Airy's integrals (§6*4).

It follows that,

when x
Jn (x) ~ - {-^^-} *1 [

(1)

and,

when

3

^

j

>

> n,

a?

^(^ÎjîOp^H.^,

(2)

where the arguments of the Bessel functions on the right are £

The corresponding formula Nicholson

;

Yn (x)

for

when

a;

> n was

with the notation employed in this work

{2

(x- n)] }/#*-

also found

by

it is

r.(.)~-p^(V-i-^-

(3)

The chief disadvantage of these formulae is that it seems impossible to determine, by rigorous methods, their domains of validity and the order of magnitude of the errors introduced in using them.

-

remedying this defect, Watson* examined Debye's integrals, and discovered a method which is theoretically simple (though actually it is very laborious), by means of which formulae analogous to Nicholson's are obtained together with an upper limit for the errors involved.

With a view

to

The method employed

is

the following

Debye's integral for a Bessel function whose order v exceeds

x(~

j/sech o)

may be

— ~——

g»(tanha— a) v

t

(v sech a)

=

= — sinh a (cosh w —

the contour being chosen so that r If r

is

argument

written in the formf

J where

its

is

1)

rx>-\-ni

e~XT dw,

/

— cosh a (sinh w — w),

positive

on

it.

expanded in ascending powers of w, Carlini's formula is obtained neglecting all powers of w save the lowest,

when we approximate by -\wn sinh a; and when a = '

tained by neglecting

all

0, Cauchy's formula of §8 2(1) powers of w save the lowest, —\wi

is

similarly ob-

.

These considerations suggest that it two terms, namely — \vP sinh o -

is

desirable to

examine whether the

first

may

£w3 cosh a,

not give an approximation valid throughout the

The

integral which

we

iirst

transitional region.

shall investigate is therefore

\e-**dW, where

r * Proc.

t This w.

B. F.

Camb. is

= - \ W* sinh a - £ W

s

cosh

96—110. making a change

a,

Phil. Soc. xix. (1918), pp.

deducible from § 8-31 by

of origin in the u-plane.

9


250

[CHAP. VIII

W

and the contour in the plane of the complex variable is so chosen that t is on it. If U + iV, this contour is the right-hand branch of the

W=

positive

hyperbola

tftanha and

+ Jtf^F

2

this curve has contact of the third order

+ni

r

/•ocexp(Jn-i)

e~ ZT

I

These integrals

with Debye's contour at the origin.

be shewn that an approximation to

It therefore has to

J

,

dw

is

-ni

oo

differ

e'^dW.

I

J ooexp( — Jirt)

by

i/:+/;H£-^K and so the problem |

{d(w— W)fdr)

which

will

j.

reduced to the determination of an upper bound

is

And

for

has been proved, by exceedingly heavy analysis

it

not be reproduced here, that

W

d (w dr

)

i

<

Sir sech a,

I

and so

<

ifj>j:H£-£i* Hence

~

e-*r

dw = J_l ATI J

ZTTl J ao-ni

where 6l j

j

<

6-7T

v

e-*rdW «,

+ -l, V

ex p (- Jiri)

1.

To evaluate the §6*4),

integral on the right (which is of the type discussed in modify the contour into two lines starting from the point at which

W = — tanh a and making angles + %ir with the real axis. If we write W= - tanh « ê ±ini on the respective rays, the integral becomes 4-

e*

ni

exp (^ tanh3 a) [ exp {- & v?

- \ v%e$ ni tanh2 a}

dl-

Jo

— e~ ini exp (Xv tanh

3

<

Expand the integrands procedure which

is

§7ri tanh a

a)

|

exp

\-%v?- |i>£e-*™ tanh

s

«}

d£

powers of tanh 2 a and integrate term-by- term and we get on reduction easily justified in

—

exp (£1/ tanh3 a)

.

[7~_j (£1/

—

tanh 3 a) - /j {\v tanh 3 a)],

and hence we obtain the formula

Jv {v sech a) =

(4)

75IT

VO

exp

{v

(tanh a

-f

3 ^ tanh a

-I-

whore d ]

x

1

<

1.

This

is

3#i

i.-

— a)} if

-1

j (

exp

[v

£ i>

tanh3 a)

(tanh o —

a)},

the more precise form of Nicholson's approximation

(1).

FUNCTIONS OP LARGE ORDER

8*43] It can be

shewn

251

whether \v tanh8 o be small, of a moderate size, or a smaller order of magnitude (when v is large) than the approximation given by the first term on the right. that,

large, the error is of

Next we take the case x (= v sec 0).

We

which the order v

in

then have

H® (v sec 0) = where

T

= - i sin

gvi(tanj3—0)

foo+Hir-p)

e~*r div,

:

Tn

J -co-ip

w - 1) — cos

(cosh

the contour being so chosen that t

The

than the argument

is less

(sinh

positive on

is

w - w),

it.

process of reasoning already employed leads us to consider the integral

e-^dW,

/ where

= -\i W* sin

r

-

£

W

3

cos 0,

W

and the contour in the plane of the complex variable is such that t positive on it. If WsU+iV, this contour is the branch of the cubic

(U*-V*)ta,n0 + $V(3U*which passes from

— oo — t'tan/9

It therefore has to

it

oo

exp^Trt.

be shewn that an approximation to roeexpini

e-^dw

I

and

=

through the origin to

foo+t(ir-/3)

The

r*)

is

is

e~XT dW.

/

difference of these integrals is

has been proved that,

when*

d(w- W)

%

\ir,

<

127r sec 0.

then

UiT

Hence

it

follows that

e-^dw^—.

-. TTIJ

where

To

|

& < j

fi

dW + zl^ v

)

1.

evaluate the integral on the right, modify the contour into two lines

meeting at real axis.

W= —

On

i

and inclined at angles

tan

The important values

of

/3

i

tan

§ 8'32 is

-f

- i tan

are, of course, small values.

of § 8-41 yield effective approximations.

proved in

and

Kir

ir

respectively to the

these lines, write

W= *

e -xr

irXJ- oo-i tan

-*-ip

+ %&\ If

/3

The geometrical property

used in the proof of the theorem quoted.

is

not small, Debye's formulae

of Debye's contour

which was


252

expand the integrands

in

powers of tan a

j3,

[CHAP. Vin

integrate term-by-term, and

it is

found that e~XT d

W=f

iri

x

= On

tan # exp (- \ vi tan3 £)

[«-*»*

J_j

7re
(4 v tan8

£)

+ e**' Jj (^ v tan

^ exp (- J i* tan

equating real and imaginary parts,

3

£)

#j<'>

s

#)]

(^ tan

3

£).

at once found that

it is

Jv (p sec /3) = £tan £ cos {* (tan /3 - £tan £-£)}. [J"_j + /j] 3

(5)

+ 3 -* tan £ sin

= | tan /3 sin

F„ (i/ sec £)

(6)

- 3"* tan £ cos

£ - £ tan £ - £)} [J. j - Jj] + 240 (tan £ - £ tan £ - £)} [J"_ j + J»] (tan £ - £ tan £-£)}. [«7"_ -
(tan

[v

2 /i/,

.

3

{i>

.

3

{*/

3

j

where the argument of each of the Bessel functions J±>. on the right is %v tan 3 £ and and are both less than 1. These are the more precise 2 3 forms of Nicholson's formulae (2) and (3); and they give effective approximations except near the zeros of the dominant terms on the right. ;

!

|

\

|

It is highly probable that the

upper limits obtained

for the errors are

largely in excess of the actual values of the errors.

8*5.

Descriptive properties* of

The contour

Jv (vx) when <

a?

< 1.

which was obtained in §8*31(1) to represent an asymptotic expansion of the function. v But the contour integral is really of much greater importance than has hitherto appeared for an integral is an exact representation of a function, whereas an

J

(v sech a)

integral,

was shewn

in § 8*4 to yield

;

asymptotic expansion can only give, at best, an approximate representation.

And is

the contour integral (together with the limiting form of

it

when x =

peculiarly well adapted for giving interesting information concerning

when

1)

Jv (vx)

v is positive.

In the contour integral take v to be positive and write «/

so that

u = log r,

v

= log {reie

},

= d.

With the contour

selected,

x sinh w — w is equal to its conjugate complex, and the path of integration flexion in the real axis. Hence v

(vx)

= r—J i __

own

re-

roo+n-t

1

J

is its

rw I

e"

" inh »-"»

,a

dw

gv(as8inhw—w) fly^

ttJo *

The

results of this section are investigated in rather greater detail in Proc.

(2) xvi. (1917), pp.

150—174.

London Math. Soc.

—

*


8-5]

Changing the notation, we

253

find that the equation of the contour is

20

1

x

r

sin

so that

and,

when

this substitution is ,

log °

made

— x sin—

(w — x sinh w)

for r, the value of

e + y/ifr-X* sin 0) 2

at — — cot.a ya(tr .

-.

a;

*m

•

a 2

is

sin 2 0).

This last expression will invariably be denoted by the symbol* F(0,x), so that

J¥ {vx)=^{'»de,

(1)

and by differentiating under the integral sign justified) it is

w

l \ JT»i ¥ (vx) =

/o\

(2)' K

f

which

is

easily

is also easily

—— —-——— Ja fl- a'Binflcosfl -

—fib*

e-» *<».*<

ttJo

This

(a procedure

found that .

x */(&* — x* sm* 0)

dv.

deduced from the equation raa+iri

1

JJ (vx) =

s—

.

\

e

»(x*inhu>-w)

sinh

w

^

Before proceeding to obtain further results concerning Bessel functions, is

it

convenient to set on record various properties! of F(0,x). The reader will

easily verify that

<

m^'^

3>

^-M)*^-^' ^ 6

-

4^.,-â^

w so that

F(0,x)>F(O,x)>F(O,l) = O;

(5)

and

also

d

ia\

Next we

V(<* „\

shall establish the

fl-g'sinflcosfl

more abstruse property

F(0, x) >F(0,x) + $ {& -

(7)

To prove

it,

we

shall first i

*

a

\

n

a? sin 2 0)/V(l

+ x>).

shew that O — aPsmO cos 6

..,

„.

This function will not be confused with Schlafli's function defined in § 4-15. supposed throughout the following analysis that O<.c^l,O^0
t It is


254

[CHAP. VIII

It is clear that

g(0,x)

= s'(\-x*)<

g(Tr,x)=\ so that, if

g (0, x), qua function of

greatest value

when

had a value

-

1

a;

2

cos

attained its greatest value at

0,

^(1+x

that value would be less than

2

between

O

_ (0 - x

2fl

2

and

si n

O

2

O

cos

2

(0o -a?sin 2

(tfo'-^sin ^)*

or

however, g(0,x) attained

If,

).

o

ir,

o

tt,

its

then

f_

~

)i

'

and therefore

g {0, x)^g (8

Hence

= V(l - a? cos 20 ) ^ V(l + «*), o

where g(0,x) attains

so that, no matter

not exceed ^(l

x)

,

+#

its

greatest value, that value does

2 )-

^W-*w^^

a

1

cos

sin

V(l+a )

and

2

so

whence


Another, but simpler, inequality of the same type

F (0, x) > F(0,

(8)

To prove

this,

'

X)

>

fe From

;

\/(0*

then the inequality

these results

J„ (vx) and

JJ (vx)

+ * 0* V(l - *").

x)

observe that

dF and integrate

is

we

are

now

- «" sin is

2

0)

> V(l - &),

obvious.

in a position to obtain theorems concerning

qua functions of

v.

Thus, since OV

IT JO

the integrand being positive by

JJ {vx)

Jv (vx) is a positive dea positive decreasing function

that

(5), it follows

creasing function of v, in like manner,

is

of v. Also, since d {**<»•*>

OV

JAva:)\ = _ 1

n (

SF{0> x)

_ F ^ 0> x)}

*-*«>. *>-"«*.*>

d0 <0,

7T.' o

the integrand being positive by

(5), it follows

function of v; and so also, similarly, is

e"

that e vF({t x

F,0,x)

'

J

v

'

(vx).

>

Jv (vx) is a decreasing


8*51]

255

Again, from (8) we have

Jv ( vx) ^

/

T

Jo

exp {- i v0°- V( 1

- a )} dd 8

e—»F(0,x) roo

<

- x*)\ d0,

exp {- £i/0\/(l

so that p -vF(0,x)

(9)

-

(1

The

last expression

(§§ 1*4,

811)

for

excess, for all

is

x*)l

^(Zttv)'

easily reduced to Carlini's

approximate expression

J„(px); and so Carlini's expression

* positive values of

The corresponding

- a? sin

0*

and replace G(0,x) by

G

always in error by

v.

JJ {vx)

result for

is

is

2

derived from

(7).

Write

=£(#,#),

for brevity.

Then

2xj;{vx)= fi

/V-™.*

-

-vF(0,x)

dG ( >^ {G(0,x)}-id0

fir«

expi-^GyVa+a^.G-MG 7T

I

.'o

e -¥F(Q,x) fao

Texp {- J^(l+^)j

IT

.

G-idG,

Jo

and so

xj;

(10)

The absence of the

factor

(vx)

^ «-'*<°. «)

^(1 —

2 a? )

(

1

+ «*)VV(2 w).

from the denominator

is

remarkable.

It is possible to prove the formula f

(11) J

*<*>*~ (2ir*y (1 -*)»

{

1

+ V (l

-W-

in a very similar manner.

This concludes the results which we shall establish concerning a single Bessel function whose argument

8'51.

We

Lemma

shall

is less

than

its order.

concerning F(d,x).

now prove

lemma

when

^ x ^ 1 and ^ 0-^sintfcos0 ^, ^ A >0. —je--{F(0,x)-F{O,x)} the

that,

dF{0,x) (1)

The lemma

will

y,^^

be used immediately to prove an important theorem con-

cerning the rate of increase of * It is

,/„

(vx).

evident from Debye's expansion that the expression

large values of

t Cf. Proc.

< ir, then

v.

London Math.

Soc. (2) xvi. (1917), p. 157.

is

in error by excess for sufficiently


256

V(^ - a

If

2

sin2 0)

= H{0, x), we

[CHAP. VIII

shall first prove that

dF(0,x) dH(0,x) d0 d0 / ,

is

a non-decreasing function of

;

that

i

— a? sin is


The

to say that

is

(\-0cot0)2 +

-a?sm*0

cos

0.

differential coefficient of this last function of

is

- 1 - £ sin 0) (1 - x*) + 2 (0 cosec - cot cosec - % sin* 0) (1 - ar ) + 2a- (1-0 cot 0) (0 cosec - cos 0) + sin 0(1-

(0 - x* sin

2

cos 0)-* [(0* cosec2

2

3

2

2

2

and

2

2

2

2 2

a,-

) ],

every group of terms in this expression is positive (or zero) in consequence

of elementary trigonometrical inequalities.

To

we

establish the trigonometrical inequalities, (i)

d+sindcosd-2^-

observe that,

first 1

sin

(ii)

6 + sin 6 cos 6 - 26* cot 6

(iii)

sin

2

when

< 6 ^ *r,

d>0,

> 0,

- 6 cos - £ sin 3 0^0,

because the expressions on the left vanish

wheu 6—0 and have the

positive differential

coefficients

2(cos0-0 -1 sin0) 2

(i)

2 (cos 6 - 6 cosec 6)\

(ii)

,

sin 6 (6

(iii)

- sin

6 cos 6),

and then cosec2 6 - Q3 cot 6 cosec 2

2

-£

sin 2 6

= (6* cosec2 6 - 1) (1 - 6 cot 2

cosec2

)

+ cosec

(sin

- 6 cos - J sin 3 6) ^ 0,

0-l-$sin 2

= 6 cosec2

(5

+ sin 6 cos - 2d"

l

sin 2 0)

+ cosec 6 (sin

-

cos 6 - £ sin 3 6)

>

0>

so that the inequalities are proved.

shewn that

It has consequently been

»(£}><> where

understood to be

the. variables are

entiations with regard to

0.

It is

now

and

x,

and primes denote

differ-

obvious that

ifiM-*£{S>* and,

if

we

integrate this inequality from

when = 0, F>(0,x)H{0x) _

Since F' and /7/i/' vanish

to 0,

we

get

this inequality is equivalent to

+

j

>Q

if (0, #)

and the truth of the lemma becomes obvious when we substitute the value of

H

(0,

x) in the last inequality.


8*52] 8*52.

We v

The monotonia, property of Jv (vx)jJv {v).

shall

is fixed,

257

and

now prove a theorem of some importance, to the effect that, if x ^ x ^ 1, then Jv (vx)/Jv (v) is a non-increasing function of v, when

is positive.

be valid only when S^.r^l, (where 8 is an some expressions introduced in the proof contain an x in their denominators; but the theorem is obvious when ^ x < 8 since e vF ®> z Jv (vx) and e -vF{o, x)/jv („) are non-increasing functions of v when x is sufficiently small moreover, as will be seen in Chapter xvn, the theorem owes its real importance to the fact that it is

[The actual proof of the theorem

will

arbitrarily small positive number), since

>

;

true for values of

neighbourhood of unity.}

Jv{vx) *M"*)_ dJ*W dJAvx)

)

Ovox

To

and,

the

be shewn that

It will first

(1

x in

we

establish this result,

when we

OV

differentiate

7T

V

Q

Cv

observe that, with the usual notation,

under the integral sign,

o

.

L\/^

•> <*

W

[

ox

\\C((l -rVh

<* *))-*

dFie X ^ >

—or

' e vn$,x)

de "

IP/A „\in(A

^W^1^L

A,)

x e-»J?ie,x)d0, if

we

integrate by parts the former of the two integrals.

Hence

it

follows that

where

by using the inequality F(yfr, x) > ^(0, x) combined with the theorem of §8*51.

;

'


258 Since CI that

is

(0,

to say,

not negative, the repeated integral cannot be negative

is

yfr)

[CHAP. VIII

we have proved

J

( vx

\

that

^H^™} _ dJAvx) dJ„ (vx ) >" dvdx

dx

'

dv

so that

\dJv (vx)

d

/

i

Integrating this inequality between the limits x and

~dJv (vx)

I

.

T

,"|*

we get

1,

A

so that

dJv (vx) Since

J

v

and

(vx)

J

v

I

,

-I Jrv (VX)

dv

(v) are

*Z

dJv (v) J* (")• —fo~ J

both positive, this inequality

may be

written

in the form

^{J

(2)

and is

y

this exhibits the result

which was

a non -increasing function of

Properties of Jv (v)

8*53. If,

for brevity,

the formulae* for

The

first

4^/(9 V3)

We

v

(v)

and

shall first

this

s

is

and

(vx)jjv (v)

v

in place of F(0,

^

1),

so that

~ Sin2 0) - cot J V(» -

n

J '{v) v

^n*

B),

are

0.

shew that

we observe

that

^~ -~ IP-* V(^ -

cot *)/*}»

6-*

v.

v

J '(v).


*"(*>

* It is to

J

is term in the expansion of F (6) in ascending powers of shall prove a series of inequalities leading up to the

F (0)1

To prove

value

be proved, namely that

and we

5

result that

J

to

v.

we write F(0)

F (0) ^ log ° +

(1)

(vx)/Jv (v)}^0,

be understood that

sin* 0)

JV (v) means

+

^ *

the value of

,,

Vi &2

_ siR2 ,.

'

dJv (x)/d.v when x has

the particular

^ FUNCTIONS OF LARGE ORDER

8-53]

259

and that d^

dd d_

f

j

1

-0 cot 0) _ ~

it

vW-sin

cosec 2

fl

+ flcotfl-

2

p

'

= _ (ffcosec'fl + flcot 0-2) sin "^VO^^-sin^) J"

a

fl))

0*

2 '

"'"

follows that

0-sin0cos0

d (F'(0)} <** I

2

fr~~~)

\

d&\ Hence

fl

#- | - * (* - sin

by inequalities proved in

2

//te

W <*

+ ecot0 ~

„v 2>

x (0 + sin

cos

.

0/1

C0Sec2 *

,

- 20* cot 0)

851.

§

Consequently

0F"(0)-2F'{0)>Q,

(3)

that

is

If

^ [OF'

to say

we integrate

3F (0)} > 0.

from

this inequality

we get

to

0F'(0)-SF(0)>O,

(4)

and

-

(0)

this is the condition that

F(0)/03 should be a non-decreasing function of

It follows that

*m

s

*

9V3'

and therefore

J

(v)< -

v

<

\

ii."

exp j-

•£(*

9lp |-

"SI-

T(i)

2*3*™*' so that Cauchy's approximation for

An

2

The truth

of this

{&*

(tf*

- sin 2

may be

always in error by

[Note.

8)

{&)

is

positive

(cf.

f

Mag.

(6)

0.

Jv

( vt )

dt

{BF'

left in

{$)

the form

- ZF{fi)\

§ 8-51).

A formula resembling those which

(6)

excess.

is

- 3 (0 - sin 6 cos 0) F{6) >

- 2 sin 2 6 + 6 sin 6 cos 6) F' (0) + (0 - sin 6 cos 6)

l

ZVtiV.

F'

seen by writing the expression on the

which each group of terms

see

(v) is

inequality which will be required subsequently

(5)

in

«/„

have just been established

—

~ J- - T

xxxv. (1918), pp. 364—370.]

,

is

%

0.


260

Monotonic properties of Jv {v) and JJ

8*54.

been seen

It has already

decreasing functions of

result

first

£r

djîoi _ dv

Sir J

we

e vFi e )de

o

Jo

|_

v

(v)

v.

observe that

^r tJo

= *Ll \$e -»FW~Y + il 07T

J

and JJ (v) are now be shewn that both v*Jv {v) and v*Jv '(v)

are steadily increasing* functions of

To prove the

(i>).

that the functions

(§ 8'5)

It will

v.

[CHAP. VIII

oTT

' f

F(d)e-.F W dd

fl

F

'

(0)

_ 3F(0)}

e~' F

^ dd

o

./

>0, since the integrated part vanishes at each limit

and

(§

8*53) the integrand

is

positive.

Hence v*Jv (v)

an increasing function of v; and therefore

is

v*J9 {v)< lim {v*J9 (i>))

(1)

In connexion with this result

J

(1)

x

To prove the second

may be

result,

o7T

dl'

it

= 0-44005,

= r(J)/(2*S*ir)- 0-44731. noted that 2«/8 (8) =0-44691.

by following the same method we

find that

JO

|_

>o, by

§

8*53 (5), and so

v*Jv'(v)

an increasing function of

is

v.

Hence v*

(2)

It is to

J: (v) < lim

t

A

JJ (v)} - 3* T (f )/(2*w) = 0-41085.

be noted that

J 8*55.

{*i

'

(1 )

= 0-32515,

4J6

theorem which

is

slightly

15

a steadily increasing function of v.

W(V)} +

,

= 0-38854.

more recondite than the theorems just proved

that the quotient

* It is

(8)

The monotonic property of v*JJ {v)jJv (v).

is

as v-*-

'

{!*/, (*)]

not possible to deduce these monotonic properties from the asymptotic expansions. If, /(*)~ (»), and if ${») is monotonic, nothing can be inferred concerning monotonic

properties of f(») in the absence of further information concerning/^).


8-54, 8-55]

To prove this result we 854 for the four functions

„U»,

use the integrals already mentioned in

-

vKTAv\

—

,

Ti/

Taking the parametric variable place of

0,

we

261

in the first

§§

853,

•

lv

and third integrals

to be

>/r

in

find that

" dv {v*J ¥ (y)

ir

J

J o J o

where n,

t) =

(0,

i

r (0>
si,,'

0)

-

^01 ^rtf F <*> V(0"-sin'0)

^^

2

>

{*«• (*) - *<*)) -

uT

^£#

yT '

KY "

»+*" <*> - *<+>

•

by §8'51. The function fi x (0, i/r) does not seem to be essentially positive (cf. § 8*52) to overcome this difficulty, interchange the parametric variables B and yjr, when it will be found that ;

Now, from the inequality just proved,

^V^-sin'^^-s^ +

^ + VrgJnVrC0B^-2gin'Vr U ^V(^

2

fl»

V

-sin2 ^)

+ flsinflcosl-2sin»l l5r

&)/(&- sin* 0) Since 0"1 ^(0*

- sin

8

0)

and

J

&F'(0)

vr '

_ '

_

V

"

vr/;

- F(0) are both (§ 8'53) increasing functions

term in the sum on the right are both positive or both negative; and, by §§851, 853, the second and third terms are both positive. Hence fl! {6, yfr) +- llj (yfr, 6) is positive, and therefore

of

6,

x

the factors of the

'

first

dv

{

which establishes the result stated,

Jv {v)

J

.


262

[CHAP. VIII

8*6.

Asymptotic expansions of Bessel functions of large complex order.

The

results obtained

(§§8'31— 8*42) by Debye in connexion with J¥ (x) where v and x are large and positive were subsequently extended* to the case of complex variables. In the following investigation, which is, in some respects, more detailed than Debye's memoir, we shall obtain asymptotic

and

Y

v

{x)

expansions associated with It will first be

J

v

when

(z)

v

and z are large and complex.

supposed that arg z\<\Tr, and we shall write |

v

= z cosh 7 = z cosh (a + i/3),

where a and # are real and 7 is complex. There is a one-one correspondence between a + ifS and vjz if we suppose that ft is restricted to lie between^ and 7r, while o may have any real value. This restriction prevents z/v from lying between — 1 and 1, but this case has already (§ 8'4) been investigated.

The

integrals to be investigated are

HJV

(z)

=

A

TTIJ

H® where f(w) =

A

(z)

e-^ w

_»

=-—

>

dw,

e-'/M

«*.'-»

dw^-~\ mJ

_.+„•

*/(* dw,

w cosh 7 — sinh w.

stationary point of the integrand

is

at 7,

and we

shall therefore in-

vestigate the curve whose equation is

If(w) = //( 7 ). we

w

+ iv, this equation may be written in the form (v - fi) cosh a cos & + (u — a) sinh a sin £ - cosh u sin v + cosh a sin /8 = 0.

If

replace

The shape {(u

by u

of the curve near

(a,

#)

is

- ay - (v -fif} cosh a sin /3 +

so the slopes of the

2 (u

- a) (v - 0) sinh a cos /3 =

;

two branches through that point are \tt

+

— i if +

£arc tan (tanh a cot

yS),

i arc tan (tanh a cot /8),

where the arc tan denotes an acute angle, positive or negative Rf(w) inw moves away from 7 on the first branch, while it decreases as w moves away from 7 on the second branch. The increase (or decrease) is steady, and Rf{w) tends to + 00 (or - x ) as w moves off to infinity unless the curve ;

creases as

has a second double-point t. * Mnnchener Sitzungsberickte, xl. [5], (1910) ; the asymptotic expansions of Iv were stated explicitly by Nicholson, Phil. Mag. (6) xx. (1910), pp. 938—943. f That is > say 0
(x)

and

A"„ (x)

*

;


8-6, 8-61]

263

If (i) and (ii) denote the whole of the contours of which portions are marked with those numbers in Fig. 19, we shall write

S,v

(z)

=—

«-*/'•">

f

.

TTI

S* (z) = -—.( 7Tt

dw,

J

(i)

.

e?fM dw,

(il)

and by analysis identical with that of § 8 41 (except that i# is to be replaced by 7), it is found that the asymptotic expansions of Sv w {z) and 8^ (z) are given by the formulae

s.»{.)~

( i)

m

s' a,

;

V

.

2)

/7

h> :"- 1 "-

\'(- \ viri tanh 7) g~ y(tanhy "

W ~ ^_ \

^

yHW

^

r( ",+ * )

s ~

T (£)

TO

tanh y))

r(«t + s i o

'

A"

(h

tanh 7) m

am

j)

r (i)

'

•

_ ^ y tanh 7)w (

,

= arg z + arg (— i sinh 7), and the value of arg (— i sinh 7) which lies between — \tr and \ir is to be taken. where

arg (— | i/7ri tanh 7)

(i)

Fig. 19.

The values of

A A ,

1}

A.2}

...

are

A = $-ji coth*y, *• = ih ~ Mr coth* 7 + Aft coth4 %

(A =l, (3)

H

1

H

n) remains to express v w (z) and (z) in terms of v and to do this an intensive study of the curve on which

It

Sv n)

(z)

and #„

( -'

(*)

//(«)- //(7) is

necessary.

8'61.

The form of Debyes contours when

The equation (1

(v

)

where

(u, v)

the variables are complex.

of the curve introduced in the last section

is

— /8) cosh o cos y8 + (u — a.) sinh o sin fi — cosh u sin v + cosh a sin /8 = are current Cartesian coordinates and

Since the equation

is

unaltered by

<

/S

<

0,

it.

a change of sign in both u and

a,

we

which a ^ and since the equation is unaltered and it — # are written for v and #, we shall also at first suppose that 0
shall first study the case in

when

it

—v

;


264

For brevity, the expression on the d(u,v)

—

it

when

follows that,

the equation in

= sinh

o sin

be called

(u, v).

Since

,

u sin

v is given, d()>/du vanishes for only

v,

one value of

u,

and so

(u, v)

= 0,

and one of these

;

is infinite

whenever v

is

a multiple

7T.

When

7r,

we have*

<£(+ oo ,v) = — oo — (a, v) = cosh a {(v /3) cos /3 — sin v + sin /3} ^ 0, (- oo

,

=—

v)

and so one root of the equation

oo

,

,

in u, v)

<£ (u,

is less

By

than a and the other

is

<£

+ oo

= 0,

greater than

a.

both becoming equal when v =

/3.

considering the finite root of the equations (u, 0)

= 0,

seen that, in each case, this root

it is

to

p — sinn .

.

u,

has at most two real roots Of

left in (1) will

,

.

[CHAP. VIII

as v tends to

greater than of the curve

or to

the equation

ir is

+

ir

— 0, and

(u, v)

(u, it)

is less

=

= 0,

than

a,

so the larger root tends

for values of v just less

shewn by the continuous

therefore roughly as

than

or just

has a large negative root. The shape

Next consider the configuration when

v lies

between

lines in Fig. 20.

and

— ir.

TTi

\ )

^

J

— TTi y-2iri Fig. 20.

When

v

is

minimum

— /3,

d(j>(u,

v)/du vanishes at

2 cosh a sin at

u — — o. There

are

now two 1

is (I) *

u

= — a,

and hence (u,—@) has a

value 13

(1

— /Q cot y8 — a tanh a)

cases to consider according as

— yS cot j8 — o tanh a

positive or (II) negative.

Since

d (a,

u)/3t'=cosha(cos/3-coBv), and this has the same sign as v-/3,

minimum value zero

at v

= £.

4> (a,

r)

has a


8-61]

The domains

of values of the complex 1

is

positive (in the strip

domains numbered responding domains

y=a

+

265

ift for

which

— >8 cot ft — a tanh a

0^#<7r)

numbered

are

1, 4,

5 in Fig. 21

;

in the

2, 3, 6a, 66, 7 a, 76 the expression is negative; the cor-

for the

complex

viz

— cosh (a 4- ift) have the same numbers

in Fig. 22.

TTl -1>

4

5

66

7b 1

5

6a

la 3

2

Fig. 22.

Fig. 21.

(I)

When 1 — ft cot ft — a tanh a is positive, (a, — ft) is essentially positive, = — ft. The only possibility therefore the curve after crossing the real axis goes off to — 00 as shewn by the

so that the curve never crosses the line v is

that

upper dotted curve in Fig.

20.

(II) When 1 — ft cot ft — a tanh a is negative, the equation (— a, v) = — and ft 27r, for has no real root between — cos v). o(— a, v)/dv = cosh a (cos Therefore has a (— a, v) single maximum at — #, and its value there is so (— negative, that and $ — 27r. a, v) is negative when v lies between

Also

(f>

(u,

/S—

so that the curve

2tt)

has a

(u, v)

maximum

=

at u

— a, and its value

does not cross v

the real axis, the curve must pass off to

00

= /S — 2
— in,

as

;

there

is

negative,

hence, after crossing

shewn by the dotted curve

on the right of Fig. 20. This completes the discussion of the part of the curve associated with S, m (z)

when a>0, O<0^$tt.

Next we have line v

to consider

what happens

to the curve after crossing the

= + ir.

Since

(a, v)

= cosh a {(v — (3) cos ft — sin v 4- sin ft},

and the expression on the right the line

u~a; (u,

is

positive

when

v

^ ft,

the curve never crosses

also

mr) = (u — a) sinh a sin ft + (mr

— ft) cosh a cos ft + cosh a sin ft,

;


266

and

this is positive

infinity

when u >

on the right must

When i.e.

when

a,

which go

so that the parts of the curve

as shewn

lie

[CHAP. VIII off to

in the north-east corner of Fig. 23.

- a tanh a + (ir - ft) cot ft > 0,

1

any of the domains numbered 1, 2 and 3 in Fig. 21, it is = 2v — ft, and so the curve after crossing ac + iri as shewn in Fig. 23 by a broken curve.

(a, ft) lies in

found that the curve does not cross v v

= 7r

passes off to

—

Fig. 23.

We now

have to consider what happens when

numbered 6a

1

and

^ (— a,

negative.

u

=—a

v)

has a

The

- a tanh a + (?r - ft) cot ft <

maximum

(— a,

the

first

at v

(2tt

positive,

= 2tt — ft, the value of (— a, 2ir — ft) being = tt, consequently remains on the right of

4tt

- ft),

of these intervals in which

(u,

;

= 2rr — ft.

+ ft,

(2Mtt

Then

domain

v) is increasing in the intervals

(/3,2rr-0), let

in the

curve, after crossing v

until it has got above v

Now

(a, ft) lies

In such circumstances

in Fig. 21.

2Mir +

2ir

— ft)

it

(4>Tr

+ ft,

6ir

- ft),

. .

.

becomes positive be

+ ft, 2Mtt +2-7T- ft).

has a

minimum

— — o, at which its value is t>= 2Mir + 2tt — ft; it must

at u

and so the curve cannot cross the line left, and consequently goes

therefore go off to infinity on the

to

-oo +(2M+l)iri; it

cannot go to infinity lower than

meet a horizontal

line in

this, for

more than two

then the complete curve would

points.

:


8-61]

When

(a, ft) is in

267

6a, the curve consequently goes to infinity at

-oo +(2M+l)Tri, where

M

is

the smallest integer for which

1

— « tanh o +

{(M

+

1) tr

— /9}

cot

#

is positive.

We can now construct a

table of values of the end-points of the contours

and $„ (z), and thence we can express these integrals in terms of (z) and (z) when (o, fi) lies in the domains numbered 1, 2 and 6a in v v Fig. 21 and by suitable reflexions we obtain their values for the rest of the complete strip in which < yS < ir. The reader should observe that, so far as the domain 1 is concerned, it does not matter whether /S is acute or obtuse.

for &„ (1) (z)

H

{1

(2)

H

>

(i)

;

If

M


is

l-o tanh a + is

positive

when

cot

£

is positive,

{(M

and

+

if

1)

N

ir

is

-

£} cot

fi


l-o tanh o - (Nir + £) cot £ is

positive

when cot#

is

negative, the tables of values of

Sv

(z)

(1)

and Sv ®

(z)

are as follows

Regions

End-points

— 00

1,3,4 2,6a 5,

76

66 7a

oo

—

-oo,

3,

7a

From

2J„

+ (2M+l)iri

— oo + iri,

oo

— oo + ir»,

— oo —

+ 2»ri,

oo

6a

-ao

+

(2J/

— ao + »rt,

{

i

e-Mvniffv V)(ze-Mni)

1

#„<*>(*)

2J.(z)

iri

Zevwi

+ l)»ri, —

l

SyW (z)

oc

oo

(z)

2e-îJ_ v z ) X»*iH e v ){ze-if*i)

End-points

4,66

76

2Niri, oo +- iri

cc

H„W(z)

oo

,

— oo —

Regions

1,2,5

+ 7Tl + iri - oo + 2m' OC

,

iri,

- oc

S„0>( 2 )

oc

2Niri

J_ v ( 2 )

eMvwi ffjM (zeMni)

e-Wvwi ffJP)

( ze

Nwi)

these tables asymptotic expansions of any fundamental system of

when v and z are both arbicomplex numbers, the real part of z being positive. The range of validity of the expansions can be extended to a somewhat wider range of values of arg z by means of the device used in § 8*42. solutions of Bessel's equation can be constructed

trarily large


268 The reader pass from

— oo

the region

1

will find it interesting to

and from - oc

prove that, in the critical case /3=£»r, the contours + ni to oo , so that the expansions appropriate to

are valid.

The

Note.

+ iri

to oc

[CHAP. VIII

between the formulae

differences

da and 66 and also for the and by "Watson, Proc. Royal

for the regions

regions 7a and 76 appear to have been overlooked by Debye, Soc. xcv. A, (1918), p. 91.

8*7.

Kapteyris inequality for

Jn (nz).

An

extension of Carlini's formula (§§811, 8'5) to Bessel coefficients in which the argument is complex has been effected by Kapteyn* who has shewn that, when z has any value, real or complex, for which z- — 1 is not

a real positive numberf, then w

exp{w\/(l--g2 )}

,g

\Jn(n*)\*

(1)

{l

+ V(l-* )} n a

2 This formula is less precise than Carlini's formula because the factor (2jrn)* (1 -s )* does not appear in the denominator on the right, but nevertheless the inequality is sufficiently powerful for the purposes for which it is required %.

To obtain the

inequality, consider the integral formula

r—

Jn {nz) = ^. in

which the contour

a

is

1

exp { \nz

(t

-

1/t)} dt,

J

w

circle of radius e

,

where u

is

a positive number to

be chosen subsequently. If

we

write

t

— eu+i8 we ,

— f exp

Jn (nz) Now,

if

get [n [\z

(eV« - eê~ i9) -u-

id}] dO.

M be the maximum value of |

on the contour,

it is

exp {\z (eV*

- e- u e~

i0

)

-u- id]

\

clear that

\jn (n*)\*M:

But

if

z

- pe ia

,

where p

is

positive

\p

is

and

this attains its

w {e cos (a

maximum

its

value

is

is real,

then the real part of

- er -u-id u cos (a e~ + 0) $)} - u,

value

tan 6

and

and a u e-*)

\z (eu eu>

when

— — coth u tan a,

then s p V(sinh u

+

sin2 a)

- u.

Ann. Sci. de VEcole norm sup. (3) x. (1893), pp. 91—120. when z approaches the real axis it follows that the t Since both sides of (1) are continuous sign may be given to the inequality is still true when «* - 1 is positive: for such values of z, either radicals according to the way in which z approaches the cuts. *

X See Chapter xvii.

' '


8-7J

Hence,

for all positive values of u,

|

We

269

Jn (npe

now choose u

im

)

j

v/(sinh 2

< exp [np

u

+ sin8 a) — nit]. may be

so that the expression on the right

by

possible in order to get the strongest inequality attainable

as small as

this

method.

The expression

+

2 p V(sinh u

has a

minimum, qua function

sin3 o)

when u

of u,

—u

chosen to be the positive root of

is

the equation* sinh u cosh u

u

\/(sinh 2

With

this choice of u it

2 V(l and,

by taking z

the ambiguity.

and

may be proved

—

s "8

)

to be real,

that

sinh u cosh u

•

1

p

sin* o)

-f

=±

— g2la ),

(cosh 2«

clear that the positive sign

it is

must be taken

in

Hence

- z*)) sinh u cosh

2 {1

+

\/(l

-z) _

V(l

u=

e2"

-

«*,

so

exp

z

log

T+

!

+

2 V(sinh a u

%

,

sin2 a)

g

s \~ V(l - z*) V(i-* )

I

e

3*

.

|

exp

- e**»

\/(l

-z) a

\

I

= 5V(l-z')-« sinh 2 m

+

sin2 a

w

sinh u cosh

= p \/(sinh 2 u + and

now

it is

sin 2 a)

— u,

clear that

rexpV(l-^) fcexp V( 1-^) |"y

Jn (nz) $

j

j

I

An

interesting consequence of this inequality

as both

j

z

'.

^

1

s exp

I

To

is

that

-v/(l

= pth,

sinh u cosh u

The previous

+ sin

2

a)

analysis shews at once that,

— z*)

[

|

gexp y'(l l+y/(l-Z>)

2 p \/(sinh a *

|

^

1 so

is satisfied,

and define u by the equation

V(sinh 2 w

then

Jn (nz)

long

— z')

construct the domain in which the last inequality

before z

;

and

This equation

is

+ sin

2

a)

_

1

"~

p

when

-

1'

— u = 0.

a quadratic in sinh 3 u with one positive root.

write as


270

[OHAP. VIII

It follows that

P*

As u

2u

=

sinh 2 m

increases from

a

sin*

'

T1997

to

creases from 1 to* 0-6627434....

— sinh u (u cosh u — sinh

...,

It

sin2 a increases from

then clear that

is

|

and on the boundary of an oval curve containing the

Fig. 24.

is

shewn

The domain

in Fig. 24

it

;

in

will

which

|

J*n (nz)

j

and p de-

to 1

* ex PV(l-^2) i

inside

u).

+ V(i-* ) a

origin.

$1

This curve

certainly does not exceed unity.

prove to be of considerable importance in the

theory of Kapteyn series (Chapter xvn).

When the order of the Bessel function is positive but not restricted to be an integer we take the contour of integration to be a circle of radius e* terminated by two rays inclined + it — arc tan (coth u tan a) to the real axis. If we take 1 = ev on these rays, we get 1

1

/""

, ,, „ T \W»)\
\amvir\

,

;

j^/l + [

i

r

s

2!L!^|

| I

w

I

cosh (u + v) — cos 2a cosh (v-u)

\/(sinh^ + sin^)

"1

- vV d, ° \

e xv{-v(v-u)}dv\

J u

)

and so |Jr(v*)|< \l

This value

+ is

\\

I

pex P V(l- g») )' 1 + ^(1 -*«) j

l/it

given by Plummer, Dynamical Astronomy (Cambridge, 1918), p. 47.

CHAPTER

IX

POLYNOMIALS ASSOCIATED WITH BESSEL FUNCTIONS 9*1.

The

of Neumanns polynomial

definition

The


The

first

On (t).

is the discussion of certain polynomials which occur in various types of investigations connected with Bessel functions.

of these polynomials to appear in analysis occurs in

Neumann's *

investigation of the problem of expanding an arbitrary analytic function f(z) into a series of the form 2an n (z). The function n (t), which is now usually

J

called

Neumanns

expansion of

W

polynomial,

— z)

l/(t

defined as the coefficient of

t=z~ Jo (Z) °° (<) + 2Jl (2) ° - S n~0

From

is

en

Jn (z)

in the

as a series of Bessel coefficients f, so that

en

l

(<)

+ 2J* < 2 >

°* <*)+•••

Jn (z)On (t).

we

shall derive an explicit expression for the function, and it will then appear that the expansion (1) is valid whenever \z\< \t\. In order to obtain this expression, assume that z < t and, after expanding l/(t - z) in ascending powers of z, substitute Schlbmilch's series of Bessel coefficients (§ 27) for each power of z. this definition

j

\

j

j


-— =- + t-z <

-

1 fr

Assuming *

°°

1

1

2 s=1

s

— 7*

** +1

„

T

for the

2

v

M

*=1 &

v

(,»i

(s

+ 2m).(s + m-l)l nl

=

r

j

-

1

moment that the repeated series is absolutely convergent*.

Theorie. der Bessel' sch en

Math,

(,\a.

m=0

lxvii. (1867), pp. 310

Function en (Leipzig, 1867), pp. 8

— 314.

derive the differential equation which will

—

15, 33 see also Journal fur assuming the expansion (1), is to be given subsequently (§ 9 12) and to solve it in

Neumann's procedure,

;

after

-

series.

t In anticipation of § 1611, we observe that the expansion of an arbitrary function by substituting for l/(t - z) in the formula

,=. f(z)r

1

224—225.

obtained

/(*+)/(«)«" (

si/

+ Cf. Pincherle's rather more general investigation, Rcndiconti R. pp.

is

1st.

Lomlardo,

(2)

xv. (1882!,


272

we is

a rearrangement by replacing

effect

a series of Bessel coefficients 1

1

»

r—- = 7c i *

*

r

S

««n«/»(')+

n=l

m=<\

(
«=0

V

1

n u\

n.(n-m-

— m!

•

f

o,(«)

r/x

1)1)

«/.(*) J

^"n.in-m-l)!

^"i,V ml(i«^

(3)

series

by the equations

n (<) are defined

Accordingly the functions /ox (2)

by n — 2m, and the rearranged

we thus get

;

3

x

,

s

[CHAP. IX

{n>l) -

-iyt

It is easy to see that

en

(4)

and the

O„(0*-îT

:

|i

+

2(2n

_

2)

+ 2i4t(2n-2)(2M _ + ...|, 4)

any possibility of a denominator factor

series terminates before there is

being zero or negative.

We l/(*

have now to consider the permissibility of rearranging the repeated series for

A sufficient condition is

- 1).

that the series

2* % % (<+2m).(«+m-l)i * iTTTi J1 * TTi »»' *

«.i

should be convergent. (4),

t

i

U-o

To prove

that this

.

. \ , «+2»> \'/ If

*/ .

;

actually the case,

is

we observe

that,

by

§ 2' 11

we have

m=0

'MI

m=0

»M

I

(*+«»»—

1)!

+ <3(J|*|)« «-»{exp(i|*P)}/(2m)!

<(il*l)*e*P(il*l

2

)-

Hence s

_J1_

/

:

(«+2m).(« + m-l)l

,

,,

»

1

,

_

|«|«

„_,.,_,*

|«|eipQi«|») '

|«|(|*|-|*|)

The absolute convergence of the repeated series is therefore established under the hypothesis that \z\ < \t\. And so the expansion (1) is valid when \z\< \t\, and the coefficients of the Bessel functions in the expansion are defined by (2) and (3). It is also easy to establish the uniformity of the

pansion (1) throughout the regions

1 1

|

^ R,

convergence of the ex\z\^r, where R > r > 0.

ASSOCIATED POLYNOMIALS

9*1]

When

these inequalities are satisfied, the

^_

x

•

+ 2m).(« + m-l)!

(8

J

.lo^^lio

"»!

sum

273

of the moduli of the terms does not

( £/•)*

+2ro

*" (*

exp

Since the expression on the right is independent of convergence follows from the test of Weierstrass.

z

exp

(fr*) \

+ 2m)!

and

t,

8

(frr )

fl-r

J-

•

the uniformity of the

was called by Neumann a Bessel function of the second now used (cf. §§ 353, 3*54) to describe a certain solution of Bessel's equation, and so it has become obsolete as a description of Neumann's function. The function n (t) is a polynomial of degree n + 1 in 1/t, and it is. usually called Neumann's polynomial of order n.

The function

n (£)

kind*; but this term

is

Neumann's polynomial is reversed by writing n is even or odd, it is at

If the order of the terms in

\n — m

or \{n

— \) — m

m

for

in (2), according as

once found that /k\ (5)

+ m-

n.fjw

* *4

/*\

r\

1)1

Q'O-ijSt dn-m)!^^ w n* n (« -2 1 _ J. a

a

r=

(6)

n (t)

=

j JE o (1

8

s

) i

i

1

-4-

,

(

(n2 -

-2 )(w — 2

9

-4») '-

"

*~

)

+ ...

__ 1Jiara=s

(n odd)

_n n(n*-l») nfo - l')(w'-3») +.... + -p + ^ 3

, t

These results may be combined in the formula 1

'

The equations

By

(5), (6)

3

j ( t» ± w ) w

ran-|m + l).(|Om+1

'

and (7) were given by Neumann.

the methods of § 2*11,

easily proved that

it is

l«»O,(0k*.(n!).(*|*!>— ^xpCil*!^

(8)

enOn (t) = l.(n\).(lt)-»-*(l

(9)

where

From

+ j m) r <>8

w T (j n

nnW (t\-"4 mvt

(*7\

j

|

these formulae

whenever the

it

<;

[exp (£

|

2

*

)

(n>l)

- l]/(2« - 2).

|

follows that the series

lan (zjt)n

+ 0),

Sa n Jn (z)

n (t) is convergent

when z is outside the circle of convergence of the latter series. a n Jn (z) n (t) does not tend to zero as n -»- oo and so the former series does not converge. Again, it is easy to prove that, as n -*- oo series

is

absolutely convergent

;

and,

,

,

*«

*

By analogy

Jn <*)

n

(t)

^

{l

-

^

+

with the Legendre function of the second kind,

—

t-z

Cf.

=

Modern Analysis,

% 15*4.

=

r.

-=, n=o

(2n

+ l) Pn {z)Qn (t).

<«-)}

,

Qn (t), which

is

such that

^


274

[CHAP. IX

and hence it may be shewn* that the points on the circle of convergence at which either series converges^ are identical with the points on the circle at which the other series is convergent. It may also be proved that, if either

any domains of values of z and

series is uniformly convergent in

the other

t,

so also

is

series.

Since the series on the right of (1)

when

analytic functions

z J

|

(-)g.(p r -^ y

(10)

<

1 1

it

\,

a uniformly convergent series of

by

differentiation! that

dPjn (z) diQn (t)

+ g)!: = T y_ 2

(t-zf + * +1

is

follows

dV

dzP

w=0

'

where p, q are any positive integers (zero included).

may be

It

convenient to place on record the following expressions:

o

o.(0«i/«.

= l/t + 4/f, t (t) = l/t + 16/* + The Otti,

coefficients in the

3

192/*

polynomial

Bern Miltheilungen, 1898, pp.

9*11.

We

(n

(1)

5 B

,

,

On (t),

for

»=0,

1, 2, ... 15,

1920/**.

have been calculated by

4, 5.

The recurrence formulae

shall

2

2

2 (t)

8

= i/< = (t) 3/< + 24/P, = (t) 5/t* + 1 20/t* + («)

1

On (t).

satisfied by

now obtain the formulae

- 1)

n+1 (0

+ (n + 1)

W _,(0

- 2 (n \~

l) n (t)

On_ (0-On+1 (0=2On

^

2n --

n?r ,

,

(2)

1

(»>1) (n^l)

(<),

-0,(«)«0.'(*)

(3)

The first of these was stated by Schlafli, Math. Ann. in. (1871), p. 137, and proved by Gegenbauer, Wiener Sitzungsberichte, lxv. (2), (1872), pp. 33 35, but the other two were proved some years earlier by Neumann, Theorie der BesseFschen Functionen (Leipzig, 1867),

—

p. 21.

Since early proofs consisted merely of a verification, we shall not repeat

them, but give in their place an investigation by which the recurrence formulae are derived in a natural manner from the corresponding formulae for Bessel coefficients.

Taking

|

z

\

<

\

1

1,

observe that, by §

(t-z)l

€n

Jn (z)

n (t)

* It is sufficient to use the is

(1)

=1= 2

and

2 € M COS \niT

.

Jn (z),

theorems that,

if 2fr n is

convergent, so also

absolutely convergeut.

t This was pointed out by Pincherle, Bologna Meviorie, Cf.

§ 2-22 (7),

n=0

7i=0

Sbjri*

91

Modem

Analysis, § 5-33.

\i) in.

(1881

is

—

2&„/n,

and that then

2), p. 160.


9'11]

275

and hence

I

z

en

n=0

Jn (z)

=I

n (0

»=0

=X since tO (t)=l.

now we

If

the expression on the right,

2

en

»=0 If

we

Jn (z)

n (t)

en

Jn (z) {tOn (0 - cos' \ntr)

€„«/„ (z) {tOn (t)

- cos2 \ntr],

Jn (z)

use the recurrence formula for

to

modify

we get

=2

{Jn- X

(z)

+ Jn+1 (z)}

[tOn (t)

- cos

2

£mr}/n.

n=l

notice that

Jn+1 (z) {£On (0 — cos2 £ n7r }/n

tends to zero as n-*>oo,

it

on rearrangement that

is clear

Jo (z) {0

- tO, (t)} + J

(t)

- \t0, (t) + £}

(z) {20, (t)

x

= 0. + XJn (Z) \20n (t) - *!*L® - *2^> + ?^!l?5l 2 n+1 n—1 re — 1 j n =2 I

Now

zasa

regard

£ remains constant if the coefficients of do not vanish, the first term which does

variable, while

the Bessel functions on the

all

left

;

not vanish can be made to exceed the sum of all the others in absolute value, by taking \z\ sufficiently small. Hence all the coefficients vanish identically* and, from this result, formula (1) is obvious.

To prove

(2)

and

(3) observe that

/d \dt

and

so, \z\

being

less

than

t

2 By

€n

, j

j

Jn (Z)

n' (t)

-J

x

+2

left

(z)

we

(t)

en

Jn

'

= 0.

(jr)}

n=l

+ 0, (t)} + i Jn (M) {20 n

'

equating to zero the coefficient of

proof of *

(t)

On (t)

Jn (Z) {0 n+1 (0 - On-, («)},

to say,

J. CO {Oo' (0

On

n

- I \J^ (,) - Jn+1 n=l

is

(z)

find that

= - J„ (,) 0, (*) - 2 that

'

we have

rearranging the series on the

5 *nJn (Z) On' («) «=0

d\ _i

+ dz) t-z~

This

(1),

we

obtain (2) and

(t)

+

Jn (z)

n+l

(0 "

on the

n _, (0}

left,

= 0.

just as in the

(3).

the argument used to prove that,

a convergent power series vanishes identically, §3-73). Ttie argument is valid here because the various series of Bessel coefficients converge uniformly throughout a domain containing z = 0.

then

is

all its coefficients

vanish

(cf.

if

Modern Analysis,


276

By combining

(1)

and

(4)

ntOn^ (t) - (w2 -

(5)

ntOn+1 (t) - (n2 -.

If

^

be written

for

t

we

(2)

1

at once obtain the equivalent formulae

0„ it) = (n - l)tOn

1)

n (t)

)

[CHAP. IX

'

(t)

= - n + 1 ) tOn

'

(

{d/dt), these

+ n sin

2

\mr,

+ n sin 2 \mr.

(t)

formulae become

- 1) (^ + n + 1) n (0 = n \tOn-* (t) - sin %rnr}, (n + 1) (^ - n + 1 On (t) = - n {tO n+1 (t) - sin |wtt} 2

(n

(6) (7)

2

)

The Neumann polynomial

of negative integral order

.

was defined by Schlafli*

by the equation 0_„(<)

(8)

With values of

9'12.

definition the formulae

this

(1)

— (7)

are valid for

all

integral

n.

The

From the (*

= (-)n On (0.

differential equation^ satisfied by

n

recurrence formulae § 9*11 (6) and

+ n + 1) (* - n + 1)

n (t)

=

— Tit

(^

-j

+ n + 2)

= — t {tOn (t) — cos (%

+

l)2

that

(7), it is clear

-L;- (^ + n + 2 1) {- ntOn+1 (t) + n sin \nnr}

=-

and consequently O n (t)

(t).

2

n+ , (0

\n-jr)

+ n sin \mr 8

+ n sin

2

\nir

t

satisfies the differential equation

n (t)

+

(<

a

- n ) On (t) = t cos \mr + n 2

2

sin2 \nir.

It follows that the general solution of the differential equation

y = On (0 +

is

and so the only solution of is On (t). It is

(1)

sometimes convenient

which

r

is

1

^„(0,

expressible as a terminating series

to write (1) in the

form

where lere (w even)

(» odd) *

Math. Ann. in.

t

Neumann, Theorie der

livii. (1867), p. 314.

(1871), p. 138.

Bessel'schen Functionen (Leipzig, 1867), p. 13

;

Journal fur Math.

ASSOCIATED POLYNQMIALS

9' 12, 9*13]

Another method of constructing the 2

f and so

z?

+z

differential equation is to observe that

l +zj Jo 6-*4 {z) °» {t)= i

£«.»**<*> °« W={*2 ^+4z + Z

Now

1

=

277

i

2z*

z

(t-zf

(t-zf

00

00

2

t^J^ (z), z= 2

fin +

enw2,7n {z)

'

t^z t-z

(2« + 1)

1

°* (0

J2n +

l (z),

00

and hence

t

+ z = t2

2 en,gn

Jn (z).

(t)

n=0

Therefore

2jn Jn (z) £{<• ~ + 3t I + On

1

+ 1* - »*} On (t) - «V» («)] ^ 0.

equating to zero the coefficient of Jn (z) on the left-hand side of this identity, just we obtain at once the differential equation satisfied by On (t).

as in § 9*11,

9*13.

Neumann's contour

It has

been shewn by

integrals associated with

Neumann* Jc f

J

c

J m (z) On (z) dz =

f Jn (Z) On (Z) dz c

(3)

G

be any closed contour,

Om (z) On (z) dz = 0,

(1)

(2)

that, if

O n (z).

(m = n and

m±n)

(ma ± n>)

0,

= 27TlA;/€n

,

J

where k

is

the excess of the

the origin over the

The larity of

The grand

is

first

m

number

number

of positive circuits of the contour round

of negative circuits.

result is obvious from Cauchy's theorem, because the only singu-

(z)

n (z) is at the origin,

and the residue there

third result follows in a similar

manner

;

is zero.

the only pole of the inte-

a simple pole at the origin, and the residue at this point

To prove the second

result,

multiply the equations

V m Jm (Z) = 0, by zO n (z) and

Jm (z)

respectively, ./„

is l/ea .

V n [z0 n {Z)\ = *9n {Z) and subtract. If

U (z)

be written in place of

W iifO|
the result of subtracting assumes the form

*U' (z) + zll{z) + (m2 - w2 ) zJm (z) On (z) = z*g n (z) Jm (z), * Theorie der

BesseVschen Functionen (Leipzig, 1867), p. 19.


278

[CHAP. IX

and hence [z

U (z)] c + (m - «*) f Jm (z) O n {z)dz=\ 2

Jo

Jo

z*gn (z)

Jm (z) dz.

The integrated part vanishes because U(z) is one-valued, and the integral on the right vanishes because the integrand is analytic for all values of z and hence we deduce (2) when 2 ± n 2 ;

m

Two

corollaries, 1

(4)

due to /(o /«> +

The

first is

1

/"<-* + /

)

(1871), p. 138, are that

m (y) dy = Jn . m (x) + ( -)m Jn + m (#),

J* {x+y)

-•

m.

Math. Ann.

Schlafli,

)

/

(5)

.

Om {x+y) Jn (y) dy=Jm _ n (z) + (-)n Jm+n (*)•

obtained by applying

(2)

and

(3) to

the formula § 2*4

namely

(1),

00

Jn {*+y)=

2 /, +p (i)/.j(y), p— — 00

and the second follows by making an obvious change of

Neumanns

9*14. It

integral for

On (z).

was stated by Neumann* that

o, (.)

(i)

We

shall

J"

where o

is

î+i:

now prove by induction the

On (z) =

(2)

+**+

= /; /•*>

UJo

variable.

-

^

+ *»

^du.

equivalent formula

exp ia [{t

+ V(l +

*

2

)}»

+ {t- V(l + * )} w] *~* dt,

any angle such that a + arg z < j

|

a

\ir

;

on writing

t

— u/z,

the truth

of (1) will then be manifest.

A

modification of equation (2)

is

roo+ta

0„ (z) = \\

(3)

w {e *

+ (-)»«""•} «-***• cosh ^0.

Jo

To prove

(2)

we observe that racexpia

/"ooexpia

O

(s)

=

e-*dt,

0,(z)=

and

so,

te-*dt;

Jo

Jo

by using the recurrence formula oo

911

§

(2), it follows

that

we may write

exp ia

0. <*)-["

Jo

gt n (t)e~ dt,

where (4)

„ +1

(0 -

2t n (t)

- <£„_, (0 = 0,

and (5) *

*!<«)

=

*.

Theorie der BtsseVtchen Functionen (Leipzig, 1867), p. 16; Journal filr Math, lxvii. (1867),

p. 312.


9*14]

The

solution of the difference equation (4)

where

The

A

and

B

n (t)

279

is

= A{t + V(* + l)} n + B {t - V(l + t*)} n 2

are independent of n, though they

A — B=\

conditions (5) shew, however, that

;

,

might be functions of t. and the formula (2) is

established.

This proof was given in a symbolic form by Sonine*, /<*> exp ia n

A

(0

*"*** dt>

who wrote

(D)

(1/z)

.

where we

D standing for (djdz).

completely different investigation of this result

whose analysis the form

is

based on the expansion of

§ 9*1 (1),

is due to Kapteynf, which we now write in

-±-=X €n Jn (S)O n (z). —

z

When

j

f

|

<

|

s

*=o

s

we have

, j

= IT\ 2 9 '">

2"

.

if

— oo (n=-<* V.n=

p

n

Jn (0\ e-»du, j

p be so chosen that

u

1

z It follows that

We shall now shew that the will

interchange of summation and integration

be sufficient to shew that, for any given values of ( and •W

f°°

n=N+\ can be made arbitrarily

3^-all

J

fu + J(u i 4-z )\ n

z

(such that |

is justifiable; it

(

|

<

|

z

|),

2

\

I

3

I

by taking

I

N sufficiently large J

;

now

\u±J{u*+z*)\^2(u + \z\), andso

{

Jn{0

\

r\^±^p^l\ J

e -u

du

^J^ fV(«+|,|)» z

2 |

I

\

1*1"

e

-«*

t

J

J LI

<|(C/*)"|exp{|«| + iim. *

Math. Ann.

xvi. (1880), p. 7.

For a similar symbolic investigation

t Ann. Sci. de VEcole norm. sup.

(3) x. (1893), p. 108.

X Cf. Bromwich, Theory of Infinite Series, § 176.

see § 6 "14 supra.


280 Therefore, since |

M

£

<

j

r-

I

.-l+i J.

j

|

•?

we have

|,

{u±JW+f^jn() (z)

£j

<

—

£

left

j

z

*

^

made

can be

j,

it is

n+

1.

\*

the integrand

^

*

.'0

9*15.

An

yjr

'

N sufficiently large

v

y)

tti

'

}

n

is

e

so defined,

(z),

expanded*

.

a polynomial in 1/z of degree

is

and integrated term by

in powers of z

n {z) with the formula § 9*1 (4).

Neumann's

— z)

is

integral.

due to Soninef; from

(§9'1) with the integral of is

Zn

and

A n be defined 1

f<°+>

ti)

Then

this general expansion,

§91 4 can be

derived without

as follows

(w) be an arbitrary function of is the function inverse to ty.

z» =

w;

and, if

(w)

yfr

= x,

let

w = ^ (®)>

by the equations%

** m

din

f°°

£k> ^«=/o

— CL

it

by taking

«" + Wj£-
Sonine's general theorem

difficulty.

Let

extremely interesting and suggestive investigation of a general type

Neumann's formula

Let

+

Sonine's investigation of

of expansion of l/(a

so that

|

Z

easy to reconcile this definition of

it is

+*

by the equation

easy to see that

When term,

{!«!

{w+^+f!)}» e~"rfu _

/„(f)

w = - ee

0„ (,) = j" and

«*P

we have

= 2

n (2;) is defined

I

\»*\{\'\-\i\r-

arbitrarily small

= 2en Jn (00n (z) where

If™

.

|

*=

I

and the expression on the when z and ( are fixed.

Hence, when

z

[CHAP. IX

—Z

= £ Zn A n

*-"*{*(*)}"<**•

,

n =-a>

being assumed that the series on the right is convergent.

Suppose that curve

for

C surrounding

* Cf.

any given positive value of x, z, and

the origin and the point

Hobson, Plane Trigonometry

J This ii.

is

connected with Laplace's transformation.

(Analysis) (1916), pp.

w > (x) w < ^ (x) \

|

\

\

-fa

|

on a closed

|

on a closed

\

(1918), § 264.

t Mathematical Collection (Moscow), v (1870), pp. 323 modified slightly, but the symbols ^ and ^ are his. Wist.

\

781—784.

— 382.

Sonine's notation has been

See Burkhardt, Encyclopadie der Math.


9-15, 9*16]

curve

c

281

surrounding the origin but not enclosing the point

—

oo

1

+ 2iriJ

\Jc

2iriJ

J

°°

e (z

f

gg»(w)-«»—

Jc

dwdx

w-^{x)

I

gZ^(W>)— a*

1

=

y%n

r

/*oo

-

•

Then

z.

—-" x

dx =

l/(o

-

z),

Jo

B(z)
provided that

and the

;

result

established

is

if it is

assumed

that the various transformations are permissible.

In order to obtain Neumann's expansion, take y}r

(w)

=

-

\ (w

vf (x)

l/w),

= x ± V(« + 2

1)»

and then Z

«

n=— oo

= i KJ»(*){4 n + (-)»A_n

}.

»=

Since

4 n + (-)» ^4_ n = ( V'* [> ± Jo

we

at once obtain Sonine notes

{jp.

Neumann's

vV + l)} n + (-)n {* ± V(«* + 1)}" W

] <*#,

integral.

328) that

Jn (z)~(hj) n ln\, *nOn (a)~n$$a)-*, and in the later part of his a\ of the expansion so that lfta-z) converges when J z memoir he giyes further applications of his general expansion.

<

\

9*16.

The

The generating function of On series

2

(-^nÔn (*),

;

\

(z).

which

is

a generating function associated with

Kapteyn* however, has

does not converge for any value of t except zero. series after the method of Borel, in the following manner

„

,

«

_

»

(n

nlomlo

_1

,

1

+

*

2

n Ion=m

_1

-

~ 2 JE

n.(n + m-$\t2n

n

*

1

v

+ $).(n+m)l

l

«"

(»-«)! (^) 2m+2 <^ (!+<«) _ !

x

(1

-'

2m+1

2 )

1

2

H

t*

-

w=

(2m+l)l 2"* + 2

(i*)

<*» + 1 (1

(l+<^

-
2m + 2 )

• (-)»».m! tm

*(l-<*)m=o{£*( 1 -< , )} m *

2n +

+

"(**)*" +

l+<

t

(n-«)!(i*)*» °° | n.(n + m-l)l

(2w) 2

On (z),

"summed"

Nieuw Archief voor Wiskunde

'

(2), vi. (1905),

pp. 49

—55. 10

the


282

/«> 7j

is

convergent so long as

There

is

no great

-

(1

—e~"du — tot

2

difficulty in verifying that the series

—

(

n=o ptotic expansion of the integral for small positive values of

may be

integral

of the theory of

9 '17. It

by

regarded as the generating function of

Neumann's function from

n

n (nz)

an d this integral

>

t

(z).

)

n(

nt

n

On (z)

is

an asym-

when arg z < n, and so the Kapteyn has built up much |

|

this result.

The inequality of Kapteyn 's type for

Neumann's

possible to deduce from

is

^

not negative.

z\t is

1'*)

[CHAP. IX

(m).

n

integral an inequality satisfied

which closely resembles the inequality

satisfied

J n (nz)

by

obtained

in § 8-7.

We

have r*

1 ° n < n *) = 2i^+i

J

H™ + V(w + z )] n +{u>- V(«/ 2

2

2

+

n

z*)} ]

e~nw dw,

the path of integration being a contour in the w-plane, and so n (nz)

j

!

~

^

z I

i

[" |

[{«,

±

y/(vfi

+ *»)} e-] |» dw !

.

where that value of the radical

, j

J o

taken which gives the integrand with the

is

greater modulus.

Now

the stationary point of

+ V(w + z )} e~ w

is

V(l

—

(!)

I

-z ),

and

2

2

[w 2

so

On (nz) ^

JL_

|

|

Lt^^p

where the path of integration

is

Jl {"

+ VK + z>)} e-\.\dw\,

one for which the integrand

is

greatest at the

stationary point. If a surface of the type indicated in § 8*3

the stationary point

w = + oo

is

is

constructed over the w-plane,

the only pass on the surface

are at a lower level than the pass

(2) |

1

Vd-*

+

;

and both

w=

and

if

2

:

)

Hence, since a contour joining the origin to infinity can be drawn when (2) is and since the integral involved in (1) is convergent with this contour, it follows that, throughout the domain in which (2) is satisfied, the inequality satisfied,

(3) is satisfied for

O n {nz)<.~ z

i

-z) z exp V(l — s ) 1 4-

some constant value of A

character as the inequality of §

8'7.

V(l

2

j"" 1

2

;

and

this is

an inequality of the same


9-17, 9-2]

Gegenbauers generalisation* of Neumanns polynomial.

9'2.

we expand

If z

283

z"/(t

by the expansion

— z)

in

ascending powers of z and replace each power of

as a series of Bessel functions given in § 5*2,

we

find

on

rearrangement that * z v+s

zv

x

=

"

2 "+*

» (v

(

+ s + 2m).T (v + s + m )

—

" (
J« iio

*— -

(i/

+ n)

.

1"

(v

+ n - m)

~~^.

•

,

We are

difficulties

.

A

Jv+n {z)

;

\

— 2m, and

the rearrangement has been effected by replacing s by n

no greater theoretical

)

.

r

it

presents

than the corresponding rearrangement in § 91.

thus led to consider Gegenbauer's polynomial

A ntV {t),

defined by

the equation

this definition is valid |

z\<

v is not zero or a negative integer

whenever

;

and when

we have

\t\,

.— =2=0 A —Z

(2)

n>v

{t)Jv + n

{z).

M

t

The reader should have no

difficulty in

proving the following recurrence

formulae 2

(3)

(v

2" (v

+ n)

+ nY -1}

$v

t\\\n +

(5)

(i/

(1/

1"

(v

V* — + n1 -

J t

+ \n- $

1 [

An

,

v

(t)

Sinz *W7T,

\)

+ n - 1) 4,,,, (0 2"(v + n)(v + n-l)r(v + ln-$) ,, ., I -, 8in2l ^ = ( y + w _l)^' i"». n> „(«) + rfln + i)

+ n) tA n_ 1
1) (v v

(6)

\(v

+ n-VAn+^M + iv + n + VA^.it)--*

+ w) ^„+1 ,„ (<) - (» + n +

—

1) (2»

—

.

,

+n-

^ /x + 2"(i/ + = -(„ + „ + -v 1)^', ,,(*)

n)(i»

1)

A n>v (t)

+ n + l)r (i» + ln + i) ... ^sin'iuTr, r(in + f) ,

4^)»2T(»+l)/t

(7) *

Wiener Sitznngsberichte, lxxiv.

(2),

(1877), pp. 124

— 130.


284

The

d*y

(R\

The

.-A

^(

J

Of these

is

)

WW

L__.l r(J

general solution of (8)

a solution

is

(n+l)( 2in -w-l)

J>-2vdy

where (9)

A n>v (t)

which

differential equation of

[CHAP. IX

+^ _I ^+n (0-

is -4 «,,(«)

results, (3), (4), (8)

and (9) are due to Gegenbauer; and he

also

proved that

_.

(10)

A n>v (t)e<*dt = 2»in T{v).(v + n)Cn »(z),

J

where

Cn v (z)

formula

is

is

by calculating the residue of (izt) m A n> „ (t) at the

easily proved

The corresponding formula

—

may

following formulae

(12)

c

(14)

I .'

The

first

(m - n)

9*3.

A

{

n arc cos

z)

be mentioned v

(z)dz

(m = nandm^n)

= 0,

(m2 ^fen2 )

z~ v

Jv+n (z) A n

v

(z)

dz =

2irik,

c

contour,

n = 0,

number

1,2,

. .

. ,

and k

is

the excess of the

of negative circuits of

last results are

proved by the method of § 913

+

m + n)

Schlafli's

\

= 0,

z~ v

Jo

V„+n

{*»-»

An

,

u

(z)}

Jv+m (z) A n,,( z ) dz =

polynomial

= z*-> gnt „ (*), z2

/

J

c

~v

gn
it is

=

0.

Sn (t).

polynomial closely connected with Neumann's polynomial

O n (t) was

In view of the greater simplicity of some of its frequently convenient to use it rather than Neumann's poly-

investigated by Schlafli.

nomial.

;

find that

(2v

properties,

number

G round the origin.

derived from the equations

V„ +m J v+m (z)

whence we

= in cos

c

and third of these

is

also

dt

this

'

of positive circuits over the

the second

izt

;

origin.

is

z-*Jy+m (z)A n>v (z)dz = 0,

f

J

G is any closed

Neumann's polynomial

A m v (z)A n

I

J

(13)

(t) e

n

Lttx J

where

for

/"< 0+)

1

(11)

The

— 2az + a*)'*

the coefficient of an in the expansion of (1

!


9'3]

polynomial

Schlafli's definition* of the

<±n {n (n

(1)

Sn (t) = 2

(2)

S.(«)-0.

On comparing

If we substitute for the functions

we

(2),

A)

(iO-"

+MW

(n

,

>

1)

see at once that

£„+!

and from the

(t)

the recurrence formulae §9*11 (1)

+ £„_! (0 - 2n«~ Sn (t) = U~ 1

l

cos2 £ mr,

latter,

- 1) #n _ (0 - 1 (n +

£ (n

n (t) in

find from the former that

(4)

If

1 \

™

£n Sn (t) = *0n (0 - cos8 \nir.

(3)

and

is

_m_

we

(1) with § 9*1 (2),

285

x

we multiply

this

(0 = nSn

1) £„+,

by 2 and add the

'

The formulae

and

- nt- 8n (t) - 2*" 1

result to (4),

Sn^{t)-Sn+1 (0 =

(5)

(t)

1

cos2 fnw.

we get

25/(0.

may, of course, be proved by elementary algebra by using the definition of Sn (t), without appealing to the properties of (4)

(5)

Neumann's polynomial.

The

definition of Schlafli's polynomial of negative order is

S_n (0 = (-)w+1 >SU0,

(6)

and, with this definition, (4) and (5) are true for

The

interesting formula, pointed out

by

all

integral values of n.

Schlafli,

Sn^(t) + Sn+1 (t) = 40n (t),

(7) is easily

derived from (3) and

(4).

Other forms of the recurrence formulae which may be derived from (4)

and

(5) are

nSn (0 - tSn (0 = 2 cos8 ^nir, tSn+1 (0 - nSn (0 + tSn (0 = 2 cos8 |n7r. *£n_, (0 -

(8) (9)

If

*

'

we write ^

for

t

(d/dt), these

formulae become

+ n) Sn (0 = tSn ^ (0 -

2 cos^wtt,

(10)

(S

(11)

(^ - n) Sn (0 = - tSn+1 (0 + 2 cos2 £ wr-

it follows that

(^

2

- n ) Sn (0 = t (^ + 1 - n) #n_, (0 + In cos8 \nir = - Sn (t) + 2t sin fnir + 2n cos \nir, 8

t

and so (12)

Sn (t)

is

*

2

8

8

2

a solution of the differential equation

du -£ + t-~ + (< - n d?v

8

*

2

)y =

Math. Ann.

2£ sin 2 £ /wr

m.

+ 2n cos

(1871), p. 138.

2

\rnr.

1

.


286 It

may be S

1

convenient to place on record the following expressions

= 2/t,

(t)

[CHAP. IX :

&(*)-4/*,

S3 (t) = 2/t +

16/t

= 8/« + 96/i = S (t) 12/t + 384/f + 7680/tf

S

3

S (t) = 2/t + 4,8/t + 768/* 3

5

5

The general descending

2

(t)

4

,

4

,

4

2

6

,

1 ;

given explicitly by Otti, are

series,

—

-2 2n(n-- 2 — )(w -4 = 2n + 2n(n + + t P 2

i

)'

^

2

2

2

)

-

i

6

.

'

t-

2 (n2

2

-

l2)

2 (wf -

2

)

(w 2

-3) 2

.

polynomial £»(<)> for n = l, 2, ... 12, have been calculated by Otti, 14 Otti's formulae are reproduced (with some obvious errors) by Graf and Gubler, Einleitung in die Theorie der Bessel'schen Funktionen, II. (Bern,

The

coefficients in the

Ztera Mittheilungen, 1898, pp. 13

—

;

1900), p. 24.

9*31.

Formulae connecting

the

polynomials of Neumann and

Schlafii.

We

have already encountered two formulae connecting the polynomials of Neumann and Schlafii, namely

\nSn (t) = t0n (t) - cos \ntr, 8n- (t) + S„ (t) = 40n(f), 2

l

l

of which the former is an immediate consequence of the definitions of the functions, and the latter follows from the recurrence formulae. A number of other formulae connecting the two functions are due to Crelier * they are easily derivable from the formulae already obtained, and we shall now discuss ;

the more important of them.

When we

eliminate cos2 |/wr from §

93 (3) and

either §

93 (8)

or (9),

we

find that

Sn _ (t)-Sn(t) = 20 n (t), Sn+1 (t)+Sn(t) = 20n (t).

(1)

1

(2)

Next, on (3)

summing

equations of the type § 9*3

Sn (t) = -2

<(i»-D

2

S'n^mr-i(t)

(5),

we

find that

+ Bin*lnir.S

l

(t),

w=

and hence n

(4)

Sn (t) + Sn _

1

(t)

= -2 2 fi^-, (*) + &(*). TO

*

Comptes Rendus, cxxv. (1897),

—

pp. 421—423, 860—863; BernMittheilungen, 1897,

pp.

61—96.

Y

F


9-31, 9*32]

287

93 (7) and (5) we have {
Again from 4

§

!

1

so that

Sn" (t) + Sn (t) - O n _

(5)

This

(t)

+ O n+1 (t).

the most interesting of the formulae obtained by Crelier.

is

Again, on

summing formulae

On (t) = -2

(6)

x

I

of the type of

0'n -2m-! 0)

§911

we

(2),

find that

+ 8^4^.0,(0 + cos $mr.O 2

(t),

5/1=0

and hence

9*32.

The

0^

+

n (t)

(7)

(t)

= -2 "%

O'n-^-,

(t)

+

0,

(t)

+

{t).

m=0

Graf's expression of Sn (z) as a sum. summatory formula

peculiar

Sn (z) = Tr

(1)

was stated by Graf*

2

m=— n

[Jn (z)Ym (z)-Jm (z)Yn (z)}

in 1893, the proof being supplied later in

Graf and most readily proved by induction it is obviously true when n = 0, and also, by §363(12), when n=l. If now the sum on the right be denoted temporarily by n (z), it is clear that Gubler's treatisef.

This formula

is

;

<£n+i (Z)

+

<£n-i (Z)

~ (%n/z)

= 7T Jn+1 {Z) + irJn_

1

n {Z)

m+1

2 m= — n-1

(z)

Ym (Z) - 7T

2 m=— n+l

-(2n7rjz)Jn (z)

n+1

Ym (z)-7rYn_

2 m=—n

2

Fn+j (z)

m=—n-l 1

Jm (z)

Y

(z)

m= — »+l

Jm (z)

Ym (z) + {2nir[z) Yn (z) 2 Jm (z). m=-n

Now

modify the summations on the right by suppressing or inserting terms at the beginning and end so that all the summations run from — n ton; and

we then

see that the complete coefficients of the

vanish.

It follows that

n+1 (z)

sums 2 Jm (z) and 2

Ym (z)

+ <£„_, (z) - (2n/z) n (z) = 7rJn+1 (Z) Yn+1 (Z) + Y_ n ^ (Z)} - 7T Fn+1 (z) {Jn+1 (z) + J_n_, ( Z)\ - -rrJ^ (z) n (z) + F_M (Z)} + 7T n_ (z) {Jn (z) + J_ n (z)\ „ _ „- {l + (_ l)»{ \Jn _, ( Z ) Yn (z) - Jn (z) Fn _ (,)}

{

{

x

x

= 4.2 -1 eos

2

^wr,

by §363(12); and so n (z) satisfies the recurrence formula which by Sn (z), and the induction that n {z) = Sn (z) is evident.

is satisfied

•

Math. Ann. XLin.

(1893), p. 138.

t Einleitung in die Theorie der BesseVschen Funktionen, n. (Bern, 1900), pp. 34

—41.


288

Greliers integral for

9*33.

If

[CHAP. IX

Sn (z).

we take the formula §9*14(2), namely r ao

On (e)

exp

ia

=h

[{t

+

V'(l

+ t*)} n +

\t

- V(l + « )} w ] e~* dt, 2

.'

and integrate by

parts,

we

find that

eft

_ cos'^wtt + " 5 Hence

it

00

_^ 2zJ

i"

«*»•

\t

+ \/(l

-

f

-

t

V( 1

+ * )} w 2

e

_rf

follows that

Schlafli,

Math. Ann. in. (1871),

Sn (z)= j"** {?*-(-)*€-'*} e-'*»*»dO

(2)

fundamental in

We

P)}»

V(l+**)

This equation, which was given by

is

4-

Crelier's researches*, of

which we

shall

now

p. 146, in

the form

f

give an outline.

write temporarily

Tn =

[t

+ V(l + *2)} n -

\t

- V(l + ?)}'\

and then so that

y"+! _ ot 4T

— IT *

T

'

and therefore C

7'

n

+ 2« + 2*+...+2e'

the continued fraction having w elements. It follows that quotient of two simple continuants^ so that

Tn ± ]Tn x

is

the

Tn+ l= K(2t,2t,...,2t)n

Tn the suffixes

n,

n—

1

It follows thatj

K{2t,2t,...,2t) n

^

denoting the number of elements in the continuants.

Tn /K(2t) n-i is independent of n; T = 2V(1 + I'), #(20o=l,

and since

1

we have

Tn = 2
421—423, 860—863 Bern Mittheilungen, 1897, pp. 61—96. 494—502. tbe continuant are the same, the continuant may be expressed by

* Comptes Rendus, cxxv. (1897), pp.

t Chrystal, Algebra, n. (1900), pp.

% Since

all

the elements of

this abbreviated notation.

1 ,

;


9*33, 9*34]

289

and hence fee

exp ia

Sn (z) = 2

(3)

K(2t)n _ e~^dt, 1

Jo

From

this result it is possible to obtain all the recurrence formulae for

8n CO by using properties of continuants.

9'34.

We

Schlafli's expansion

now

shall

Sn (t 4- z)

+ z)

as a series of Bessel

obtain the result due to Schlafli * that,

coefficients.

when

|

z

|

<

1 1

\

,

can be expanded in the form

Sn (t + z)=

(1)

is

of Sn (t

I Sn _n (t)Jm (z).

m= — 00

The simplest method of establishing this formula for positive values of n by induction f. It is evidently true when n = 0, for then both sides vanish

;

when n =

1,

the expression on the right

S, (0 Jo (*)

+ X

{Si-m (t)

= 2Oo (t)J

is

equal to

Jm (*) + S^ (0 J_ m (z)} 2 {Sm^W + SîtyJ^-z)

{-z) +

*»=1

by §91(1) and

Now,

if

0, 1, 2, ... n,

=

2

=

2/(*

I e m Om (t)Jm (-z)

+ z) =

(t

(t

+ z),

§93 (7).

we assume the we have

Sn+i

S,

truth of (1) for Schlafli's polynomials of orders

+ z) = S^ (t + z)- 2Sn

-

(t

+ z)

(z)-2

2

X

Sn-r»-i(t)Jm

2

\Sn. m-,{t)-2S'H . m (t))Jm (z)

m= — »

=

'

wi= — 00

S'n _,n (t)Jm (z)

30

m= — *> and the induction

is

established

we have used the obvious

S„'(t *

Math. Ann. in. (1871), pp. 139

left to

t

the reader

(cf.

The extension

;

to obtain the second line in the analysis,

result that

+ z) = ^SH (t + z).

— 141

;

the examination of the convergence of the series

is

§ 9-1).

to negative values of « follows

on the proof

for positive values,

by

§ 9*3 (6).


290

[CHAP. IX

The expansion was obtained by Schlafli by expanding every term on the right of (1) in ascending powers of z and descending powers of t. The investigation given here is clue to Sonine, Math. Ann. xvi. (1880), p. 7 ; Sonine's investigation was concerned with a more general class of functions than Schlafli's polynomial, known as hemi-cylindrical functions <§ 10-8).

When we make

use of equation

(2)

n (t

§ 9*3 (7), it is clear that,

+ z)=

—223, who expanded

the obvious formula

[cf.

is

Wiener Sitzungsberichte, lxvi.

powers of

* (-) m pCm .On - p+2m

^§r-)==

easy to deduce Grafs* results (valid

Sn (t-z) =

(5)

On (t-z)=

The

definition of

The problem

<

1 1

\,

z

(2),

(1872),

by Taylor's theorem, used

(t),

series.

(4)

9*4.

\

£ O n.m {t)Jm (z).

n (t+z) in ascending

and rearranged the resulting double It

z

$ 9-11 (2)]

2P

(3)

|

»»= — 00

This was proved directly by Gegenbauer, pp. 220

when

when \z\<

\t\),

I Sn+m^Jmiz),

i O n+ m (t)Jm (z).

Wl =

Neumanns

— oo

'polynomial ft n (t).

of expanding an arbitrary even analytic function into a

series of squares of Bessel coefficients

Neumann f by

formulae of

series of this type.

§ 2*72,

was suggested to which express any even power of z as a

The preliminary expansion, corresponding

to the expansion of l/(t

the

—

z)

given in § 91, is the expansion of l/(F — z2 ); and the function ft n (£) will be defined as the coefficient of €n Jnz (z) in the expansion of l/(tfa — z*), so that C1)

ir-r, t*-z

= «# (*) Oo (0 +

2

^

8

<*) fli

(0

+

W

<*)

n. (0

+

.

• .

= 2 c/.'Wfl.^ n=0

To obtain an panding *

l/(£s

— z})

Math. Ann.

explicit expression for fi n (0>


xliii. (1893), pp.

141

—142;

mit Bessel'schen Functionen (Bern, 1894).

z,

tft

ke

I

z

\

<

* I

|»

aQ d, after ex-

substitute for each power of z the

see also Epstein, Die vier Rechnung*operationen

[Jahrbuch

iiber die Fortschritte der

Math. 1893—1894,

pp. 845—846.]

t Leipziger Berichte, xxi. (1869), pp. 221—256. [Math. Ann. in. (1871), pp. 581—610.]

I


9-4]

§ 9*1,

we have

1

|

«

a

Neumann

given by

series of squares of Bessel coefficients

291 (§ 2-72).

As

in

z 2*

-s ~,=o«M+a 2

when we rearrange the presents no

ment

by writing

series

greater theoretical difficulties

n—s

m

for

rearrangement

this

;

than the corresponding rearrange-

in § 9*1. is

denned by the equations

(n

- *)! (2s)! (i

Accordingly the function fl„ (t)

W ~ 4 ,r

V>

Un

(3)

n i (o-i/«'.

On reversing

the order of the terms in (2)

we

M+a

(n>l)

'

find that

* n.(2n-tn-l)l{(n-m)!} » ~ m!(2n-2/rc)!(£02M 2w+2

1

(n>l)

nW ~4 mto

<*)

while, if (2) be written out in

1.2.3 4n a (4na - 2 2 )(4tt2 -4a )

1.2 4n'(4n»-2')

14na

1

assumes the form

full, it

"nW-i+oir+o ^3.4 2

(5)

«

+4, s.«

/6

/L

**

'

+••••

Also @4«4

®A* (6)

2.(2w-l)(2n-2)

-

+

2"

Hen*

)

+ 1.2...n.(2n-l)(2n-2)...nf where 2 n-l

«.= ~

@ = (2n-l)(2n-3) 2n(2n-2) 4

2/i

'

_ (2»-l)(2n-3)(2n -5) "' W6_ 2n(2n-2)(2n-4)

(7)

(2m-1)(2w-3)...1 2n(2w-2)...2

0»« = Since

< Bm <

1, it is

(8)

and,

j

easy to shew by the methods of §

e„

n» it)

\

<*

2*» j

t

\

- 2»- 2 (n 2 exp !) (

«

when n > 0,

(9)

where

en

nw (0 » 2W <- "2

|

0\

S {exp|*| a

2

(n!)a (l

+ 0),

- l}/(2n- 1).

211

2 |

),

that

'


292

By

reasoning similar to that given at the end of § 91,

[CHAP. IX it is

easy to shew

1an Jn* (z) iln (t) and %an (z/ty*

that the domains of convergence of the series are the same.


the curious formula, due

difficulty in verifying

Kapteyn*,

to

°-(')- s

ao)

5/r^(a^)«»-

The recurrence formulae for

9*41.

The formulae corresponding 2 (1) K)

n'(t) lln{t)

Qn

911

to §

(2/<)rV(o

(3)

(2/0

n

'

(t)

n+

= - 2n, (t) + 2n

obtained these formulae

is

Take the fundamental expansion

2 Jo («)

We find by

J„' (z)

en

{n>4 < n >2) >

(t).

The method by which

§ 9*11 (1).

and observe that

2*5,

= - z 2 JV, (*) - ./Vi (*))/»• {

differentiations with regard to

-2t/(t*-z>Y= 5

2n 'Q n»-l'

that described in § 9*11.

§ 9*4 (1),

by Hansen's expansion of §

that,

1

= *n (0-**MO,

There seems to be no simple analogue of

and

(3) are

**«•»<*>

n-1

(2)

and

(2)

= n »-^

t

Neumann f

(t).

and with regard

t,

to

z,

that

Jn*(*)fV(0>

n=0 a»/(ff

- *y = 2 j

(#>

=* I n=l

On comparing *-»

On

I

en

j; to

{/»*.,

*

+

«

s

these results,

{

J'n., (*)

- J»n+1 (*)} n n (o/n

clear that

it is

Jn*(z)a n \t)+ i {Jv to-^

we

91),

(0

to -/Vito){n»(*)-n. (*)}/«•

1

selecting the coefficient of

(cf. §

flo

Jn

a

(z)

2

» + ito}.{«»(0-n„(o}/n=a

on the

left

and equating

it

at once obtain the three stated formulae.

Ann.

Sci. de

VEcole norm. sup.

(3) x. (1893), p. 111.

+ Lnpziger Berichte, xxi. (1869), p. 251. [Math. Ann.

m.

(1871), p. 606.]

to zero


9-41, 9*5]

293

Gegenbauer's generalisation of Neumanns polynomial

9*5.

Q n (t).

"l(t — z)

in ascending powers of z and replace each power If we expand z of z by its expansion as a series of products of Bessel functions given in § 55, we find on rearrangement (by replacing s by n — 2m) that ;L+

gfl

* gH + v + S

+v

2*i+ " + *(

«

_

x

t/

i=0 (,m=0

X

it is

\z)

M+ j«+»i

*•

»=0

^ r(/j.+§s+l)r(v+%s + l)(fji+i>+s+2m)r(iJ. + v + s+m) J v+ ig +m

(z)

y

*n— swi + i &

(/A+y + n)r(/i,

+ ^w-m +

supposed that z |

<

\

|

£

|,

+ |n-m + + + w-2»i + l)

1) r(i/

m!r(/tx

l)r(/A

+ y + M -w)

i/

and then the rearrangement presents no greater

theoretical difficulties than the corresponding rearrangement of § 91.

We

consequently are led to consider the polynomial

Bnillt¥ (t),

defined by

the equation

m (-

1)

R ;**(*) -»n

+ i>+w) = 2»+'+»(m ^T+l X

r(fjL

,„t

+ %n-m + l)r(v+$n-m + l)r(fi + v+n-m) m!rV + v + n-2wi + l)

This polynomial was investigated by Gegenbauer*; it recurrence formulae, none of which are of a simple character. It

may be noted

on

£,„;

0,0

(0

= *«*"« (*)•

They are obtained by expanding the Bessel

and calculating the

(3)

(4)

'

satisfies various

following generalisations of Gegenbauer's formulae are worth placing

record.

series

,

that

(2)

The

W

t-"J„

2^. t~ >

2^. J

functions in ascending

residues. (2t sin

)

B2n +

1;

» >v (t)dt

= 0.

J

J, (2t sin

)

Bm

.

M> „

(t)

dt

_ 2* + "((j + v + 2n)r(fi, + l)r(fj, + v + n)8m''
In the 1

(5)

^

2 *J sin

).

r(o+)

t-*Jp {2tam4>)Bm

J

This formula

.

v, v

may be still further

(t)dt=2*(v+n)r( v )sm*Cn*(co82).

specialised by taking

* Wiener Sitzungtbcrichte, lxxv.

(2),

(1877), pp.

equal to ±tt or £tt.

218—222.


294

The genesis of Lommel's* polynomial

9*6.

[CHAP. IX

RmiV (z).

The recurrence formula

Jv+1 (z) = (2v/z) Jv (z) - /„_, (z) may

obviously be used to express

J ^{z)\ v

and the

Jv+m {z)

linearly in terms of

coefficients in this linear relation are

We

which are known as Lommel's polynomials.

Jv (z)

and

polynomials in \jz

proceed to shew

how

to

obtain explicit expressions for them.

The

Jv+1 (z), Jv+3 (z),... Jv + m_

result of eliminating

x

(z)

from the system of

equations

Jv+P+1 (z) is easily

1

(z)

x

=

(p = 0,

0,

1,

. . .

m - 1)

seen to be -2z- 1

«^ + m(2),

Jv .

+ p)/z] Jv+P (z) + Jv+P_

{2 (v

+ m-l),

(v

+ m-2),

1,

0,

0,

1,

0,

0,

0,

Jp('),

0,

0,

0,

0,

(z)-(2v/z)Jy

(z),

=0.

1,

-2z- l

0>

(v

1

-22- 1

("

+ l)

1

By expanding

Jv+m (z)

is

in cofactors of the first column, we see that the cofactor of unity; and the cofactor of (-)"*- 1 Jv {£) is

-2?- 1 (v + m-l),

i.

0,

0,

1,

-23~ 1 (v+m-2),

1,

0,

°»

1,

°» °»

The

cofactor of (-)m

of the last

The

~1

The

0,

_2*-J(„ + l),

0,

0,

1,

denoted by the symbol (-)m RmtV (z); and called Lommel's polynomial. It is of degree m in 1/z

is

also of degree

~

1

J

v

m

in

(z) is

v.

effect of

suppressing the last row and column of the determinant by is equivalent to increasing v and diminishing by and so the cofactor of (-)m ~ ./„_, (z) is (-) mr " 1 iC-i.r+i (z).

m

a

;

1

-22-1

determinant modified by the suppression

Rmiy (z) is defined

which unity

0,

«/Iî (z) is this

cofactor of (-) w,

it is

0,

row and column.

Rm,» (z), thus defined,

and

-2«-i(i/+«i-3),

o

Hence

it

follows that

Jv+m (z) - J„ (z) Rmt *

Math. Ann.

v

(z)

+

J_ v

y

( z)

iv. (1871), pp.

R m_

1

„ +1 ( z )

108—116.

= 0,


9*6, 9*61]

that

is

to say

Jv+m (z) = Jv (z) Em>

(1)

It

295

easy to see that*

is

Rm

,

the numerator of the last convergent of

(z) is

¥

- J^ (z) R m-h „ +1 (z).

(z)

,

the continued fraction

2*rl(v

+m~

1

1

1

1>

>-2z->(v+m-2)-2z-*(v + m-3)-...-2z->v-

-R,»,„(z) was denned by Lommel by means of equation (1). then derived an explicit expression for the coefficients in the polynomial by a somewhat elaborate induction ; it is, however, simpler to determine the coefficients by using the series for the product of two Bessel functions in the

The function

He

way which It

will

be explained in

Bessel, Berliner Abh., 1824, p. 32, that, in consequence of the

had been observed by

recurrence formulae, polynomials

'.<«)-

where

[cf.

§

962

§ 9'61.

-2

'

4

A n . x (2), 2

'^- r"

Z?n _i (2) exist

2)

such that

^n-l (*) JiW- *»-!

M*

<*».

(8)]

^n_i

(2)

BH (2) - A n (2) 5B _

1

(2)=

2K^...(2n-2f2n

'

It should be noticed that Graft and Crelierf use a notation which Lommel's notation ; they write equation (1) in the form

W-^CT

Ja + «>

9*61.

The

series for

1

<*>

differs

from

Ja <*) " K>-1 <*> '.-1 (*>•

Lommel's polynomial.

m It is easy to see that (-) J- r -m

(z),

qua function of the integer m,

the same recurrence formulae as J,+m (z); and hence the analysis of §

satisfies

96

also

shews that (1)

(-)* J_,_„ (z)

Multiply this equation by

= J.

¥

(z)

Rm, v {z)

+ J.v+1 (z) R m- ltV+1 (5).

J^ (z) and § 96 (1) by J"_„+1

(z),

and add the

results.

It follows that (2)

J¥+m (z) J_„+1 (z) + (-)m J- ¥-m (z) /.-1 {z) = Rm „ (z) {Jy (z) J-^ (z) + /_„ (z) /„_, (z)} ,

2 sin vtt Rm,v{z), TTZ * Cf. Chrystal, Algebra,

11.

(1900), p. 502.

t Ann. di Mat. (2) xxm. (1895), pp. tionen, n. (Bern, 1900), pp. 98—109.

t Ann. di Mat.

(2) xxiv. (1896),

pp.

45—65; Einleitung 131—163.

in die Theorie der Bessel'sehen Funk-


296

by

§

32 (7).

j v+m {z)

But, by

j_ +1

J- v - m (z) Jv -

X

§

w(z)

[CHAP. IX

541, we have

z

opir{v +

= 2

l)r{ _

m+p+

v

+ 2+p y

n\T(—v- m + n + 1 V (v +

n~l

)

H

(~) (- »»

_ S n =o n\

+ i

T(—

v

—

+

{\z)- m+tn

«-)n

m+n

+

~

n) 1

_

r(y + n)

1)

mw

(|*y»+g+* (-v»+*-» P + i) + p + l)!r(-i/ + 2+p)r(i/ + w+p + l)' last summation by m-f jo + l. Now it is clear

P =o(r/i

when we

replace

in the

??

that (jo

+ l)m+p+i = (m + 2p + 1) = (m + p + p$m+p+$\ p\ !

2)p

(m-\p+l)\ and

so,

when we combine the

series for the products of the Bessel functions,

we

find that

2 sin vir

R

__

_ smvir

^~

> \m

the terms for which n

(—

m + n) n {\z)- m+m-

(-) m+w (-

«

<*» (-)n (m Mr

-n)\r(v + m- n) (%z)-m + m w!(m-2n)! r(v + w)

1

'

vanish on account of the presence of the factor

+ n)n in the numerator. When v is not an integer, we

in

infer that

<*» (-)» {m

p

— But the

(»i-2»)!

C

(—\ n

-

r(i/ -

-

w) {\z)- ™+™

+

n)

drZ\~ m +-n

R

m „ (z), by means of a determinant, shews a continuous function of v for all values of v, integral or not by an obvious limiting process, we infer that (3) is a valid expression original definition of

R m>v (z) is

that

and

N

- n)\ r (p + m n!

M =o

so,

R mv (z)

even when v is an integer. be necessary to replace the quotient for

1

T(v

+ m-n)

When T(-

y

T(v+n)

y

'

v

is

V

a negative integer

it

ma}'

-n + l)

r(-v-m + n + l)

in part of the series.

The series (3) was given by Lommel, Math. Ann. iv. (1871), pp. 108—111; an equivalent result, in a different notation, had, however, been published by him ten years earlier, Archiv der Math,

An

und

—

Phys. xxxvn. (1861), pp. 354

355.

interesting result, depending on the equivalence of the quotients just

mentioned, was

first

noticed by Graf*, namely that

RmA z = (~)m Rm, -,-«.+!

(4)

)

*

Ann.

di

Mat.

(*)•

(2) xxin. (1895), p. 56.


9'62]

In the notation of

Pochhammer

(cf. §§ 4*4,

4*42),

297

we have

Rm (z) = (v) m {% z)~m J\ (\-\ m, - \ m v,~ m, l-v-m; Since R mv {z)/z is a linear combination of products of cylinder functions of

(5)

„

z*).

;

.

m and v — 1, it follows from § 54 that it is annihilated by the operator + m y + (v~ l) ^ + + my -{v- \Tf] + 4s (^ + 3^ + 2) where ^ = z (d/dz) and so Rmv (z) is a solution of the differential equation [(* + m) (* + 2v + m -2)($-2v- m) (^ - m - 2)] y (6) +

orders v

O* -

2

2

{(i;

2

2

}(i/

}

2

;

+ 4^-^(^+l)3/-0. An

equation equivalent to this was stated by Hurwitz, Math. Ann. xxxm. (1889), p. 251; and a lengthy proof of it was given by Nielsen, Ann. di Mat. (3) vi. (1901), pp. 332 334 ; a simple proof, differing from the proof just given, may be obtained from formula (5).

—

Various properties of Lommel's polynomial.

9*62.

We

proceed to enumerate some theorems concerning

Lommel

published by

in his

Rm
which were

memoir of 1871.

In the first place, § 9*6 (1) holds if the Bessel functions are replaced by any other functions satisfying the same recurrence formulae and, in particular, ;

Yv+m (z)= Yv (z)R m v {z)-

(1)

,

whence

it

Y^^R^îz),

follows that

Yv+m (z) J„ (,) - J^ m ( Z ) Y^ (z) = R m {z) Yv {z) J _, (z) - Jv (z) Yv ^(z)}= - 2R m
(2)

,

v

v

{

961 (2), take m to be an even integer replace m by 2m, and v by v - m. The equation then becomes (3) Jv+m (z)Jm+1 _ v (z) + J_ v _ m (z)J_m _ i+v (z) = 2 (-)m sin vir. R^m v - m {z)\{TTz\ and, in the special case v = £, we get Next, in

§

;

,

J'^ (z) + J3_ m _

(4)

that

is

j Wi („ + ./._

(5)

This order

is

}

(z)

= 2 (-)- R2m>h ^n (z)/(ttz),

to say

is

2 5 vr~v»-»)i{»*-w W - t^w=o — |(ra

_

the special case of the asymptotic expansion of § 7*51 when the



(«)

n)!} 8 .n!

M + y_ W - 2

J>

(i

+

»

+ •),


298 Formula

(5)

[CHAP. IX

was published in 1870 by Lommel*, who derived

at that time

it

by a

direct multiplication of the expansions (§ 3 4) #

followed by a

As

somewhat lengthy induction to determine the

96 (1) and

special cases of §

Jm+i (*)

=

2 \4

/ ]

(-)'" [

By

J-m -i (z)

R\

Finally,

i

— m, we

we deduce from

we

replace

by

interesting result, pointed out

algebraic in

z,

identity of the type 2

we can

at once infer

2 fm {z) m=0

^

r

by writing the postulated identity s

Nielsen,

fm (z)J

v+

m=o the two

sin *

.

J

(z),

i^

lf

,

(*).

odd integer 2ra

+

1

and then

2

(*)s0,

m=0 the form

in

-m-i (f) = 2 (-)"» sin vTrR%m+Ai „_m (z)/(ire).

.4 ran. rfi J/a*. (3) v.

m (z) = 0, where

(1901), p. 23, is that

the functions

fm (z)

are

identities /„, (*) i2m _!, „ +

1

(*)

= 0,

/.w (/,(i)Ji;,(»)-/,.i(i)ii..i,» t i(*)}^

m=0 and observing that, by § 474 combined with algebraic function.

li§

(4) thatf

m by the v

we have any

# w_

get

n if

.

2 \*

/

Jv+m+1 (z) J- v+m+1 (z) - J-y-m-i (z) J

(9)

cos z

+ R*m-,, i (z) = (-)'" R™, i-m (4

(z)

in § 9*61 (2),

if,

replace v by v

(—J

z.Rm ,i (z) + ^ —

cos

—J

^

squaring and adding

(8)

An

=

we have

§ 9"61 (1),

— Y sin z.RmA (z) -

(

coefficients in the product.

§ 3*2 (3), the quotient

Nielsen points out in this memoir, and

Jv -i(z)lJv («)

is

not an

its sequel, ibid. (3) vi. (1901),

—

340, that this result leads to many interesting expansions in series of Lommel's polynomials; some of these formulae will be found in his Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), but they do not seem to be of sufficient practical

pp. 331

importance to justify their insertion here.

Recurrence formulae for Lommel's polynomial.

9*63.

In the fundamental formula

Jv+m 0) = J* replace

m and

Jv +m (z),

we

v

by

m+1

(*)

and

Rm, v (?)

v

—1

;

~ J*-\ (*) Rrn-1, H-l (z),

on comparing the two expressions

see that

,/„_! (Z) {i? m _ liH -i (*)

+ Rm+1,,-1 (*)} =

{«/",

(*)

Math. Ann. n. (1870), pp. 627—632. t This result was obtained by Lommel, Math. Ann.

+ «/,-«(*)}

Rm,u(z).

*

iv.

(1871), pp.

115—116.

for

—

)

»


9-63]

Divide by

J ^ (z), which v

Rm-1,,+1 (*)

(1)

To

299

not identically zero, and

is

+ Rm+l

,

w - l <*)

=

^-— Rm }

,

z

we

obtain another recurrence formula,

it is

replace

apparent that

v (z).

m in § 9 62 (2)

by

-

m — 1, and use the recurrence formula connecting Bessel orders v + m — 1, v + m and v + m + 1 it is then seen that 2 iC->, (*) + R m+ (s) = ^^1 Rm> (z\ (2) z and

w+1

functions of

;

,

i,

and hence, by combining

R m-i

(3)

t

„

and

(2),

962 (2)

§

differentiate

R'm, v (^)

(4)

and

We

it.

so,

by

~

— Rm,

z

Rm, v (*)•

\z»-i F„_,

(*)}-{*--'»

Y9m (s)}[r-*J w

l

(z)},

(*)

v

+ Rm+i, *-i (*) ~ Rm+i, v (^)»

(2),

= — — i2 TO

(5)

R'm> „

(6)

R' m< v (z)

(7)

R'mA Z) = ~

(s)

1

_

z

deduce that

=

and

(3), (1)

(m +

in the form

(z)={z-»- m Jv+m (z)}

7T

and

2

+ R m +i, v (s) — Rm-i, v+i (t) — Rm+i, v-i (*) =

(z)

Again, write

-z-*»-*R mtV

(1)

v

9

Z

=

,

,,

(z)

Rmt

z

y

T"" Z

+ Rm (z)

lt

v

(z)

- Rm-i, v+i (z)>

— R m-h „ +1 (z) — Rm+

RmA Z

)

lt

y

(z),

+ Rm+hv-i (z ) + Rm-h* ( z

)-

of these formulae were given by Lommel, Math. Ann. IV. (1871), pp. 113 due to Nielsen, Ann. di Mat. (3) VI. (1901), p. 332; formula (2) has been used by Porter, Annals of Math. (2) in. (1901), p. 66, in discussing the zeros of Rm> v (z).

The majority

116, but (6) is

It is evident that (2)

m is zero or a negative of m,

we

may be used

to define

integer; thus, if (2)

is

Rmv (z),

when the parameter

to hold for all integral values

find in succession from the formulae

R%v\ z) =

4,v(v

+ l)

~i

JLv

p 3 )=— î,A .

1>

that

R

(8)

0<

R_ltV (z) = 0, R^y(z) = -1,

y(z)=l,

and hence generally, by induction,

R- m

(9)

,

y

(Z)

= (-)—* Rm-^y (z).

This formula was given by Graf, Ann. di Mat.

If

we compare

(10)

When

(2) xxiii. (1895), p. 59.

(9) with Graf's other formula, § 9*61 (4),

we

find that

Rm,y(z) = (-)™" R-r^^y (z) = - R-.m^ v+m+1 {z) = (-)" Rm ,.y-m +i (z). 1

the functions of negative parameter are defined by equation

formulae (1)

—

(9), all

(7) are true for negative as well as positive values of m.

the


300 9'64.

It

is

Lommel

Three-term relations connecting

[CHAP. IX

polynomials.

possible to deduce from the recurrence formulae a class of relations

which has been discussed by Crelier*. The relations were obtained by Crelier from the theory of continued fractions.

shews that Jv+m (z) and Rmv (z), qua functions same recurrence formula connecting three contiguous functions; and so a repetition of the arguments of § 9*6 (modified by replacing the Bessel functions by the appropriate Lommel polynomials) shews that First observe that § 9*63 (2)

of

satisfy precisely the

in,

Rm+n

( 1)

2

v

(z)

— R m> y(Z) R n< „ +m (Z) — J£ m—

Next in §9*63(2) replace m by (m + v)/z from the two equations it ;

**t», v

\Z) -tim.v+i

\

z)

~~ -^"in+i.v z \ )

it

\Z)

v

in

—

is

then seen that

is

§96)

(cf.

More generally, if in § 9*63 (2) we had we should have similarly found that \Z) -tim-n+i, v+n \Z)

t

v+1 \ z )

— Rm, v\ z

and, since its value

;

m-

lt

v+1 (z)

replaced

~ Rm+i, v \ z ) Rm—n, v+h \ z ) = ivm _ „ \Z) xv m_ n „+ n yZ) — iim

Rm>

v

(z)

Rm

-.

n+ i „ +n (z) t

Rm-\

(4)

we

If

m and wbym-1 v

(z)

;

and since

= 0.

and v by v

+ n,

v\Z) tim—n—i, v+n \z h

t

its

1 in this equation,

#m _n-i, K+M+i (z) — Rm, p

m into m=

value

1,

v

m into m=n

when

\Z),

Lommel +.

and n +

rewrite this equation with

when

unaffected by changing

— Rm +i, v (z) Rm-n, v+n \ z ) = -Rji-

a result given in a different form by

Replace

is

Rm—2, v+1 \ z )>

1,

m by m — n

li

(3)

=

in the special case v

and so the value of the function on the left m — 1. It is consequently independent of m is R n - i
)

unaffected by changing

Rm< „(z) Rm< v+1 (z) - R m+hv (z)

a result essentially due to Bessel

v

\Z)'

v+m+i

Rm-i,v+i \ z )

— Rm—1, v \Z) Rm~l

R,n,

i,

and v by v+1, and eliminate

1

and so the value of the function on the left m — 1. It is consequently independent of m is unity, we have Crelier's formula (2)

Kn —

v

(z)

and

it is

R n (Z eliminate Rm ^hv (z)

Rm-n-2, v+n+1 \ z )~

in place of n

and

found that

,v

J-

be-

tween the two equations, we see that

R

(z) R m -P - v+p+i (z) — RPt „ (z) rt m _w _j v + n+i \z) — Rm, v (z) [ Rm-p-2, v+p+i (z) Rm-n-i, v+n+1 ( z ) ~~ Rm-n-l, v+n+i ( z ) Rm-p-^ v +p+i \Z)] — Rm, v \Z) Rn-p—i, v+p+i \ z

n> „

Jt

i

)>

by

(3).

If we

we obtain (5)

R

transform the second factor of each term by means of § 9*63 (10),

Crelier's result (loc.

ni

„(z) jRp_ m_i,„+ m +i (z)

cit.

p. 143),

— Rp

,

v

\Z)

Rn-m—i, i/+m+i \ z )

= Rfn v \Z) Rp—n—i,v+n+i \ z t

*

Ann. di Mat.

(2) xxiv. (1896), p.

136

et seq.

f

Math Ann -

-

Iv<

(

1871 )'

P-

U5,

)'


9'64]

This

301

the most general linear relation of the types considered by Crelier

is

connects any three polynomials

R m>v (z) Rn>v (z), RPiV (z) which have )

parameter v and the same argument symmetrically

it

The formula may be written more

z.

+ Rp, v (z) Rm -n-\, v+n+i (z) + Rm v ( z ) Rn-p-i, K+p+i (z) = 0,

Rn, v {z) Rp-m-i, v+m+i (z)

(6)

;

the same

,

that

to say

is

2 Rn,Az)Rp- m -hv+m+ i(z) = 0.

(7)

m, n, p

A

similar

result

may be

functions whose orders differ

obtained which connects any three Bessel

by

If

integers.

we

eliminate

Jv+m -i (z)

between

the equations*

"c+n \Z) ^Jy+p \Z)

we «/

= J»+m \Z) K>n—m,v+m \Z) — Jy+m—i \ z ) -">n— m-i, p+m+i (*)> — Jy+m \Z)

ltp—m,v+m, \ z )

— J v+m-i \Z) Rp—m-\,t>+m+\ \Z),

find that

— Jv +p (z) R n -m-i,v+m+i (z) ~ J »+m \Z) L-ftn-m, v+m \Z) -ftp-m-i, v+m+i \Z) — -Hp-m, f+m {Z) Rn—m—i, v+m+i (*)] *= J»+m (Z) Rp-n-i, f+n+i (z)

M-n (*) Rp-m-i,v+m+i (z)

'»

the last expression

m,

n, p,

v

by

0,

n

is

obtained from a special case of (5) derived by replacing

— m, p — m, v + m

respectively.

It follows that (

2 Jv+n (z) Up-TO-i,

8)

„

= 0,

+m+1 (*)

m,n,p

and obviously we can prove the more general equation m,n,p

where

^ denotes any cylinder function.

The last two formulae seem never to have been previously stated explicitly, though Graf and Gubler hint at the existence of such equations, Einleitung in die Theorie der BmeVschen Funktionm, n. (Bern, 1900), pp. 108, 109. [Note.

If

we eliminate J^-i i

(

from the equations

z)

J»+m («) = J* (*) Rm, v («) - Jv -

1

(«)

Rm -

Wv + m-l («) s= «^k (*) Rm-1, v («)—•/_! and use

(2) to simplify the resulting equation,

we

(*)

1.

p+

1 (*),

Rm-2, v + 1

Jv(z)= -Jv + m (?) Rm-2, r + l(») + ^ + »-l (*) ^m-1. k + and

so, replacing v

by

v

(*)i

find that 1 (*)>

- m, we have

«A--m(«)= -Jy(t) Rm-2,v-m + l(z)+Jv-i(*) ^w-l,v-m + l

By

using § 9-63 (10),

(')•

we deduce that

Jv-m i?)=Jv (*) R-m, v

(«)

-Jv -l («) R-m-h v +

l

(*),

say that the equation § 9*6 (1), which has hitherto been considered only for positive values of the parameter wi, is still true for negative values.]

that

is to

* It is supposed temporarily that result is symmetrical, this restriction

m is the

p ; but since the final See also the note at the end of the section.

smallest of the integers m, n,

may be removed.


302

Hurwitz' limit of a

9*65.

We

shall

now prove

Lommel polynomial.

that <

lira

(1) V '

[CHAP. IX

*-.

**)^™.^(*> = j r(i/

(,).

+ m + l)

—

This result was applied by Hurwitz, Math. Ann. xxxm. (1889), pp. 250 252, to discuss Jv (z) when v has an assigned real value (§ 15*27). It has also been examined by Graf, Ann. di Mat. (2) xxiii. (1895), pp. 49 52, and by Crelier, Bern " Mittheilungen, 1897, pp. 92—96. the reality of the zeros of

—

From

961

§

(3)

we have

(m-n)\r(v + m-n + l) + n + 1)' (m-2n)\r(p + m + l) K+an

r(v + m +

Now

n= on\V{v

l)

'

write (v + m — n + 1) _ (m-2n)!I> + m + l) =

(m — n)\T

Q ° ( ™' n)

>

so that

n

.

.

(m, n)

=

(m —

7^

(i/

If

now

—n—

w)(ra r-P

1)

...

/

-.v

+ ra)(i' +

(m - 2n +

m-l)...(i/

N be the greatest integer contained in

numerator of 6 (ra,

ri) is

numerically less

denominator, provided that n

Hence, when

n> N,

1) TV 1)

+ m-n +

.

>

\

then each factor in the than the corresponding factor in the |

v

,

\

> N.

m > 2N,

and

\0{m,n)\
when n has any fixed

value,

lim

is

absolutely convergent,

lim

t

W

it

hZY+" is

e
,

«=o»!

$z)" +m

rr is

ru+»+i) ,

of values of z (by the test due to Weierstrass),

also uniform in

= i

theorem* that

Jf^T-

established.

Again, since the convergence of 2 ° °

is

1.

follows from Tannery's

and the theorem of Hurwitz

to its limit

=

(->;(*'>•+"

i

Since

(m, n)

/^

it

uniform in any bounded domain t

follows that the convergence of

+ 1 (2)/r(»

+ m+l)

any bounded domain of values of z.

* Cf. Bromwich, Theory of Ivjinite Series, § 49. f An arbitrarily small region of which the origin excluded from this domain when R (v) < 0.

is

an internal point must obviously be


9*65, 9*7]

From

the theorem of Hurwitz

fraction for

by

§

Jv-

X

963 (1). On

easy to derive an infinite continued

it is

when

For,

(z)/Jv (z).

Jv (z)

=jfe

0,

we have

^# =

lim

r^t^l

-

lim

L^-l/I^M-l,

carrying out the process of reduction and noticing that

R* v+m-Az)

m_

=2

z

_

x

_

1

2z

î,„+m(*)

we

303

l

+ m)

(v

find that

^Hi^-2^» ttm,, + i (z)

and hence

-

2

(i/

+ 1) z- 1

^W--

(2 )

J

v

(z)

\

,

2

(i/

+

1 2) r-

*

2(v

-

-2

. . .

+ m) z~*

J

+ l)z- -2(v + 2)z-

1

1

-...

This procedure avoids the necessity of proving directly that, when m-*-ao, the element of the continued fraction

2(» + l)*-»-...-2>+»)*-i-

Jv {z) may

be neglected

9*7.

;

the method

is

due to Graf, Ann. di Mat.

Jv + m (z)

xxnr. (1895),

(2)

last

p. 52.

The modified notation for Lommel polynomials.

In order to discuss properties of the zeros of Lommel polynomials, it is convenient to follow Hurwitz by making a change in the notation, for the reason that Lommel polynomials contain only alternate powers of the variable.

Lommel

Accordingly we define the modified equation *

m

n ^•'

(1)

M-^ W- :

ro-

C ntB

+ m-n + I> + n+l)

(-)*r(„

o

tt

polynomial gm>v

(z)

by the

l)*" *

so that

Rm

(2)

,r +1

By making

- (hz)~™ gmA±*)-

the requisite changes in notation in

will easily obtain the following

§§ 9*63,

964, the reader

formulae

= (" + m + l)gm>v (z) - zgm_hv (z), #,„+,, r-i (z) = vgm „ (z) - zg^, „ +1 (z),

0«+i,r (*)

(3)

(4)

,

(5)

IF*

bTz

P^'W = *9Â*) +gm+h

^m ^ (,-m-i gmv

(6 )

(7)

(z)

gm

,

,

(z) gm+i< „+1 (z)

( z ))

«

m+1 „_, (*)

<,

- gm+2t „ (z) gm _

1%

„ +1

* This notation differs in

,- 1 (z),

[J

9-63(7)]

gm+1
[§

9-63 (4)]

(z) = z m g

0<

[A

963 (2)] [§ 963 (1 )]

[§

„

(z) glt ¥+m+1 (z).

special case of § 9*64 (5).]

unimportant details from the notation used by Hurwitz.

:


304

These results

will

be required in the sequel

down the analogues

of

all

;

it will

the other formulae of

not be necessary to write

9'6

§§

[CHAP. IX

— 964.

The result of eliminating alternate functions from the system some importance. The eliminant is (f

+ m) gm+2> „ (z) = cm (z) gm% „ (z) - (v + m + 2) z*gm _

where

cm (z)

We

=

(v

2>

(3) is of

„ (z),

+ m + 1) \{v + m) (v + m + 2) — 2z).

thus obtain the set of equations f

+ 2)giy (z)- (v + 4>)z'g0tV (z), (z) = c4 (z) gitV (z) - (v + 6) *g%w (z), (v + 4) g

(v

St

(8)

-

v

= Ca, (z)

- {v + 2s + 2) z g„^ „ {z), 3

(v

+

(y

+ 2m - 2)gvn,* (z) =
2s) gv+ % y (z)

<7M)

„

(z)

t

2

9*71.

We

The reality of the zeros of g^îz) when v exceeds

shall

now

give Hurwitz' proof of his theorem* that when v

a ^ real; and also that they are in which case one of them is negative.

zeros of gtm,v{z) are

— 1 > v > — 2,

After observing that gmt,v(z)

shew that the

— 2.

is

> — 2,

all positive, except

the

when

a polynomial in z of degree m, we shall ... g2iV (z), g^ v {z) form a set

set of functions g*m,*(z), 3W-a.*(s)>

of Sturm's functions.

be the case are (i) the combined with (ii) the theorem that

Sufficient conditions for this to

existence of the set of relations §

97

(8),

the real zeros of gvm-^vi?) alternate with those oi gimv {z).

To prove that the

zeros alternate, it

is sufficient

to prove that the quotient

a monotonic function of the real variable z, except at the where the quotient is discontinuous. denominator, zeros of the

9*m,»( z )l9im--2,*(z )

We

is

f^ %9

have

(z)

w(

*M -£ j dz {{gtm-^Kz))

-

= - 389**, »*-*,

Wi r - 9r.* (z) g\ » (*) - 9:* (*)#V (*);

where

,,

and from

%

§ 9*7 (3) it follows that

B9m-i.m-8 -

* SK&m-s,*^ + iv + 2m - 2) g\n^y (z),

so that

m

X

i {v + 2r) z™-™ tf^ m™,^* - g>*n-*,„ (z) + (v + 2m) r=l and

therefore, if

when

v

m > 1,

5KK2IW,*W_8

is

expressible as a

sum

> — 2. * Math. Ann. xxxin. (1889), pp.

254—256.

9 (z),

of positive terms


9-71, 9-72]

305

The monotonic property is therefore established, and it is obvious from a graph that the real zeros of g^^, (z) separate those of g2mv (z). It follows from Sturm's

any

theorem that the number of zeros of g^nîz) on

number of alternations

interval of the real axis is the excess of the

g^^

in the set of expressions g*m tV {z),

of sign

...,g0l/ (z) &t the right-hand

„ (z),

end of

the interval over the number of alternations at the left-hand end.

The reason why the number

of zeros is the excess and not the deficiency in that the a decreasing function, and not an increasing function of z, as in the usual version of Sturm's theorem. See Burnside and Panton, Theory of Equations,

quotient gr^

i.

„ (z)/£2„,_2, „ (z) is

(1918), § 96.

The arrangements

— oo

,

of signs for the set of functions

00

The upper ;

z has the values

2m

2m- 2

2m -4

+

+

+

...

+

+

+

+

+

...

±

+

(->*

~ (~)m l

...

-

+

— 00

negative

when

are as follows

0, oo

2

_)m-2 (

or lower signs are to be taken according as v

and the truth of Hurwitz' theorem

is

+1

is

positive or

obvious from an inspection

of this Table.

9*72.

Negative zeros of g
Let

be

v

less

than

— 2,

and

let

p< — 2.

the positive integer * be defined by the

inequalities

-2s>v>-2s-2. It will

now be shewn

when v lies between — 2s and — 2s when v lies between — 2* — 1 and — 2s —

that*,

has no negative zero; but

that,

has one negative zero.

Provided

that v

+ 2m

that, in

each case,

m is taken to

1,

gsm ,(z) t

g-mtV (z) be so large 2,

is positive.

It will first be

shewn that the negative zeros

(if

any) of

g^^z)

alternate

with those of g*m -%,y{?)' * This proof differs

(1921), pp.

266—272.

from the proof given by Hurwitz

;

see Proc.

London Math.

Soc. (2) xix.


306

By means

of the formulae quoted in § 9*7,

M

it is

[CHAP. IX

clear that

M

= ^2m-2, zg*m-i, + 9*m+i, v-i( z )} - 9*m, »( z ) lâm-i, —1(2) ~ S'am-i, •-(*)} = (1/ + 2m)0 tM -i,r(«) + gmi~2, v {z)g*m»,v-\(z )-g*m-i,v-i(z)gvmAz) = (l/ + 2m)5r 2m_ „(^) - ^nrWft.rtW-lW {

,

2

1>

>o, provided that

i>

+ 2m

is

and z

positive

Therefore, in the circum-

negative.

is

stances postulated, the quotient

is

a decreasing function, and the alternation of the zeros

The

existence of the system of equations § 9*7 (8)

is

evident.

now shews

that the set

of functions g*m,v(z), g*m-2, v(z)>

• •

g»+t, *( z )>

•

.g*»,

*( z )>

-g**-*A z)>

+g*»-4,v( z ),

•••,

(~Ygo,v( z)

form a set of Sturm's functions.

The

signs of these functions +, +,

and

when

....

z

is

-

are

oo

+,+,-,+,

...,

When

there are s alternations of sign.

(-)*,

z

zero, the

is

signs of the

functions are ±, ±,

•-.,

+,-,+,

±,

•••,(-)*,

the upper signs being taken when — 2s > v > — 2s — 1, and the lower signs being taken when — 2s — l>v> —2s— 2; there are s and s + 1 alternations of sign in the respective cases. Hence, when — 2s >v > — 2s — 1, #»»,„(£) ^ as no negative zero but when — 2s — 1 > v> — 2s — 2, gzm,v(z ) nas one negative ;

zero.

The theorem

9*73.

As

Positive

stated

is

therefore proved.

and complex zeros of g^^îz) when v<

in § 972, define the positive integer s

- 2s > It will

now be shewn*

has m — 2s positive zeros;
that

has

v>—

that when v but that,

by the

2.

inequalities

— 2.

lies between.

when v

m—2s—l positive zeros.

2s

—

lies

— 2s and - 2s — 1, gtm, v (z) — 2s — 1 and — 2s — 2,

between

Provided

that, in

each case,

mis so

large

m+v is positive.

* This proof la of a more elementary character than the proof given by Hurwitz paper cited in § 9*72.

;

see the

,


9-73]

In the

first place, it

307

follows from Descartes' rule of signs that, in each case,

9tm,v{z) cannot have more than the specified number of positive zeros. when v lies between — 2s and — 2s — 1, the signs of the coefficients of

z\ ...,z*>, z»+\ z-»+\ *•+»,

1, z,

are

+, +, +,

ffsm v (z) ,

+, -, +, -,

...,

zm in

...,

For,

...,

(-)»;

and since there are m — 2s alternations of sign, there cannot be more than m — 2s positive zeros. When v lies between — 2* — 1 and — 2* — 2 the corresponding set of signs

is

_ _

_ _ _

m — 2s—l

and since there are than

w — 2s —

1

_

a.

(— V»-

alternations of sign there cannot be

more

positive zeros.

Next, we shall prove by induction from the system of equations § 9*7 (8) many as the specified number of positive zeros.

that there are as

When

v lies

between - 2* and

the coefficients in gVtV (z) have no +) and so this function has no positive zeros.

alternations of sign (being all

On

— 2s —

1,

the other hand

0«+ 2 ,(0) > 0, gu+3f „(+ ,

ao )

=-»

and so gu+2lV (z) has one positive zero, aM say and, by reasoning already given, it has no other positive zeros. Next, take <7«+4,i,(s); from § 9*7(8) it follows that its signs at 0, a M + ao are +,—, + hence it has two positive zeros, and by the reasoning already given it has no others. The process of induction (whereby ;

;

,

we prove function)

that the zeros of each function separate those of the succeeding is

now

evident,

and we

infer that

gm

y (z)

,

has

m— 2s positive zeros,

and no more. lies between — 2s — 1 and have no alternations in sign (being all

Again, when v -

5 4«+s,i'(^) no positive zeros.

On

2s ),

—

2,

and

the coefficients in

so this function has

the other hand

#«+4,r(0)

< 0,

and so g4a +4t ,(z) has one has no other positive zero.

5T« +4(K (+ 00 )

positive zero,

By

— —

=+

CO

,

and by the reasoning already given

appropriate modifications of the preceding reasoning

cession that gu+^viz),

that g2m,r(z) has

gv+8tV (z),

m — 2s — 1

By combining

...

have

2, 3, ...

v< —

in suc-

and in general

positive zeros.

these results with the result of

theorem, that, when

we prove

positive zeros,

it

2,

has 2s complex zeros, where s

and mis so large is the integer

§

that ni

such that

- 2s > v > - 2s -

972, we obtain Hurwitz'

2.

+

v is positive, gsmfV (z)

CHAPTER X FUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS 10* 1.

The functions J„(#) and E,

(^) investigated

by Anger and H. F.

Weber.

In this chapter we shall examine the properties of various functions whose

by certain representations of Bessel functions. by integrals resembling Bessel's

definitions are suggested

shall first investigate functions defined

and Poisson's

gral

integral, and, after discussing the properties of several

Yn (z) we shall study a class of functions, first defined

functions connected with

by Lommel, of which Bessel functions are a particular

The

first

It is defined

function to be examined, J,

case.

suggested by Bessel's integral.

(z), is

by the equation J„ (z)

(1)

We inte-

-=

-

f 'cos (v$

- z sin 6) d6.

7T J o

Jn (z)

This function obviously reduces to

when

It follows from § 6*2 (4) that,

when

v has the integral value n.

an integer, the two functions are distinct. A function of the same type as J, (z) was studied by Anger*, but he took the upper limit of the integral to be 27r; and the function J,(z) is v is not

conveniently described as Anger's function of argument z and order

A

similar function

was discussed

E„ (z)

investigated the function

later

v.

by H. F. Weberf, and he also


E„ (z) = - ('sin (v0 - z sin 0) dd.

(2)

In connexion with this function reference should also be made to researches by Lommel, Math. Ann. xvi. (1880), pp. 183-208. It

may be

noted that the function

— 2m

/"2"-

1

/

cos {?6

- z sin

6)

dd which was actually

Jo cussed by Anger is easily expressible in terms of J„ (2) and E, 2tt — 6 in the right-hand half of the range of integration, we get

— 1

*T

/"2t /

fn

1

cos(v0-2Sin0)cW=—

/

*w J

J

— cos

2

vit

I

;

for, if

we

replace 6

by

If' cos(2vir-vd+zain8)dd

cos (p6 - z ain 6) dd + 7T-

I

2,1t

J

.

J„

(z)

+ sin

vit cos

vn

.

E„ (z).

* Neueste Schriften der Naturf. Ges. in Danzig, v. (1855), pp. 1

Vv

(z)

dis-

cos (vO - z sin 6) d0 = (z

Additions a la Conn, des Temps, 1836, p. 15 name to the function.

(cf.

-

v)

—29. It was shewn by Poisson that

sin v jt,

§ 10*12), but as

he did no more it seems reasonable

to give Anger's

t ZUrich Vierteljahrsschrift, xxiv. (1879), pp. 33 nition of

Ev (z).

— 76.

Weber omits the

factor 1/x in his defi-

ASSOCIATED FUNCTIONS

10-1]

To expand J„ (z) and E„ (z) sinm 6 sin

I


and proceed thus

in the integrals

vdd$ =

JO

309 z,

write %tr

+

for

:

cosm

I

sin (^vtt

+ v) d(f>

J-Jir

=

2 sin \vir

cos"* <£ cos i><£d -'o

it

by a formula due

to

.

m

sin ^yn-

!

Cauchy*.

In like manner, Sin TO

0COSl>0d0=_—-=7-;

r—

:

:

—

:

.

But, evidently, v

J„(*)=- 2 Tr

:,

/c m =o (2m)! Jo ;

sin2m

0cosv0d0+ - 2 hr 7r m=0 (2m +

2™* 1 rrr sin l)!Jo

sin vddd,

so that

J r (^) = cos^7r 2

(3)

/

jTT

\:l

;

Tx

(_)m/l^)m+i

co

+ Sill &V7T 2

=r-

.

„,

„

T— Ti

,

.

and similarly

w

w

B„(^) =

(4)

sinii/7r

*

2 TO -

"7 ^ r(m-| /+l)r(m + .->-

—

'

i

£i/

I

+ l)

2 =-

-cos*i/7r

„vVi/

1

5--

i

These results may be written in the alternative forms

._ sini/7r

_

,-. 1

( AZ) ~

'

sin

+

inr

|_

w["

2

7T

L

12

VTT

_

1

+

cos virY tt"

2

2

+ -i^

z6

z^

(2

2

-^)(4

2

-i/

2

)~(2

2

2

2

"1

2

-i/ )(4 -i/ )(6 2

.2?

2

-y ) 2

+

,

"'J

5

-»;2_ (l 2 -»'2 )(3 2 -va ) + (l 2 -i'2 )(3 a -i/2 )(5 a -y2 )~'

1 s E,(,)= -i:^

(6)

z*

I"

^[l-^L+r 2 -i>

,-...1

2 2 2 (2 -i/ )(4 -i/2)

3

z

[l 2 -!/2

^

"

2

2

z

~ (l - v ) (3 2

2

2

2

i/

)

+ (l

*"j

z* 2

-i/2 )(32 -i;a )(5 8 -^)

~

'

Results equivalent to these were given by Anger and Weber.

The formula corresponding to (5) was given by Anger (before the publication of his letter to Cauchy which was communicated to the French Academy on July

memoir) in a 17,

1854 *

;

—

135.

sur les integrates difinies (Paris, 1825), p. 40.

Cf.

see Comptes Rendu*, xxxix. (1854), pp. 128

M&m.

Modem Analysis,

p. 263.


310

For a reason which

will

be apparent subsequently

[CHAP.

(§ 10*7), it is

X

convenient

to write

Z3

Z

(7)

(8)

(*)

So,,

«_,,„(*)

=

p—^ ~ (p _ y

1 =-- + 2

and, with this notation, -

/

%

=

(9)

J,

,,~x

„ E„ (z) =

(z)

/

(10)

3a _

„a(2»

_ ^ (32 _ ^(52 _ ^ ~

18

_ „*) (4 2

2 i/

+

'

' • ' >

"

)

we have ,

—

v sin vir

.

s 0t v (z)

l+cosj/ir

-

^

(

z*

_ ^) ~

a ^2

2?_

^+

z2 yi

sin vrr

cos vB

I

(

,

.

—

i/(l

-

(^)

So, „

,

*_,, „ (*),

— cos i/7r)

.

.

*-i. , (z).

deduce the following formulae from these results

It is easy to

(11)

iy

cos (z sin

.

6)d6 =

— v sin vir

s_hv

.

(z),

Jo sin vd

(12)

.

cos (z sin 6)

dd

= - v (1 - cos

.

sin (z sin 6)

d6

= sin vtt

.

sin (* sin 0)

d0 =(1

vjt)

.

s_i

„ (z),

Jo

(13)

sin v 6

I

.

s0< „ (*),

Jo

f cos Jo

(14)

i>0

Mir

(15)

cos

I

v

.

cos (z cos )d(f>

+ cosi/7r)

= — v sin îrr

.

.

s0> „ (z),

S- h „ {z),

Jo /•in-

cos v<£. sin (z cos <£)d<£

(16)

= cos £i/7r.s

,

•.(*).

Jo Integrals

somewhat resembling the s™°

[e

integrals discussed in this section,

COS

(ne-cos8)dd,

sin

J

have been examined by Unferdinger, Wiener

Sitzungsberichte, lvii. (2), (1868), pp.

Also, Hardy, Messenger, xxxv. (1906), pp.

according as v

10*11.

that,

— 11

when

flO

sin

v is real, it is

is positive, zero,

(vO-zsin

6)

-r=-,

equal to

\it

2

from the formulae

J_„ (z) =

(1)

J„ (z)

+

(2)

J„ (s)

- J _, (s) =

rj

n

Jn (z),

where

tf n

is 1,

or -

or negative.

Weber s formulae connecting

It is evident

611—620.

158—166, has investigated the integral

/oc I

and has proved

namely

4

A.

§

his functions with

101

— —

(9), (10),

Anger s functions.

(15) and (16) that

cos \vtt fi"

gin ftvir

/

f* I

cos

v

cos (z cos

cos

v

sin (2 cos <£)

) d,

Tr

d,

1


1011, 10-12]

E„ (2;) + E_„ (.s) =

(3)

E„ (3)

(4)

-

—

a

T

cos

sin (s cos

v

) cUf>,

Jo

^^

E_„ (s) =

311

cos

T

v

cos (2 cos

<£) d.

.'0

It follows on addition that

J, (z)

= I cot |i/tt {E„ (*) - E_„ (*)} - £ tan |i/tt {E„ (z) + E_„ (5)}

so that sin vir

(5)

J„ (z)

.

= cos vir E„ (2) - E_„ (z), .

and similarly sin vir

(6)

The formulae

10*12.

(5)

and

.

E„ (z) = J_v (z) — cos vir J„ (2). .

(6) are

due to Weber.

Recurrence formulae for J„ (z) and E„ (s).

The recurrence formulae which are satisfied by the Weber have been determined by Weber.

functions of

Anger and

It is evident from the definite integrals that „_! (z)

+ Jr+i (z)

—

- J„ (z)

=-

(cos

I

cos {v0

0--J

- z sin 0) d0

\* ^-{sm(v0-zam0))d0

d0

irz J

_

2 sin vir irz

and B„_, (s)

2i>

+ E, +1 (z)

2 f'/ cos E» (s) = f

- -iA sin (v0 -zain0)d0 J

J

=—

-J* {cos (i>0

J

— z sin

2(1— cos vir)

_

irz

It is also very easy to prove that

fJ-,(*)- J„+1 (*)-2J/(*) = 0,

]_, (z) - E„+1 (z) - 2 E/ (*) = 0. From /i

\

(1 )

(2)

(3) (4)

these results

we deduce the

t / \ J,-! (*>

+ Jt r+1 /(0)\ = .

eight formulae

—Z J,t 2l/

\

/

(z)

w ——

2 sin

,

TTZ

l v^{z)-J v+l {z) = 2J v'{z),

+ v) J„ (z) = zJ v _, (z) + (sin w)/ir, (^ - v) J„ (z) = - zJ v+ (z) - (sin V7r)/7r,

(^

i

0)}

d0


312

(5)

L m„(.) + m„iM)-±m.(M)- *<

(6)

B r _ (#)-« r+1 (jr)-2« l

[CHAP.

X

-™"\

r (*),

(^ + v) E„ (z) = Z E„_, (z) + (1 - COS 1/7T)/7T, O - v) E„ (z) = - ^B„+ (z) - (1 - cos i/7r)/7r,

(7) (8)

i

where S, as usual, stands

for z(d/dz).

Next we construct the (^

2

differential equations; it is evident that

- v>) J„ (z) = O - v) \z J v _! (s) + (sin w)/ir} = z(^ + l — v) J v _! (*) — {v sin vji)\ir = — z J v (z) + (z sin i>9r)/w - (v sin tnr)/7r, 2

so that

v.J.(,)-£^™~.

(9)

We

also

(^

9

have

- v*) E„ (z) = (*-v) {*£„_! (z) + (1 - cos i*r)/w} = Z (^ + 1 - v) E,., (2) — (1 — COS V7r)/7T = — Z E„ (2) — Z (1 + COS I/7r)/7T — (1 — cos v7t)/tt, J/

2

so that

v >B , ( ,)

(10)

Formulae equivalent to

(9)

10*13.

formula

;

—

and

Naturf. Ges. in Danzig, v. (1855), (1879), p. 47, respectively

1/

i±J!

<—'')""• .

7T

7T

(10)

were obtained by Anger, Neueste Schriften der

17 and by Weber, Zurich

p.

had been discovered

(9)

Vierteljahrsschrift, xxiv.

earlier

by Poisson

Integrals expressible in terms of the functions of

(cf.

§ 10"1).

Anger and

H. F. Weber. It is evident

from the definitions that

J, (*) ±

(1)

i

r exp {±i(v$-z sin 0)} d$.

E„ (z) = 7T

Jo

—

of this result, combined with formulae obtained in §§ 6'2 622, it possible to express numerous definite integrals in terms of the functions of

By means is

Bessel,

Anger and Weber. Thus, from

§ 6*2 (4)

-^— T e-"-»i"h< dt = Sin V7T

(2) V '

{ l

we have

J„ (z)

- Jv Or)},

J

when jarg0|<^7r; the R(v)>Q.

result

is

valid

when

jarg.r|

= |7r,

Again, we have

•

(3)'

f" J

e"

t

-"inht dt =

-^— \Jsini/7r

V

(z)

- J-„ (*)},

provided that


10*13, 10*14]

when we combine

so that,

and

(2)

(3),

(* e~** inht cosh vtdt=\ir tan \ vir {J„ {z)

(4)

313

-

Jv (*)} - \ ir {E„ (z) +

-

J„ (z)}

F„ (a)},

Jo

f" e-zaiaht sinh

(5)

itf

d<

=

|tt cot £i/tt [Jv (z)

- %w

{E„ (*)

+ Yv (z)\.

Jo

The

e-zcosht cosh

integral [

^^ has already been evaluated (§63);

but

Jo

[" e-zco* ht smhvtdt Jo

does not appear to be expressible in a simple form; its expansion in ascending powers of z can be obtained from the formula of § 6*22 (4), 2siny7r M -z^ht sinh vtdt} I_ v (z) + Iv (z) = - PVcose cos v d6 + ( e 7

7r

7TJo

but, since [*

..

(-)m sinv7r

* co8 «*«**

- V(v + m)

.

-

cosm j

-

* '

•'O

m_

+ "2~

v

/

Fl {- m>

;

i/

-f

ra

2-'

*

\

"V'

*. the integral under consideration cannot be evaluated in any simple form

The formulae 9*33 we have

(2)

—

rent -

(6)

(5) are

z * inht

nugatory when v

dt

is

an

integer,

but from

§§ 6*21,

= l{Sn (z)-TrE n (z)-TrYn (z)},

Jo

I" e- M

(7)

- zainht

Jo

The

dt

= \ (-)n+1

{Sn (z)

+ 7rE n (z) + ir Yn (z)}.

associated integrals

08 tC l" e-* °*(x cosh t)dt t)dt, ("e-^8m (xaiuh v am Jo Jo xx. Journal, (1885), p. 260. have been noticed by Coates, Quarterly

Various integrals of these types occur in researches on diffraction by a prism Whipple, Proc. London Math. Soc. (2) xvi. (1917), p. 106.

10' 14.

of

J± v

see, e.g.

Asymptotic expansions of A nger- Weber functions of large argument.

It follows (z)

;

from

when

§

1013 (2) that,

\z\ is large

and

in order to obtain the |

args| <%tt,

it is

asymptotic expansion

sufficient to obtain the

asymptotic expansion of the integrals

Jo

out this investigation we shall first expand cosh sinh i/t/cosh t in a series of ascending powers of sinh t.

To

carry

i;£/cosh

t

and

* See Anding, Sechsstellige Tafeln der Besselschen Funktionen imaginaren Arguments (Leipzig, Takeuchi, Tdhoku Math. 1911) [Jahrbuch Uber die Fortschritte der Math. 1911, pp. 493—494], ard

Journal, xviii. (1920), pp. 295—296.

11

W.

B. V.

j


314 If en

= v, we

manner of §

have, after the

M«+,l/«+)

1

[CHAP.

X

7*4,

ui

(

i/M *

)

so that

_ 1 f<«+. !/«+.!+) "~ fi"-*(f-l)d£ ~ 2tti 2w»7J (^l)2 -4fsinh

cosh vt COshy<

:

cosh*

(«+, i/«+,

i+)

2^7

^ r j>-i 2«i*

»

3

<

gm sjn h»» t

(?-l)2m+1

L«-o

2*p

p sinh * 2

*

(f-l^-iftf-l^-^sinh

]dr-

2

*}

Now (i+)f*»+m-*d£

(^-l)am+1

$7rt J 27rtJ

+ m + f)

r(£i/

r(|i/-m+£).(2m)! OT _ (-) cos^7r

r(m-t-.^

+

^)r(m+|-^)

'

(2m)!

7T

and,

we take p

if

to be that 1

shewn

R(p + $±$v) > 0,

so large that

in Fig. 15 of § 7 '4,

/•(«+, l/t»+,l + )

2m

fp

we

and then take the contour

find that

+ * v ~irf^

(f- 1)^" {(£- iy 1

- 4fsinh 2 «}

~ If v

and

*

7r

Jo

are real, the last expression

may be

1

+ 4
2

«

'

written in the form

<2p)I

where

^

dx <

1, since 1

It follows that,

COSh COSh

+ 4#

when

I/*

COS £ V7T

t

7T

— x) sinh <^ 1. 2

(1

R (p + \ ± ^

j/)

^ 0, we have (2m)!

m=o

+ ,, (zm£±i+M£<£±irW (2 sinh (2p)!

,

r]

For complex values of v and < this equation has to be modified by replacing < X ^ 1 by a less stringent condition, in a way with which the reader will be familiar in view of the similar analysis occurring in various sections of Chapter vn. the condition

Similarly

we have 1

u h»

_ u -* = _L

f(« + ,l/M+)

2-iri-J

(

z» *

1 J— - _L_

k-w

\

)

£-1/4

d£,

L

315


10-14] so that

~ 2iri J sinh 2t

(£ - I) - 4f sinh 2

r(«+, i/«+, i+)

1

"as J whence sinh

vt

it

= sinîrn- p-

1

(t-ir»

we take p

(-)w

^ ^2

i;

is real

and z

<

J

- )" r(m+ *

o

the (p

It follows from §§ _

+

—2—

2

J r (*Wr (*) +

(1)

,

v

(2)

(3) I2

1

trz

- r (*)~-F„(#) x v/n /

_

1

J

(ro+

"^

*

+ j»Q r (m + 1 -

& y)

2m+2

is

same sign

of the

is

combined with

-"2

(I 2

as,

and

is

R(p + l

so large that

numeri-

± $v) ^ 0.

§ 10*11 (6) that

~ v*) (32

L.....1

trz

y(2»-y»)

sin vir \v

/ox (2)

;^

(f2)

when p

and

sin vir

'

these asymptotic expansions possess the

is positive,

1013

1

OT= o

l)th term

+1 1

sinh

(

property that the remainder after p terms cally less than,

± \v) > 0, then

v

+ l)!

sin^Tr * (-)OT r (w +

e-* 8inh * cfe~

27T

If

R (p + 1

,*.

follows that

it

.

.

"I

so large that

(2p

* r^.a~qgJg:

sinh vt

2» £p sinn 2* £ * {
+

+a

ui y)r( P + i-^)

(-)^( P +

integrating these results,

cosh

*

r(m + 1 + \v)Y(m + l -jiQ ginh tym+1 (2 v (2m + 1)!

|_«t

7T

2

r p-i 22w* w sinh 2"1 t f

r L:

follows that, if

cosh*

On

$.#

+ ,i+) 1_ r(u+,ilu

sinh vt

1

?

|_i

+ CQBinr

^—

-cosier Vv irz

r,1

_

— l

2

-"

- v ) (4a - y )

+

*

2

;

y (2 2

-

v*)

'"

[z

z*

z*

1

"'J'

(!*-*){»-*)

,

-y )

2

a

?

2

ii-

y(2 2

v (22

+

(4

1

-J 2

- i/ ) _ 8

"I

"J

These results were stated without proof by Weber, Zurich Vierteljahrstchri/t, xxiv. 48 and by Lommel, Math. Ann. xvi. (1880), pp. 186—188. They were proved as special cases of much more general formulae by Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), p. 228. The proof of this section does not seem to have (1879), p.

been given previously.

Since the only singularities of cosh vtjcosht and sinh vt/cosht, qua functions of sinh t, are at sinh t = ± i, it is possible to change the contours of integration into curves in the tf-plane on which arg (sinh t) is a positive or negative acute angle; and then (1)

and

we deduce

in the usual

(2) are valid over the sector

|

manner

arg z\
(cf.

§6*1) that the formulae


316

[CHAP.

X

Asymptotic expansions of Anger- Weber functions of large order and

10*15.

argument.

We

now obtain asymptotic

shall

expansions investigated in Chapter

when

v

j

and z are both

|

|

|

expansions, of a type similar to the

which represent J„ (z) and E„

viii,

(z)

large.

In view of the results obtained in

§ 10*13, it will

be adequate to obtain

asymptotic expansions of the two integrals

-1

f* /

e T M-*Bnh«

fa

TTJo

As

in

Chapter

we write

viii,

v

where

^ ft ^

We

(I)

and 7

it

first

— z cosh (a + ift) = z cosh 7,

not nearly equal* to

is

tri.

consider the integral

lf°° vt ~za niit If* g— «('<»8hy+8inht)^ — e~ dt = — ^

/

I

Wo in which

it is

""Jo

It is easy to

shew that

t

t

where n positive

number

imaginary

them

since

is

{

;

— iff),

y

is

t

increases from

a monogenic function of t, except possibly

to oo

;

we

when

= (2n + 1) id cosh y ± sinh y + y cosh y,

an integer; and, when coshy

is

When cosh 7 is positive,

supposed temporarily that v\z is positive.

cosh 7 + sinh t steadily increases from to 00 as shall take this function of t as a new variable t.

t

none of these values of r is a real cosh y does not vanish, and, when y is a pure the singularities are on the imaginary axis and the origin is not one of for,

not equal to

The expansion

rri

iti.

of dt/dr in ascending powers of t

»T 1

f<

0+ >

a-=2^j

where

is positive,

when y is real, (2n + 1)

is

m=

^^ 1

(ft

'

,

«*

T

If

(0+)
~ &S J

.jjm+i

»

and so a™ is the coefficient of I ft in the expansion of t-2™-1 in ascending powers of t. In particular we have

a _

1

°~l+cosh7'

ai

_ ~

_ 8,3

From

~"

_

1

2(1 225

the general theorem of

+

cosh 7 )4

— 54 cosh 7

4-

9

-

cosh 7

* ~ 24 (1 + cosh 7

'

'

cosh2 7

720(1+ cosh 7)10 § 8'3,

we

are

now

in a position to write

the expansion

* Expansions valid near

r

)

y = iri are obtained at the end of

this section.

down


10-15]

This expansion

valid

is

on the hypothesis that

when arg z

|

v/z is positive j

<

by

but,

\tr,

|

z

<

\

been established

a process of swinging round the

contour in the T-plane, the range of validity

domain in which arg

has, so far,

it

;

317

may be extended

to cover the

ir.

Next, we consider the modifications caused by abandoning the hypothesis that cosh 7 is real. If we write t = u + iv, the curve on which t is real has for its

equation

u sinh a sin ft + v cosh a cos

The shape

#+

cosh

u sin

v

— 0.

examined by methods resembling those

of this curve has to be

For brevity we write

of § 8'61.

u sinh a Since 4>

(u, v) is

sin

£ + v cosh a cos /3 + cosh u sin v =

v).

unaffected by a change of sign of both u and

> 0.

study the curve in which a

we

a,

first

It is evident that the curve has the origin as

its centre.

w,

when

any assigned and so the equation in u

follows that,

value of

= sinh a sin /9 + sinh u sin v,

33> (u, v)ldu

Since it

v has

has, at most,

multiple of

When

two

real

u

the value of

(—

oo

the

,

v)

If this is negative,

contour does not

= — oo

maximum

is infinite

whenever

v is a

(u, v)

4>

(+

oo

value of

<1>

,

,

v)

= — oo

(u, v),

;

qua function of

u, is

at

then being

- a tanh a 4- (it - ft) cot ft}. = has no real root, and so the (u, fi — the equation — = meet the line v @ ir or (by symmetry) the line

- cosh a sin fi

v

=

roots; and one of these

> v > — tr, we have

when v = £ - rr,

= a,

one

it.

3>

and,

(m, v)

value, 34>/3tt vanishes for only

{

I

-rr)

= ir — j3. Hence provided that the point

(a, /3) lies

in

one of the domains num-

v)= lies as in Fig. 25, bered 1, 2, 3 in Fig. 21 of § 861, the contour when a is positive contour the continuous curve indicating the shape of the <£

(it,

Fig. 25.

and the broken curve the shape when a increases is marked by an arrow.

is

negative; the direction in which t


318

when

It follows that the expansion (1) is valid

domains

[CHAP.

0)

(a,

lies in

X

any of the

I, 2, 3.

Next, we have to consider the asymptotic expansion when

(or, /8) does not any of these domains. To effect our purpose we have to determine the which passes through the destinations of the branch of the curve 4> (u, v) =

lie in

origin. first the case in which a is positive and /3 is acute. The function maxima at v — (2n + 1) ir — $ and minima at v = (2w + 1) tr + fi, — tt) is now each minimum being greater than the preceding; and since (a, — positive when v is is greater than positive, it follows that

Consider

(a, v)

has

Hence the curve cannot

cross the line a

=

a above the point at which

v = — 7T. and similarly it cannot cross the line u = — a below the point at which v — tr. The branch which goes downwards at the origin is therefore confined to the strip — a < u < a until it gets below the line v— — 2Kir + tt — #, where

K

is

the smallest integer for which 1

- a tanh a +

{(2K +

The curve cannot cross the line line u = a and goes off to infinity Hence,

if

a

is

positive

and

v

1 ) ir

— — (2/£'+l)

is

acute,

we

and

is

0.

and so

7r+j8,

it

crosses the

= — 2Kir.

get

*

we get

acute,

1

*

6

ttJo

By combining

>

,

7»m=0

w. + .« (3)

cot J3

in the direction of the line v

TTJo

while, if a is negative

- 0}

- (2m)l

J-*

*m-o

a. 1 '

these results with those obtained in § 8*61,

we

obtain the

asymptotic expansions for the domains 6 a and 7 a. the branch which goes below the = u o below (a, ir — /3) and it does not axis of u — ao along the line v = — (2L + 1) tt, go to must so it again, cross the w-axis for which integer where L is the smallest If,

however,

is

obtuse and a

is positive,

at the origin cannot cross the line

1

Hence,

if

a

- a tanh o -

is

positive

IT

Jo

while, if a is negative

(5)

if

and

{(2L is

+ 1) ir +

obtuse,

j3\

cot

/8 is

obtuse,

> 0.

we get

~ m=0 and

/8

z

we get

r'«--«^

S».

'


10*15]

By combining

319

we obtain the

these results with those obtained in § 8*61,

asymptotic expansions for the domains Since formula (1)

is

4, 5,

the only one which

66 and

is

76.

of practical importance,

we shall

not give the other expansions in greater detail.

An approximate

am when

formula for

m is large and y is zero, namely (->mr <*>

-~

was obtained by Cauchy, Comptet Rendu*, xxxviii. (II)

Next consider the

If The

integral

00

grt-tnnht fa

j

(1854), p. 1106.

— _If* g-«(-tcosh y +mnh*» fa I

only difference between this and the previous integral

the sign of cosh 7; and so, when 7 5 in Fig. 21 of § 861, we have

1 rV-iainht^^l X J*™)***' ir mZo 2am+1 Jo

(6) K '

is

in any of the regions

lies

the change in

numbered

1, 4,

'

ir

where a^'

is

derived from a„, by changing the sign of cosh

_,_

1

2

/

l-cosh7'

This expansion

fails to

pansion (1) failed

To

(1

be significant when 7

when 7 was

we

deal with this case v

after the

If

e,t-z»inht

+ cosh 7

- cosh 7)'

small, just as the previous ex-

iri.

r = t — sinht,

e" e~ tzt

ttJ =s

9

24 (1


dt= -l

7rJo

,_

so that

write

If -00

00

is

nearly equal to

= z{\ — e),

method of § 8*42.

i

^

-cosh 7)*'

7,

dt -T-

dr

__Lj~°V £

dr

6**+*

m =o

oir.'o

B m (-ez).(~.y*»-*dT

and hence l fv-,.,»b((ft

O)

A

TT'Q result equivalent to this

p. 313.

1 37T

1 m=

<->"

r (f'"+t> (m+l)

i;"-<">

(if)*

has been given by Airey, Proc. Royal Soc. xciv. A, (1918),

:


320

Hardy's generalisations of Airy's

10*2.

The

integral considered

by Hardy* If s

in the following

= sinh

,

[CHAP.

X

integral.

by Airy and Stokes manner:

(§ 6'3)

has been generalised

then

= 4s + 2 2sinh3£=8s + 6s 2cosh4<£ = 16s4 +16s + 2 5 1 2 sinh 5 = 32s + 40s + 10s, 2 cosh

2

2

3

2

.

3

and generally

the cosh or sinh being taken according as n

Now

is

even or odd.

write r»(«,

a)- *.,*•,(- i»,*-in;l-n;-4q/*),

so that

7 8 («,o) = e2 T

T

3

(t,a)

+ 2a

= t* + 3at

= e +4at + 2a T (<,a) = t +5at + 5a T

Z 4 («,a)

4

2

2

6

s

2

6

Then the

«

following three integrals are generalisations f of Airy's integral

(1)

Cin («)

= J°°cos Tn (t, a)dt,

(2)

flK„ (a)

-J" sin T, (*,«)<**»

Ein (a) =

(3)

J"

exp {-

Tn (t, a)} dt

may be shewn J that the first two integrals are convergent when a is But the third integral positive or negative) if n = 2, 3, 4 (whether real fairly obvious that Ein (a) is indeed and it is complex is when o converges It

;

an integral function of

a.

an even integer, the three functions are expressible in terms but when n is odd, the first only is so expressible, the function of H. F. Weber. the other two involving

When n

is

of Bessel functions

;

Before evaluating the integrals,

we observe

which reduce to Cin (a) and Sin (a) when a Gin (a)

+ iSin (a) =

is

that integral functions exist

real

;

for take the

[" exp {%Tn (t, Jo

combination

a)} dt.

* Quarterly Journal, xli. (1910), pp. 226—240. + The sine-integral in the case n=3 was examined by Stokes, Camb. Phil. Trans, pp. 168—182. [Math, and Phys. Papers, n. (1883), pp. 332—349.]

J Hardy,

loe. cit.,p.

228.

ix. (1856),

.


10*2, 10-21]

By

321

when taken round an

Jordan's lemma, the integral,

arc of a circle of

R

with centre at the origin (the arc being terminated radius with complex coordinates R, Re*11"), tends to zero as R-~cc

by the points

.

And

therefore /•ooexp(}irt/»)

Cin (a) + iSin (a) =

exp [iTn (t,

a)} dt

Jo

= e W»

Tn (r, «T*)) dr,

f °°exp {-

Jo

where r = te~^ln and the last integral is an integral function of o. The combination Cin (a)-iSin (a) may be treated in a similar manner, and the ;

result is then evident.

10*21.

The evaluation of Airy-Hardy integrals of even order.

suppose temporarily that a

in the integrals,

we

— 2a* sinh (ufn)

by

find that,

§

—

2a* f

Gin (a)

+ iSin (a) = n

621

exp (2a*n i cosh u) cosh (u/n) du

J o 1

H

****»

1/n

v

(2a* w)>

to say

Cin (a) + iSin (a) If

(10),

°°

= iria* wis

and then,

is positive, t

that

Ein {a) when n is even, we making the substitution

evaluate the three integrals Ctn (a), Sin (a),

To

we equate

«

real

nsi

^

and imaginary

^(.)- E

{e*«>»

/w)

J. Vn (2a*») - «rW^ JVn <«*•)}

parts,

3=S^ 5

we have

jlJ-*(W-)-^(«^}.

In a similar manner,

Ein (a) = so that,

by

§

—n

2et* f

00

exp(-2a*n cosh u) cosh

(u,'n)du,

J o

622 (5),

(3)

#iw (a) = (2a*/n) #„„ (2a*).

These results have been obtained on the hypothesis that a is positive; and the expressions on the right are the integral functions of a which reduce to Cin (a), Sin (a) and Ein (a) when a is real, whether positive or negative. Hence,

when a is negative the we have Gin (a) =

equations

(1), (2), (3)

2n sin flw/n) | j!o m! T (m + whether a be positive or negative.

1

are

still valid,

so that, for example,

- 1/n) " " Jo ™\ T (m + 1 + 1/»)J

'

"

.

;


322

Hence, replacing a by

— /9, we

see that,

when

ft is

[CHAP.

positive

and n

is

X

even,

then Ci <" »

<*>

» " 2n

&„ (-0) = 2);

(5)

Ein (-/3) =

(6)

in«U) [J~ " (2 l

8

J^

(

/ra)

^g^

^ + '» )

(2/Si0)1 '

J-lln (*»-) - J-1M (20*)),

(/-„. (2/3*) + /„. (2/?»«»

}

431

It follows from §

(9) that,

when n

is

even, the functions Ci* (a)

and

Sin (a) are annihilated by the operator

and that Ein (a)

is


In the case of the

first

two functions

it is difficult

to obtain this result*

directly from the definitions, because the integrals obtained

by differentiating

twice under the integral sign are not convergent.

10*22.

To

The evaluation of Airy-Hardy integrals of odd order.

evaluate Cin (a)

when n

is

odd,

we suppose temporarily

that o

is

positive, and then, by § 6*22 (13),

Cin (a)

—n

= 2a*

f

00

cos (20*" sinh u) cosh (u/n)

du

Jo

= 2a*cosOWn)

2ftin)

n

That

is

to say,

CTn(g)

(1)

= 2a*o8 (^/n) Jfi/<|(2at, ) n

7ra*

2nsin(^7r/n)

Using the device explained

«•<- »

<2 >

(2a*)-/1/n (2a*»)}.

{J_ 1/n

in § 10*21,

we

see that,

n& W (J-'»

It follows that the equation § 10*21 (4) is true

and, whether n be even or odd, Cin (a) is

^+ for all real values of * It

/9 is

positive,

+ *'» <**"»•

whether n be even or odd

annihilated

by the operator

^

(-)n »2 «n

a.

has been proved by Hardy,

integrals.

when

loe. eit., p.

229, with the aid of the theory of " generalised

323

ASSOCIATED FTTNCTIOHS

10-22]

Next we evaluate Ein (a) when a is positive making the usual substitution, by §1013 (4), ;

we

find that,

=

Ein (a)

-

Hence the any value (3)

—n

jo

—

{tan (*7r/n)

exp (- 2a*n sinh

J 1/n (2a*») - B 1/n (2a*»)}

which represents Ein (a) when n

series

du

is

odd and a may have

is

Ein (a)

-

n cos (i7r/r?)

J

7T

+ «sin;(ir/n) and hence

u) cosh (u/n)

it follows

g

[

|m =

r (m + 3 _ i/n) r (m + f + 4/f|)

o

(-)^ttmw

w!

T(m + 1 - 1/n)

(-)m amM

g

-2

na on

jj^ +

|

—

2a* f"

+ % Sin (a) = n =

.'

^ +

is

temporarily assumed to be

exp (2a*" * sinh w) cosh (u/n) du

o

(tan

Qwln) J 1/n (- 2a*»t) - B 1/n (- 2**i)}

.** ,

n sin (7r/w), { 7ra*<

l

J-un (-

n+1 U"

2a*»t)

- J-1/n (- 2a*»i)} amn

"

n cos (£t>>) m=0 T (m 7ra* + n sin .,(-Tr/n) /x («M* 7_

+ f - */n) r (m + f + £/n)

1/n

(2o*«)

- «-

W

/1/B (2a*»)},

and therefore Sin (a) -

-

7ro* (w+1)

"

wcos(Ww)

<*

J

o

mn

r(m + t-i/n)r(TO + f + i/n)

2wcos($7r/w)

whence (6)

it

follows that,

Sin (- 0)

-

'

Ein («) = wofK""3'.

Next consider Cin (a) + iSin (a), where a positive. From § 10"13 (4) we deduce that

(5)

)

M=0 *»! T(m + 1 + l/n)J

that

(4)

ta'n (a)

a

when # > 0,

—^ ^^

J

o

r (m + f _ i/n) r (^ + | + */»)

W 2 + 2/iCOs(j7r/») ^fl /.x (^/» ( 2# ) " ^/» ( ^ 9

W )}>

.


324

and hence,

X

for all real values of a,

(7)

|

-1 - n

2

an

~2

Si n (a)

j

= - not

<

This equation was given by Stokes in the case n = It should

be noticed that

Sin (a)

(8)

[CHAP.

+ (-)*<«+» Ei n (a) =

^^

{sin

n -3

>

3.

(**/»)

+ (- 1 )*<+»}

x{/_1/n (2a*«) + /1/n (2a*»)}

« —Z^-j-^ {sin (*,r/n) + (n sin (717/1)

1 )»«•+»>}J

x{/_ 1/n (2£*»)-J1/n (2/3*«)}, where

= — a,

ft

and a and

The formulae his

in

are real.

/9

of the preceding three sections are due to Hardy, though

methods of obtaining them were the special case n = 3.

10*3.

and he gave some of them only

different

Cauchy's numbers.

In connexion with a generalisation of Bessel's integral which was defined (see § 10*31), it is convenient to investigate a class of functions known as Cauchy's numbers.

by Bourget, and subsequently studied by Giuliani

The

typical

number, N-n,k,m,

the term independent of

t

is

defined by

1\* 1\*/

/


Cauchy*

IV

It is supposed that n, k,

t.

as the coefficient of

in the expansion of

and

m are integers of which

the last two are not negative. It follows from Cauchy's theorem that 1

(i)

/'

(0+ »

JU*--5RJ

1\*

/

<~'H)

2tt

,

e~ nie cos* 6 sin™ -v

\*\ \e- ni0 Jo

7T

2»»+*

cos (^rriTT 7T

6d6

+ (-)m eni0

}

cos*

sinw

6d6

- n6) cos* 6 sinTO 0d6.

I

N-n^m

is

zero

if

—n+k+m

is

odd or

a negative integer.

* Comptes Rendu*, xi. (1840), pp. pp.

dt

Jo

It is evident from the definition that if it is

l\ m

/

(«")

682—687, 850—854.

473—475, 510—511; xn.

(1841), pp.

92—93; xm.

(1841),

+

325


10-3]

From

(1) it is seen that

#-*,*,»

(2)

= (~)m

Fn,k.m

= (-)"-* &n.k,m-

These results, together with recurrence formulae from which successive numbers may be calculated, were given by Bourget*.

The recurrence formulae

are

+ " -n-i, *-i, m N-a t,m = N-n+i,k,m-i ~ "—n—i,*,i»-i>

-^-n, k.m

(3)

(4)

— ^-n+i,

Jfc-i,

to

>

t

and they are immediate consequences of the identities

+ 1/0* (t - l/0m - *- (* + 1/0*" (t - 1/0™ + 1-»~* (t + 1/0*- (* - l/0~ " m_1 - t-n ~ 1/0* (* - I/O" (* r- (* + 1/0* (* - i/0m - p- (* + 1/0* (* - V0 1

1

t~»(t

1

l

By means

N-n

terms of numbers of the types

A

number

of these formulae any Cauchy's

ultimately expressible in

is

r t

due

to Bourget,

owes

existence to the equation

It follows that

by a

On

partial integration.

(5)

(m +

1)

K)

rn

^.*--2^TTi)/ +

l)J

d*l

«/

V

dt

dA'-i) */

V

J

performing the differentiation we see that

N-n k,m = nN-n> *_!, m+ - (k i

1 ) iV_„, *_2, m+2 ,

t

and similarly (6)

(k

+ 1) iV_n>i m = nN^ k+h m-i - (m - 1) iV_„ *+2 m _2 p

,

Developments due to Chessin, ^nnafa o/ Math.

x.

(1895—6),

pp.

.

,

1—2, are

8

2 Cr m= r=Q

2r- n+s -.2r,

k-$,

(7)

-^-n,k,

(8)

#-»,*,m = 2 (-y,Cr .N-» + ,-tr.

t

These may be deduced by induction from

^-«.

(9) s

where p=\ (k+m—n).

.

m>

*.

m-,-

r=0

Another formula due to Chessin

This

is

(3)

and

(4).

is

*.

-

t,
different class of recurrence formulae, also

2w»(ro

1

2 m" r=0 (

r )

k

Cp - r m Cr .

,

proved by selecting the coefficient of «n in the product

(t+l/0*x(«-l/«r• Journal de Math. (2) vi. (1861), pp.

33—54.

its


326

X

The functions of Bourget and Giuliani.

10*31.

Jn
The function J,.

where w

[CHAP.

is

W = ^ P *~

an integer, and k

is

by the generalisation

defined

{*

+

))'

of Bessel's integral

«P }'*« (' - j)} *

a positive integer.

is

It follows that •A..

* (*)

=

1

f*

2,r

ex P {~

I

nB ~ z sin

* (

0)\

•

(2

cos

W de

>

and therefore (2)

•/«.*(*)

= - r(2cos0)*cos(n0-ssm0)d0.

T Jo

The function «/„,*(*) has been studied by Bourget, Journal de Math. (2) vi. (1861), pp. 42 —55, for the sake of various astronomical applications ; while Giuliani, Oiornate di Mat. xxvi. (1888), pp. 151 171, has constructed a linear differential equation of the fourth

—

order satisfied by the function.

[Note. An earlier paper by Giuliani, Giornale di Mat. xxv. (1887), pp. 198—202, contains properties of another generalisation of Bessel's integral, namely 1

-

i'

v

cos (nd

/

- z*> sin p 6) dB,

but parts of the analysis in this paper seem to be incorrect.]

If

we expand the integrand

<8>

of (1) in powers of z,

•/«,*(*>

= I »»=o

and

it is

Jn

Again from

§ 10*3 (2)

;

and

(z) ,

= Jn (z).

(3) it is evident that

J-nA*) = i-)n- k Jn,k{z\

(5) (6)

if

§ 10*3 that

evident from (1) that

(4)

and,

i\U,*, m ^Jf ml

we deduce from

Jn,k (*)

we take k —

= •/«-!, *-i (z) + Jn +i,k-i (Z)

',

1 in this formula,

Jn ^z) = ^Jn (z).

(7)

These results were obtained by Bourget; and the reader should have no difficulty in

proving that 2J'n
(8)

= Jn .hk (z) - Jn+1 ,t(z).

Other recurrence formulae (due to Bourget and Giuliani respectively) are (9)

(10)

-A*+2 (*) =

~ A*+i

(*)

-

* {•/»-!, *

(?)

- «/»+,,*(*)},

4/\*_ CO = /«,*(*) " *Jn,k-, {Z\


10-31]

The

most simply constructed by the method used

differential equation is

by Giuliani

;

327

thus

V n Jn *(*)=-

f"

ir

.'o

=

4 {- (n + z cos 0) sin (nd - z sin

du n

2k

C

7T

Jo

(n

0)} (2 cos

0f d$

+ z cos 0) sin (n0 — z sin 0) (2 cos 0)* -1 sin 0d0

= - 2kz J' nk (z) + —[\ffi cos (n0 ~ z sin *)} = - 2kzJ'n k (z) - — 7T

'

cos (n0 ./

- * sin 0)

o

^

(2

cos 0)*"1 sin

{(2 cos

0?-* sin

ed6

0} d0,

and so

V„ Jn.* (*) = - 2kzJ'n
and hence we have (11)

It

z*

was

J\

t (z)

1

,

and using

"*

W + 2kzJ ^ + ^Jn n k '

-

k

(10), it follows that

W -*(*-!)

CO.

Giuliani's equation

+ (2k + 5) zJ'\ k (z) + {2z* + (k+ 2) - n8 J\ k (z) + (2fc + 5) zJ' n>k (z) + (z* + k + 2 - n2 ) Jnk (z) = 0. a

}

also observed

«*"»»• (2 cos

(12)

+ '

G& + *) {Vn Jw .

d* -r-9

by Giuliani that

0)*-

5

€*n J*n,k(z)
n=0 ao

+i t

€M+1

»=o this is verified

by applying Fourier's rule

(cf.

Jm+hk (z)sm(2n + l)0;

§ 2*2) to the function

on the

right.

A somewhat similar (1883), col. 1

—

8.

J(z;

(13)

function J{z\

This function

is

whence

j<„

(16)

it

k) has

been studied by Bruns, A»tr. Nach. civ.

by the

series

»»*)-J^ w!r (, + 2 * +m )( v + 2*-2)(» + 2A)(»+2*+2» + 2)'

The most important property (i4)

v,

defined

,,

of this function is that

q-Jto

..

*+V= {y +£_%\*+l + 2y

follows that

J(z; v,*)=

2

2* «/* + »»(*)

=t (i/ + 2m-2)(i»+2m+2)*


328 10*4.

Now

The that

[CHAP.

X

of Struve's function H„(s).

definition

we have completely examined

resembling Bessel's integral,

it is

the functions defined by integrals

natural to investigate a function defined by

an integral resembling Poisson's integral. This function is called Struve's function, although Struve investigated* only the special functions of this type of orders zero and unity. The properties of the general function have been examined at some length by Siemonf and by J. Walker*.

H„(4

Struve's function

H

(1 >

-

<*>

provided that R(v)

By

of order

v, is

-

r£$m

"

r^hfel) .C


^

-

(1

f.

" *

8in

•

sin ( " cos $) sin*

m

> - |.

analysis similar to that of § 3*3,

we have

r (v + 1) r (f) ,„-o (2m + 1)! Jo r (-)m zim+1 .ml _(i*) | r(|),»-o(2w + l)!r(i/ + m + f)'

so that

H"^ )= J

(2)

The function H„ (z)

R (v) exceeds — $

is

or not.

o

r(m + f)r(, + m + f)-

defined by this equation for It is evident that

all

H„ (z) is an

values of

and, if the factor (\z) v be suppressed, the resulting expression tegral function of It

is

whether

is

also

an

in-

z.

easy to see

[cf. §§

211

(5),

3121

(1)] that

H'<*>=r<#r£W)< 1+

< 3)

v,

integral function of v

«>'

where

w and

(3)

j

i»i*#« P AIit -i}, {T

v

+f

|

is

the smallest of the numbers

!

|

v

+f

I,

*> I

+f

|>

^ I

+$

l»

••••

* Mim. de VAcad. Imp. des Sci. de St P4tersbourg, (7) xxx. (1882), no. 8 ; Ann. der Physik, xvn. (1882), pp. 1008—1016. See also Lommel, Archiv der Math, und Phys. xxxvi. (1861),

p. 399.

t Programm, Luisenschule, Berlin, 1890.

[Jahrbuch uber die Fortschritte der Math. 1890,

pp. 340—342.]

—

J The Analytical Theory of Light (Cambridge, 1904), pp. 392 395. The results contained in and (11), are there given.

this section, with the exception of (3), (4), (10)

'

))

'

329


10»4]

We can

obtain recurrence formulae thus

W = Jo5

d

H dz ^

"

d

_

<£

H

t*

" (

(-)m (2m

_ S =

* )}

„,

.o

2" +

= ,„r_i

= comparing these

+ 2to + 1) z2v+im r (to + 1) T (i/ + m + 1)

(-)TO (2u 2" +2m+1

and similarly

On

+

:

+ l)g2w

- +1 r (m + f

sF*"* 5 r (to +

r („

)

r ( v + to + 4

1) ,'

we

H +l( * ''

2*r(» + f)r(iT*~

results,

+*

to

)-

find that

- H, (5) +

(5)

HU

(*)

+ H„ +1 («) =

(6)

H^

(,)

- H„ +1 (,) = 2H/ (z) -

z

(7)

(^+ v )IH v (z) = zIl^

(8)

(* - V ) h„ (*)

1

Y^\Wl\)

'

p (7^|yf(J)

(z),

* H " +1 ( * } = r( y +t)r(f ) " (

*

In particular we have

^ {^

(9)

(,)}

Again, from (7) and

(*

so that

H„ (z)

2

p. 218.

(11)

Ii„

(a)

= | - H, (*).

{*!!_,

^

(z)}


W - r(

.

(t)

which bears the same relation to Struve's function as /„

studied (in the case

This function

(z)}

we have

(8),

V.H.

The function

± {H„

2

satisfies

Jv {z) has been

(z),

- v ) H„ (*) = (* - v)

(10)

to

= *H

is

v=0) by* Nicholson, Quarterly Journal,

(z)

bears

xlii. (1911),

denned by the equation

L„ («)-

i

r(. + | )P(r+(l+t)

= _2(|^__ f^sinh^cos^sin 2 "^^, the integral formula being valid only


when R

(p)

> —J.

difficulty in obtaining the

fundamental properties of this

function. * See also Gubler,

Zurich Vierteljahrsschrift, xlvii. (1902),

p. 424.

g

(

"

1


330

The loop-integral for

10*41. It

was noticed in

R(v)^ — \, because

[CHAP.

H„ (z).

10 4 that the integral definition of -

§

X

H„ (z)

fails

when

We

the integral does not converge at the upper limit.

can avoid this disability by considering a loop-integral in place of the definite integral.

Let us take (V

-

iy-i sin

zt

.

dt,

Jo t* — 1 vanishes at the point on the right of t — 1 at which the contour crosses the real axis, and the contour does not enclose the point

where the phase of *

= -l. If

ment

we suppose

that

R (v) > — \

,

we may deform the contour we find that

into the seg-

(0, 1) of the real axis, taken twice, and

r(i+) 2

(t

-

1)"-* sin zt

.

dt

ri

— 2i cos wr

Jo

2 )"-*

sin zt

.

dt,

— t* is zero. when R (v) > — \, we have = r <* 1

h, ( .)

(1)

-

Jo

where the phase of Hence,

(1

I

in

yr

1

(.£ )

Jo

v - D-* *. « ^ .

Both sides of this equation are analytic functions of v for all* values of v so, by the general theory of analytic continuation, equation (1) holds for ;

and all

values of

From (2)

v.

this result,

j

v

(z)

To transform

combined with

+ «„ <*) =

§ 6*1 (6),

we deduce

that

(1+)

r

^rVi)*^ /

ei

*

(

'°

lr

*

dt

a> be any acute angle (positive or negative), between — \ir + a> and \ir + o>. We then deform the contour into that shewn in Fig. 26, in which the four parallel lines make an angle — a> with the imaginary axis. It is evident that, as the lines

and

let

this result, let

the phase of z

lie

move off to infinity, the integrals along them tend to The integral along the path which starts from and returns to 1 + oo ie~ iu is equal to ff„w ( z ) and on the lines through the origin we write t = iu, so that on them parallel to the real axis zero.

;

(P

_ i)*-4 = e =F (r-« « (1 + u*y-l.

It follows that

Jv (z) + ill. (z) *

The

isolated values J, f

undetermined form.

r

,

«

|,

(z)

...

+r (y + 1) r

e— (1 +^)"-*^, |} j o

are excepted, because the expression on the right

is

then an


10-41]

where the phase of

1

+

2 it

has

H.(«)-r. W+r(>+<«

(3)

This

result,

z for which

-

which

ir

<

is

arg z

ptotic expansion of

and hence

principal value;

its

> )

331

erîl r(i) /t

+ u*y-*du.

v, and for any value of be applied immediately to obtain the asym-

true for unrestricted values of

< ir,

will

H„ (z) when

|

z

|

is large.

Fig. 26.

A

was obtained by J. Walker*, who assumed that R(z)->0, so that a> might be taken to be zero. In the case = z/ 0, the result had previously been obtained by Rayleighf with the aid of the method of Lipschitz (§ 7 "21). result equivalent to (2)

R(v)> — \,

If,

we

as in §6*12,

replace

by args — /S,

a>

it is

evident that (3)

may be

written in the form

H.(«).y.(, )+

(4)

where

— \tt <

/8

<

\ir

<*) r(|

and —\ir +

j;

e-«(i

$< arg z < \ir +

This equation gives a representation of

+7.)

(3.

H„ (z) when

j

arg z\
a representation valid near the negative half of the real axis, for unrestricted values of

(5)

and use

we

define

H„ (z)

arg z by the equation

H„ (zem' ) = «*("+«« H„ (z), {

(4) with z replaced

by ze*™.

* The Analytical Theory of Light (Cambridge, 1904), pp. 394—395. t Proc. London Math. Soc. xix. (1889), pp. 504—507. [Scientific Papers,

m.

(1902), pp.

44—46.]


332 we write z=ix

If

where x

in (3),

is positive,

we

we deduce

and, by considering imaginary parts,

[CHAP.

X

when R(v)<%>

see that,

that °°

L„ (*) = /_ „ (*) -

(6)

r(y

+yr

/

( |)

siu (*»* )

•

^+

a result given by Nicholson, Quarterly Journal, xlii. (1911),

tt2 )"" i

<**'

p. 219, in

the special case in

which i>=0. 10'42.

We lating

The asymptotic expansion of

H„ (z) when

\z\is large.

now obtain an asymptotic expansion which may be used for tabuStruve's function when the argument z is large, the order v being fixed. shall

Since the corresponding asymptotic expansion of

Y

v

(z)

has been completely

investigated in Chapter vn, it follows from § 10*41 (4) that determine the asymptotic expansion of /•ooexpi/3

As

in | 7*2,

/

take

for

which

p

t***\»-*

so large that

V

22 (v — p — £) < 0, and

'

|args-/3|^7r-o\

|/3|^tt-S,

— 7T +

W

V

take $ to be any positive angle

so that z is confined to the sector of the plane for

We

28 ^ arg z ^

ir

which

— 28.

then have (

1

±

——J

^sinS,

arg

1

1

±

—

J --

<

tt,

so that

^V" " 1

1

I

say,

where

where

+

(l

£

\

j

^

gftrl/WI Sin ^)2iJW-2p-l (

= Ap>

/ j

A p is independent of

It follows

to

we have

(p-.l)!*w

We

it is sufficient

z.

on integration that

j

B.

I

$

=

=f

v* l

(z-w).

M

;*

e— «*dtt

+


10-42, 10-43]

We

deduce

when arg z <

that,

R(p — v +^) > 0;

provided that

and z |

is large, j

——^ ^

H„ (,) - Yv (z) +

(1)

tr

\

j

333

^-^

+

{z

but, as in § 72, this last restriction

^ may

be

removed.

may

This asymptotic expansion

b.

( 2)

w - y.

also be written in the form

r

(.)

1

£ n + ^S*r— +

°

(z

„

"^-

It may be proved without difficulty that, if v is real and z is positive, the remainder after p terms in the asymptotic expansion is of the same sign term neglected, provided that as, and numerically less than the first ~& — established This be by the method used in § 7 32. 0. may R (p + \ v)

The asymptotic expansion* was given by Rayleigh, Proc. London Math. Soc. xix. (1888), 504 in the case v — Q, by Struve, Mem. de I' Acad. Imp. des Sci. de St Pe'tersbourg, (7) xxx. (1882), no. 8, p. 101, and Ann. der Phys. und C/iemie, (3) xvn. (1882), p. 1012 in the case v = l; the result for general values of v was given by J. Walker, The Analytical Theory of Light (Cambridge, 1904), pp. 394—395.

p.

If v has any of the values \,\,

terminating series and

when

Y

v

...,

then

(1

v is half of

i

2

(z) is also expressible in

an odd positive integer, of elementary functions. In particular

that,

+ u /z ) v ~l

is

expressible as a

a finite form.

H„ (z)

is

It follows

expressible in terms

Hj(*)=(^)*(l-cos*), (3)

§

.™-(s) 10*43.

We

K)-®V*~0-

The asymptotic expansion of Struve s functions of large order.

now

shall

obtain asymptotic expansions, of a type similar to the

expansions investigated in Chapter

H„ (z) As

when usual,

|

v

and z are both j

|

we

|

vm, which

represent Struve's function

large.

shall write

v

and, for simplicity,

we

= z cosh (a + lyS) = z cosh y

shall confine the investigation to the special case in

which cosh y is real and positive. The more general case in which cosh y is complex may be investigated by the methods used in § 86 and § 1015, but it is of no great practical importance and it involves some rather intricate analysis. *

For an asymptotic expansion

of the associated integral

/;s(-r:r«». see Rayleigh, Phil.

Mag.

(6)

vm.

(1904), pp. 481

—

487. [Scientific Papers, v. (1912), pp.

206—211.]

.


334

The method of steepest descents has type,

and

f

(3),

we

one of Bessel's type.


dw

„v

/-•

X

to be applied to an integral of Poisson's

not, as in the previous investigations, to

In view of the formula of § 10*41

[CHAP.

which we write in the form

where t =

w — cosh 7

.

log (1

w

-f

2 ).

qua function of w, has stationary points where w — e iy equal either to o or to ifi, two cases have to be considered,

It is evident that t,

so that, since

which give

7

is

,

the stationary points

rise to

(I) e ±a ,

Accordingly (II) in (I)

we

(II)

e±*

consider separately the cases (I) in which zjv

which z\v

When 7

is

is

greater than

is less

than

1,

and

1.

a real positive number a, t is real when w is real, and, as w -2a to e~a — cosh a log (I + e t first increases from ),

increases from

to 00

then decreases to

e*

.

,

— cosh

+ e**) and

a log (1 .

finally increases to

+

00

In order to obtain a contour along which t continually increases, we suppose that w first moves along the real axis from the origin to the point e~°, and then starts moving along a certain curve, which leaves the real axis at right on which t is positive and increasing.

angles,

To

find the ultimate destination of this curve, it is convenient to

make a

change of variables by writing %— £

w — sinh f

;

where

£,

i\

and £ are

+

cosh f sin it

t\

is real,

has for

,

its

— 2 cosh a arc tan (tanh f tan

has a double point* at |

We now

= sinh f

real.

The curve in the £-plane, on which t

and

e~ a

it],

equation

17),

-

write v£>

V) -

2 arc tan (tanh g tan cos yi £

and examine the values of F(g, 17) as % 0, A, B, C have complex coordinates arc sinh

0,

As tg°es from increases from

to

A, F(^,

1,

traces out the rectangle

arc sinh 1

i/)is

tj)

sm ^

+

\tri,

\tri.

equal to 2 sinh £/cosh 2

£,

to 1. * Except

when a=0,

in which case

it

has a

whose corners

triple point.

and

this steadily

—

.


10-43]

When

f

is

on AB, F(g,

V2 and

~

rj) is

.

equal to

arc tan

—

(

cosec

.

J

this steadily increases from 1 to trj^l as

To

Note.

establish this result, write tan

dt

because

—

-=

-5-

{—r

aroten

7

335

=

i>

t

*J2

=
— arc tan t, which vanishes with

t,

nj

17,

increases from

to

%-rr.

and observe that

li+^

arcten

7 ^°'

has the positive derivate

—~—

^

.

When f is on BC, F(g, ij) is equal to ir sech £, and this increases steadily from tt/V2 to 7T as f goes from £ to C; and finally when £ is on CO, F(^, rj) is zero.

Hence the the rectangle

curve, on which F(^,

OABG,

rj) is

equal to sech

a,

cannot emerge from

except at the double point on the side

part of the curve inside the rectangle

must pass from

OA

;

and so the

this double point to the

singular point G.

The contours in the w-plane for which a has the values broken and continuous curves respectively.

0,

\ are shewn in

Fig. 27

by

Fig. 27.

Consequently a contour in the w-plane, on which t

is real,

consists of the

part of the real axis joining the origin to e~* and a curve from this point to

the singular point to

+

i;

and, as

w

traces out this contour, t increases from

00

It follows that, if the expansion of d£/dr in

d

l-

£

h

Tm

powers of t

is


336

[CHAP.

X

then

* m!&„ «^»

—

2.

1=0 *

and hence, by 10 4 #

(1),

we have

h.(.)~-
(1)

T7

^^ 2=&. >

It is easy to prove that

6,-1,

=2coshy,

When 7

(II)

from

61

to 00 as

w

is

&2

= 6cosh2 y-£,

a pure imaginary (=

i0),

= 20 cosh8 7 - 4 cosh 7 r

is real

travels along the real axis from

dw I *~ <* + "^ " j. Hence, from § 1041 (8)

it follows

H,(,)^Fy (,) +

(2)

6,

«~"

and increases steadily and so

to 00

;

{TOT*) 3FJ

dT

-

that

r(y + i)r(i)

jS o

^

r>

provided that arg z\<\ir. This result can be extended to a somewhat wider domain of values of arg z, after the manner of § 8*42. j

From the corresponding results in the theory of Bessel functions, it is to be expected that these results are valid for suitable domains of complex values of the arguments. In particular, we can prove that, in the case of functions of purely imaginary argument, 1* v

(3)

when

v

|

is large,

|

arg v

I

\

(vx)~Iv (vx)

< \ir, x is fixed, and the error is of the order of magnitude of

i[ -±4±^w w(i + i

(

*°>}j

times the expression on the right. a [Note. If in (I) we had taken the contour from w=0 to w=*e~ and thence to w= - i, ufv {£). This indicates (z) in place of containing v we should have obtained the formula that we get a case of Stokes' phenomenon as y crosses the line /3=0.]

U

10*44.

When

The relation between the order n

is

H„ (z)

and

B B (z).

a positive integer (or zero),

we can deduce from when n

(z) by a polynomial in z; and n (z) differs from § 10-1 (4) that in 1/z. is a negative integer, the two functions differ by a polynomial

E

H„


10-44, 10*45]

when n

For,

is

a positive integer or

zero,

337

we have

n +m

p— (lz) = v = _«r(|m+l)r(£m + w + l)' \mtri

,

and

^w

BW and

therefore, since

M=0 r(|w+l)r(£»n + n+l)' Jn (z) — Jn (z), we have gi(m—l)tri/l^\n—m

n

m=\ " (1 - Jm) T (n + 1 - |/n) that

say

is to

^= 1 e E-^>

^ î

n (i)\ In

—

manner, when

like

n

TAe

10*45.

si^ra

Ic*'""

(

w

1

H

±ili (z) r(n + i-t»)

,,i

a negative integer, 1 '

r (n -

m - j)

.

(js)-"*-^

H"w( ^

r(m + |)

of Struve s function.

now prove the

shall

1>r

is

n+1

—^

/., _ (~) B -«(*)

(2)

We

'

interesting result that H„(a;)

is

positive

when

a?

is

and v has any positive value greater than or equal to \. This result, which was pointed out by Struve* in the case v = 1, is derivable from a definite integral (which will be established in § 1347) which is of considerable importance in the Theory of Diffraction.

positive

To

obtain the result by an elementary method,

parts and then

«•

we

1

(h®)"' "-W- rfr'tW /

^

-

we

see that, for values of v exceeding [*"

integrate § 10*4(1) by

J,

..,-,)jg—i«*~edB

d cos (x cos 6)

.I.

(

l> tf)r(i) {[eos(xcos9)sin-«J - (2i/ -

1)

|

'cos (x cos 0) sin 2'- 2

'*?

cos

0dd

Jo

§*Y ^"p V

= ( i*?

1 (i>

.

1 1

- (2v - 1)

1(

iîV ./„ + £)r(£)

f'sin2

*- 2 -'?

f

cos

^cos (* cos 0) sin2"- 3 cos 6dd\ II

- cos (x cos

-9)1

^0, since the integrand is positive. *

Mem.

de VAcad. Imp. des Sci. de St P€tersbour
proof given here

8,

pp. 100—101.

The

,


338

When

v is less

than

J,

[CHAP.

the partial integration cannot be performed

;

when

and,

i/

X

= J, we

have

Hj(*)-(^.) 4 (1 -cob *)>(>, and the theorem

is

completely established.

A comparison of the asymptotic expansion which was proved in § 10*42 with that of Yv (x) given in § 7*21 shews that, when x is sufficiently large and positive, H„ (x) is positive

H

>

K (x) is not one-signed when v \ and that asymptotic expansion of H„ (x) is if v

(ir)

F -»

r(,+|)rd)

or

/2\*

U>

asi*>£ori»<£. The theorem

according

that Struve's function

is

.

<£

dominant term of the

for the

.

.

sin

;

,

.

^-i—i-)

of this section proves the more extended result

positive for all positive values of

x when

v

>\

and not merely

for sufficiently large values.

The theorem functions

values of x,

10*46.

indicates an essential difference between Struve's function

for the asymptotic expansions of

;

Jv (x) and Yv (x) are

Chapter

vn shew

and Bessel

that, for sufficiently large

not of constant sign.

Theisinger's integral.

we take the equation

If

*j

{/o(«)-l»(*»-"f

j^r'—dt

and choose the contour to be the imaginary «» +ttan |0, we find that

j {h (x) - Lo (*)} = f

*

axis,

cos (* cot

indented at the origin*, and then write

)

^^

log tan (i*r+|tf>)

and so 7 (*)-I*(jf)

(1)

=^ J

*cos(#tan<}i)logcot(^) (

^

, >

a formula given by Theisinger, Monatshefte fiir Math, und Phys. xxiv. (1913), If

we

replace

x by *' sin 6,

multiply by sin

6,

and

integrate,

we

find,

p. 341.

on changing the

order of the integrations in the absolutely convergent integral on the right, iW

t

E, (* tan

)

log cot (£<£) ^$-r

=

*'

| f

{/„ (x sin 6)

-L

(x sin 6)} sin

6d6

so that 1

(2)

X

/o*'E (^tan0)logcot(^)^ = |. -^1

on expanding the integrand on the right in powers of x.

,

This curious result

is also

due to

Theisinger. *

The presence

indentation.

of the logarithmic factor ensures the convergence of the integral round the

— ASSOCIATED FUNCTIONS

10-46, 10-5] 10'5.

The

:

339

Whittaker's integral.

integral a*

e

I

izt

P

v _i(t)dt,

is a solution of Bessel's equation only when 2v been studied by Whittaker*.

which

from

It follows

V, L*

(1)

f

*

§

61 7

<$*

an odd integer, has

is

that, for all values of v,

P„_x

(t) dt\

=2

=

0*e*« (1

lira

—cos

* vir .z*e

- P) P;_j(0]

-& .

7T

If

we expand the integrand (multiplied by

integrate term-by-term +

Ji

.,_

(2«r)m .ro!

2

„, e^P^CO^-aê-" S fn— rr (r»/ _l * ^ * + f + y)v i-i M = r(m+|-p)r(w

The formula

and then

,

v

.

,

we

of § 3'32 suggests that

(2), or

by using recurrence formulae

(3)

W_

1

(z)

(4)

W_

a

(z)

v

+

W ^{z)

-

W

r+1 (z)

W

2v

v

z

= 2W„

,

(»—)w.w

(6)

— iw

asymptotic expansion of

for

Legendre functions 2ii-"zê- iz

(Z)

'

(z)

p + ,) w, (,) = ,w... (.) +

5)

write

easy to verify the following recurrence formulae, either by using

it is

the series

An

powers of s and

e") in ascending

found that

2*1

(2)

(

it is

-

j(2Tr)r(§-v)T(%+v)\' 2ii- v z~le- iz

^2ir)r^-v)r^ + v)

^

2i*~ y z*e~ iz ),

r(t . t)r(i + ,) 2i*~ y zi e~ u

H ,(.) + V(a|r) r(t _ >)r(t+y)

London Math.

.

_

W„ (z)

for large values of z |

|

may be

deforming the path of integration after the manner of Lipschitz * Proc.

'

Soe. xxxv. (1903), pp.

obtained by

(§ 7 21).

198—206.

t By a use of Legendre's equation the recurrence formula

may *"

(4

be

and we

verified;

- *. 4 + "

5

1

;

4

- 4')

can prove

that

in ascending powers of 1

/

-

P^ (t)dt= p 1,

+

p

,

_

.

by expanding

and integrating term-by-term.

+

S


340

The

function

is

it is

known

iV^(*)-t*i(*

e"

M p-<-'>*-^-/

«-

-

that*, near

-** +

_ fcos*7r\ P ^<-*>— ^-^"j

<=

».;

M *-.«>*

1,

i;

i-10. • r(m-y + £)r(m+i/ + £ )

8

*\

i

X

thus seen to be equal to

— -355-/, Now

[CHAP.

.*•

/l

- «\m

VY)

(ml?

\T~) ~ 2^ (m + 2 ) + ^ (w ~ v + £) + ^ (™ + * + $)}

x log |

>

and since

we

obtain the asymptotic expansion

W,(#)~ *#„«(*)

(7)

g-i <*+**»+*»)

COS V7T T •

(if,

m)

,

,

.

,

+ ^(w + ^ + y) — ^r(m+

x

l)~log2^-^7rt'}

.

Some noticed

functions which satisfy equations of the same general type as (1) have been by Nagaoka, Journal of the CoU. o/Sci. Imp. Univ. Japan, iv. (1891), p. 310.

The functions composing

10*6.

The reader integral order,

v

/

will

remember that the Bessel function of the second

may be

%

\

1

The

series

which has (1)

„?.

kind, of

written in the form (§ 3*52)

(n-m- 1)1 "*

»»=0

+

Tn (z).

WM M ,

,

j

^(^)r

<

2

10

« <*'> ~ *
x

>

- * <• + m

«)•

on the right may be expressed as the sum of four functions, each of fairly

simple recurrence properties, thus

nrYn (z) = 2

{log

n (z)

+ Tn (z) -2Un (z),

* Cf. Barnes, Quarterly Journal, xxxix. (1908), p. 111.


10*6]

341

where "2

Tn (z) = -

(2)

^-'--f^1 (^)-

n+2wl

w =o ra!(n + ro)! and

(cf. §

3*582)

g'<')-.!, ( ;i(,7m)

<3>

The

147,

>-»< 1 "-

Un (z) have been studied by Schlafli, Math. Ann. m. (1871), though he used the slightly different notation indicated by the equations

Tn (z) = 2£fn (z), Un (z)= -En (z);

Sn (z)=-2Gn (z), more recent

The

investigations are

function

Tn (z)

due to Otti* and to Graf and Gublert.

most simply represented by the

is

Tn (z) = -

(4)

To

1

functions 7^(2) and

—

pp. 142

<»<» + '» + l

f'(J 7T

-

-

0) sin (z sin

definite integral

nd) dd.

establish this result, observe that

(-ra*r*~ 1 r ( ,\ - [l I jfnW ~La€ m >riw _ i r(m+i+6)r(n+m+i-6)J e „o _j_ra | (-)w (|^)w+wt /•«•+> (i + ty***

i

,

-&r* [&«>-*»-*

e | __2 f' ~iriJom>-in-i

where

t

+

2w)!

nie

(- 1> sin (n

has been replaced by e *""r,i (

m+1+e

J_,

(-)w (jg)w+2OT

-

i_

(n

/•

(o+)

w+2w»

fl)

J,.o

*

(i

.

w+2w log t tj)

(0

+ 2m)\

- £tt)

,

^

.

It follows that

Now 3

(—

_ ttt . t m>—in~i

ta"

sin 0)n +*»l

(n

+ 2m)\

_

= and

(— 1> sin 0) (- iz sin 0) sinfl -« + (_)n eiz sin |{ e (cosh

(n even)

(sinh

(n odd) «}

(

so

-L P(0 _ 1^) yn (s) = 7rt

f

e

»»-»8in#

+ em(e-r)+izsin«} <£&

Jo

* Bern Mittheilungen, 1898, pp. 1—56. t Einleitung in die Theori-e der Bessel'tchen Funktionen, iv (Bern, treatise, pp.

77

—87, should also be consulted.

1900), pp.

42—69. Lommel'a


342 If

replaced by

is

ir

—

which

(4),

obtained at once.

The corresponding

integral for

Un (z)

is

obtained by observing that

(-rq*)w+2w r (i + 6) Ide „Zo m T (n + »« + 1 + e) J'.-

un ^ W = -fi -and

so,

(5)

from

§ 6*2 (4),

[I

?

i

!

{<**)-

r (i +

«)

Jn+t (*)} j^,

we deduce that

Un (z) = {\og$z)-,fr(l)}Jn (z) + - r$ sin (n0 - z sin 7T

From

Tn_

1

§ 10*6 (4) (z)

we

d$

+ (-) n

Jo

I

" e-nc-*«nhf

Tn (z)

and

Un (z).

see that

+ Tn+1 (z)-(2n/z)Tn (z)

=-

I "(h™

~ 0) sin (z sin - n0)

.

{2 cos

- 2n/z} d0

7T J o

=-—

P(W -0)4b {cos (z sin a"

- n0)} d0

irz J o

= -4 cos hmr 2 8

z

4 z

Jn (z),' v

on integrating by parts and using Bessel's integral.

Thus

Tn.

(1)

x

(z)

+ Tn+1 (z) = (2n/z) Tn (z) + 4 {cos \mr - Jn (z)}/z.

Tn

(z)

Again

'

2

=7T

!"($ ir-0) sin

cos (s sin

(9

- n0) d0,

Jo

and so

r-iW-ZV+ito-MV^).

(2)

From

these formulae

it

follows that

(3)

C*

+ n) Tn (z) = zT^ (z) - 2 cos

(4)

(*

- n) T„ (z) = -z Tn+1 (z) + 2 cos

2

and hence (5)

Jo

Recurrence formulae for

10*61.

0)

(cf.

§

1012) we

\nir 2

X

by considering only the is due to Schlafli, is

in the integral obtained

second of the two exponentials, the formula

[CHAP.

+ 2/„ (s),

\nir

- 2Jn (z),

find that

2 2 ^n Tn (z) = 2{z sin \mr + n cos \mr) - 4??

Jn (z).

.


10*61, 10-62]

343

With the aid of these formulae combined with the corresponding formulae Y„(s) and Sn (z), we deduce from § 10*6 (1) that

Jn (z),

for

Un_, (z) + Un+l {z) - (2ft/*) Un (z) - (2/z) Jn (z), Un _, (z) - Un+l (z) = 2 Un (z) - {21z) Jn (z), (Sr 4 n) Un (z) = zUn ., (z) + 2Jn (z), (*-n)Un (z) = -zUn+1 (z), [cf. §§358(1), V n Un (z) = -2zJn+l {z).

(6)

'

(7)

(8) (9)

(10)

The reader may It

is

from the

verify these direotly

106 (4).

T_ n

=-

(z)

we

If

vr

f "(Itt

—

ir

-

6) sin (z sin

7r

- 0) sin

of negative order, by the

in the integral

+

we

find that

nO) dd

Jo

2 [*

=

by

replace

definition, § 10*6 (3).

T_ n (z),

convenient to define the function

equivalent of §

358(2)]

(1

I

it

Jo

— n0 + mr) d0,

(z sin

and so

T_ n (z) =

(11)

We

now

define

U-H (z) by

(-)"+*

Tn

(z).

supposing § 106 (I) to hold

for all values of

n

;

then found that

it is

U-n (z)

(12)

10-62.

We

Series for

shall

now shew how

Un (z) - Tn (z) + Sn (z)}.

(-)» {

Un {z).

and

to derive the expansion

Tn (z) = I I

(1)

from

Tn (z)

=

§ 10*6 (4).

{Jn+2M

The method which we ,

2 7r

„

—

- Jn _ im (z)}

shall use is to substitute 2,

= Z

a

{z)

sin

2m0

Tn (z), and then integrate term-by-term. This procedure needs justification, since the Fourier series does not converge uniformly near

in the integral for

=

= ir,

and

values of

To

and, in fact, the equation just quoted

justify the process*, let 8

series converges uniformly

when I

8

and

e

..

M

...

The analysis immediately

257—262. The value

2 ro=l

I

*

untrue for these two

be arbitrarily small positive numbers.

% $ ^7r-o\ we can

(*tr-d)-

pp.

is

0.

find

Since the

an integer m^ such that

sin2m0| |
m

I

following

is

due to D. Jackson, Palermo Rendiconti, xxxn. (1911),

of the constant

A

is 1-8519...

r

.


344

throughout the range 8 ^ 8 ^ v — 8, for and n, we have

values of

all

[CHAP.

M exceeding m^.

X

Again, for all values

of 8 between

Ljt-02 2 m =i

=

{l+2cos2< + 2cos4* + ... + 2cos2Jft}(fc

/

ye

«i

_ —

i1* sin(2i/"-H)f J

—

/

^»r,

By drawing the graph of .r -1 sin# f n sin x exceed \n dx in absolute value * x / o I

\Tn {z)

2

siu

|

«"»»,=iyo

i

21

<

where

-{/:

+

bound of

|

10*63.

£/sin

t

is

be called \ rrA, we have

if this

— n8) dd

— n8)

!!

î-'-»«--«'-

\

arbitrarily small, it follows from the definition of

is

(* sin

2m6

J

an

{t/n +»» fc)

, sin (z sin n .

.

- w0) dd

— «/n-2m (^)}»

''*

established.

be remembered that

power

,

eta:

that*

series of Bessel coefficients

that, in §

;

sin (z sin 8

m=l

as a

x

easy to see that the last expression cannot

/r*aiK-«-i

= 2 —

and

/"(i»f+J)T sin /

I

2 £ mn (z) = — 2 T

It will

it is

*>i

2(AB + e) Bis

and the result

,

Jjt

["{{ln _ 6 )- 2 -*^lsin(* S intf-«tf)^|

5 is the upper

Since

.

a«=

by the second mean-value theorem, since

sin 8

(z

,

t

8

,

infinite series

sin

(2JT+ l)t f i* sin 1

.

= Anfor some value of between 8 and a monotonic (increasing) function.

_J_ '

t

9

Un (z) has already been

defined

(§

3*581) as a

by the equation

3582, this definition was identified with the definition of

series given in §

106 (3).

Graf's expansion of

Tn (z + 1)

as a series of Bessel

coefficients.

It is easy to obtain the expansion

Tn (z + t)= 2 T^^Jmiz),

(1)

Wl= — on * This expansion

was discovered by

Schlafli,

Math. Ann.

m.

(1871), p. 146.

Un (z)


10*63, 10'7] for,

from § 10-6 (4),

it is

345

evident that

Tn (z + t)=- P\\tt - 6) sin(* sin 6 -n0 + z sin 6) dO TT J

V

=

J_ 771

=

'

- 6) £ Jm (z) {e'(t8in»-n9+me) _ e -*(în9-n«+»»»)J

(\ir

\

ffl

»i=-oo

o

? ["(!„.

_ 0) I

«/,„

(0) sin

{*

- (n - w) 0} d0,

sin

under the integral sign is uniformly convergent, the order of summation and integration may be changed, and the

by using

§

21

since the series

;

result is evident.

The

the formula given by Graf, Math. Ann. xliii. (1893), p. 141, depends on the use of the series of § 10*62 combined with § 2*4.

it

;

There seems to be no equally simple expression

Un (z +

for

more com-

t).

$M> „ (z) and

The genesis of Lommel's functions

10'7.

A

is

pi-oof of

plicated

sM) „ (2).

which includes as special cases the polynomials z0 n (z) and Sn (z) of Neumann and Schlafli, was derived by Lomniel, Math. Ann. IX. (1876), pp. 425 444, as a particular integral of the equation function,

—

V v y = kz»+\

(1)

where k and

fi

are constants.

,

(2)

*

r

z*

,

1

,

+i

z*+*

[> + If - S

air

shew that a particular integral of powers of z beginning with z** is

It is easy to

this equation, proceeding in ascending

{(/T+

If-

(-)m

£

m==0 (i> -$V +

1/

2 }

[i>

+ 3)* - v*\

+

]

•

'

J

(^r

|)„,+i

.

(\p

+ ±P + |)m+i

r^-^+m+Dr^+^+w+i)

wt

For brevity the expressions on the right are written in the form ks^ v {z). is evidently undefined when either of the numbers an odd negative integer*. Apart from this restriction the general

The ft

±

v

function sMt „0)

is

solution of (1)

is

y -#,(*)

(3) In

evidently

like

manner

+ *»**(*)•

the general solution of

is

y = *-*<•-!>

(5)

* .

The

F.

{<$ v

(«)

+ *V „ (2)}.

solution of the equation for such values of n

and

v is discussed in § 10*71.

12


346

Next

X

us consider the solution of (1) by the method of "variation of We assume as a solution*

let

parameters."

y=A(z)J

A (z) and B (z)

+ B(z)J_ v {z),

(z)

v

where

[CHAP.

are functions of z determined by the equations

= 0, J\ (z) A'(z) + J'_ v (z) B' (z) = kzr-\

Jv (z) A'(z) + J.

On

using § 3*12

A (z) = v

'

Hence a

(2),

we

-

f J

solution f of (1)

it

sin vir

where the lower

v

y

(7)

B'

B(z)

(z) dz,

(z)

= - ^~i^2 sin

[

vir J

VJ

(z) dz.

v

is

J

J

\_

limits of the integrals are arbitrary.

Similarly a solution of (1) which integers or not,

(z)

see that

^ V Jw

2 sin

v

valid for all values of

is

i>,

whether

is

= \1ctt

lY, (z)

[V J

v

(z)

dz

- Jv (z)

\* z*

Y v (z) dz\

.

numbers fi ± v + 1 have positive real and (7) may be taken to be zero. If we expand the integrands in ascending powers of z, we see that the expression on the right in (6) is expressible as a power series containing no powers of z other than ^ +1 z» +3 z***, .... Hence, from (3), it follows that, since neither of the numbers /jl±v is an odd negative integer, we must have It is easy to see that, if both of the

parts, the lower limits in (6)

,

,

,^ r

(8)

\*z*J_ (,)*_[,/,(,) l sin vir

In obtaining this result

we

{z)dz-J_ v (z) Fz»J,(z)dz\

v

jo

j o

|_

it

was supposed that

introduce functions of the second kind,

we

.

J

v is not

an integer

but

;

if

find that

2

8Â*) =

(9)

and in

this formula

\v\yAz)\ z»JAz)dz-Jv {z)^ z*Yv (z)dz\, we may proceed

It should be observed that, in

(10)

^ %(,)

y

to the limit in

making

Pochhammer's notation

tl)(^> +

* Cf. Forsyth, Treatise on Differential Equations (1914), §66;

integer.

(§ 4*4),

it is

|i'

+ f; -i*

2 )-

supposed temporarily that

not an integer.

f

of

an

l)

x/,(i; i^-ii' + |,i/* + v is

v

z,

generalisation of this result, obtained by replacing zn+i in (1) by an arbitrary function was given by Cbessin, Comptes Rendus, cxxxv. (1902), pp 678—679 and it was applied by

The

him, Comptes Rendus, cxxxvi. (1903), pp. 1124 Bessel's equation.

;

;

— 1126, to solve a sequence of equations resembling


10-71]

The

associated function

£Mi „ (2)

derived from a consideration of a solution

is

We now

of (1) in the form of a descending series. solution and investigate its properties.

A particular integral

of the equation § 10'7(1), proceeding in descending

powers of z, beginning with z*~ l

is

,

- 1)* - v (fl + P >

y

proceed to construct this

The construction of the function S^tV (z).

10*71.

(1)

347

= kz^

1

,

j (/

- 1)* _

,/>}

|(„

- 3)* - V>

a-.].

*

This series, however, does not converge unless it terminates but a solution of § 10*7 (1), and it will be called kSîz). ;

if it

terminates,

it is

The

series terminates if

odd positive integer, and

in

ft

—v

In the former case we write q

—

(r \

K

When

vanishes,

(2)

fi

r(i»

m=o

— v±=2p +

and

so,

//,

+

if /*

v is an

case.

= v + 2p +

1,

and then we have

(-rr(ij*-iF+|)r(jj» + ii' + l)

*

'

an odd positive integer, or

is

no other

when

l,

fi

l)r(i;

+ m+l)

the function

—

s> ,(*)^,(*) + x ' f. \ /

+

v is

an odd positive integer, we have

2'~ir(i

''-*"

+ * )r(i '' +

i ''

+ l)

Sln V1T

'

X

[COS %(fJL

—

v) IT

.

J_„ (z)

— COS |(/Ll +

V) IT

.

Jv (z)].

Since both sides of this equation are even functions of v, the equation is true also when /x + v is an odd positive integer, so that it holds in all cases in

Sllv (z) has, as yet, been defined. We adopt it as the general definition $M( „(z), except that, when v is an integer, we have to use the equivalent form

which of

(3)

S^(z) = *„,„(*) +

2"" 1

r (§/* - I v

4-

\) r(|A*

x [sin \{yk — v)ir.Jv (z)

in § 1073 that Sîz) has a limit negative integer, i.e. when sMi „ (z) is undefined

It will be

an odd two functions is

latter.

+ £* +

shewn

sM)V (a)

and S^

v (z), it is

|)

— cos % (fi —

v) ir

when fi+ ;

and

so,

.

F„

(*)].

v ov

fi-v

of Lommel's

frequently more convenient to use the


348

[CHAP.

X

It will appear in § 10*75 that the series (i),

defined

when

by means of which Sîz) is numbers ft ± v is an odd positive integer, is still when the numbers fi ±v are not odd positive integers. It an asymptotic expansion of S^ v (z) valid for large values of

either of the

of significance yields, in fact,

the variable

z.

Recurrence formulae

10*72.

from

It is evident

§

107 (2)

satisfied

**'Wthat

is

by Lommel's functions.

that

+ iy-*

fr

»

to say

W.., (z) = **

(1)

Again,

it is

+1

- {i> + 1) 8

:*»}

M (4

i

easy to verify that j

TZ^ S

*' "

^=

+ V ~ X ) ^ «#-l. r-1 (*).

(/*

so that

8\ w (Z) + (v/z) SM> „ (z)=(fl + V-

(2)

1) V-1..-1

(*)>

and similarly (3)

On

s' ilLiV

(z)-(v/z)s^ v (z)

= (fi-v-l)s

_ltV+1

ll

subtracting and adding these results

we

(z).

obtain the formulae

= (/M + v-l) s^ „_j (z) - (fi - v - 1 ) $„_i. m-i (*)> 2s'M „ (0) = (/i + v - 1 ) v-j, .-i (*) + (/* - v - 1 ) Sp_ ,+1 (z).

{2v/z) SrtV (z)

(4) ( 5)

lt

,

The reader

deduce from § 10*71 (2) that the functions of may be replaced throughout these formulae by functions of

will find it easy to

the type SntV (z) the type #Mi „ (z)

;

W

so that

= **+1 - {(fi + 1) - #M( , (*), S^ v (z) + (*/,) SM,„ <«) - 0» + * - 1) flMlPU4 (#), 6\, (#) - (v/z) £„,„(*) - 0» - v - l)8^ v+l (*), (2*/*)^ „(*) = (,* + »/- 1)^^_ ,,_ (^)-(m-i'- l)iSU.r*(*). 2S'^(z) = (M + v -l)S^. (z) +
(6) (7)

(8) (9)

2

(*)

1

(10)

,

2

1/

}

1

1

These formulae may be transformed in various ways by using (1) and (6). They are due to Lommel, Math. Ann. ix. (1876), pp. 429—432, but his methods of proving them were not in

all

cases completely satisfactory.

Lommel's functions Sfl>v (z) when fi±v is an odd negative integer. The formula §1071(2) assumes an undetermined form when fi — v or fi + v 10'73.

an odd negative integer*. We can easily define Sv-i,»(z) by a repeated use of § 10*72 (6) which gives

is

n\ (1) *

Since

*-i

J.

£>*+*.,(*)SMl „

odd negative

(z) is

integer.

o

(-fi^»

2^(- J

an even function

of

,)m+1 ( v

v, it is

-p)tn+1

+

Sv-vp-\ >v (z)

in terms of

(-)*&.,,,(*)

2^! (1-,)/

sufficient to consider the case in

which

ix

-v

is

an


10*72, 10*73]

We

next define

Sv-h ,(z) by

the limiting form of §

^w^,[

(2 )

x>

1072 (6), namely

.-Cifey+ i)]-

The numerator (which is an analytic function of when /* = i/— 1, and so, by L' Hospital's theorem*

Now

it is

fi

near

= v — 1)

/*

vanishes

easy to verify that

pW»

1

L

_U,-i

3/*

349

(-)"(^)"

-i-iw U + 1,n I

.> +

m

l)ir(» + m + 2)

x {2log z + yfr(l) + yfr

(p

+ 1) - ^ (m + 2) - yfr(v + m + 2)}.

Also

[^

{2m+i

r (i/* - ** + f) r (^ + 1 * + f) cos j i> + v) »}]^

r_ x

= 2' r (v + 1) sin i/7T {log 2 + |^ (1) + |^r (v + 1) + £tt cot vv},

and

[^{2^r(^-^+f)r(^ + ^+f)cosH/*-^)^]^_ = 2^7rr( v +i), and hence (3)

it

follows that

s^y-i'-rwsJrlfSli)

and

this formula,

integer,

is,

\z - ty (p + m +

-

- 2"-

ir

which appears to be nugatory whenever v

is

x

{2 log

in effect, nugatory only

when

v

1)

=

yjr

;

(m + for

1)}

2

when

v

=—n

V (v) Y¥ (*), a negative (where n

a positive integer) we define the function by the formula

^—n-i,—n \8) = £>_„_!« (#), in which the function on the right

To discuss

the case in which v=0,

is

we take the formula

-V«Wwhichgives

Since

Sr

'

W

defined by equation § 10-73(1).

(/t+1)2

-'.o(*)=^[p{^ +1 -^ +S.o(«)}]

()-*« {r (J, + f

)}»Jo

on reduction, that

(4)

<8L,,o(*)=5 * * Cf.

_i

^t^

+2* + i {r (*/*+$)}* (cos &** it follows,

•

Mss

.

r

(z)

- sin &**

.

J

(«)},

^&#^[{log(^)-f (m+l)}2-4^'(»i + l)+i^]. Bromwich, Theory of Infinite

Series, § 152.

is


350

[CHAP.

X

Functions expressible in terms of Lommel's functions.

10*74.

From the descending series given in § 1071 (1) it is evident that Neumann's polynomial On (z) is expressible in terms of Lommel's functions by the equations 0.im (z)

(1)

and

polynomial

Schlafli's

flU (*)

(2)

0^ 0) = {(2m +

= (1/*) S^ n i», Sn (z)

is

1)/*}

similarly expressible

by the equations

&t*n (*) = 2S0>2TO+1

= 4™ S-i. »»(*)»

S0tWl+1 (z),

(*).

It is also possible to express the important integrals

j

d_

{z»

Jv (z)

.

zl ~>

l {Z ~" «/"„_! (Z)

.

5M_

§

(3)

„_ 1 (*)}

f

Y

z"

v

(z)

dz

thus we have

= zJv^ (z) /SM _ ,„_ (g)+(jt-v- 1) zJv (z) SM_2,, (*)> 1

8^^-^z)

eliminating

1072 (6), we

]>

;

(z) dz,

1

SM> „ (*)} = - Zjv (Z) Sîz) + (fl+V-l) zj^ (Z)

*-*

dz

On

v

Lommel's functions

in terms of

dz

#J

S^^

(z).

from the right of these equations, and using

by integrating that

find

P ** Jv (z) dz =

(ii,

+ v-l) zJv (z) 8^,h „_, (z) - zJv-x {z) Sp „ {z\ t

and proofs of the same nature shew that (4)

Yv (z) dz = (p + v-l)zYv (z) /S^^-i (z) - zY^ (z) £„,„(*),

[* z»

and, more generally, (5)

j* z»

^

(

v

(z)dz

= ( i + v-l) z<@v (z) 8^ „_, (*) - *#_, (*) 6V, „ (z). f

Special cases of these formulae are obtained by choosing

the functions on the right reduce to

(6)

f z9g^

(7)

f Sf

(*)

dz = Z*

9Bl+ x (*)

Neumann's or

L*™ j K,n (Z) Otm-

dz = £* {#»+,

(*)

l

p and

i>

so that

Schlafli's polynomials,

(*) -

#»_, (*) Om

thus

,

(*)J

&m (*) - #m (*) 8am+l (*)}.

contained in Lommel's paper, Math. Ann. ix. were given by Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), p. 100, but his formulae contain some misprints.

Of these

(1876), pp.

results, (1), (3), (4)

425—444;

It should

(6)

and

and

(6) are

(7)

be noticed that Lommel's function, in those cases when

it is


10*74, 10*75]

equivalent to Gegenbauer's polynomial of § 92. formulae connecting the functions are*

expressible in finite terms,

The

351

(8)

T(v+m + l) v+2m+l

2 v+1

..

.

r(y + m) v+ 2m Q

2r

/n

is

III

\

a

.

It follows that the most general case in which the integral (5) is expressible in terms of elementary functions and cylinder functions is given by the formula

/'

(0)

^r-wm

(*)

dz ~

ml z*- v tvT (y + m) I

>M-2m \ z ) ™-2m—i,v\ 2; )

v

+ 2m-l (

_

&,+tm-l (z) v

The function denned by the

w

*

A^

v

{z)

+ 2m

series

r(v+2m+i)

r(v-i)

,+l,lw

has been studied in great detail by W. H. Young t this Function possesses many properties analogous to those of Bessel functions, but the increase of simplicity over Lommel's more ;

general function seems insufficient to justify an account of

The

integral v

f* I Jo

J

(t) '.

,

t(z — t)

dt has been studied (when v

them is

here.

an integer) by H. A. Webb,

Messenger, xxxin. (1904), p. 58; and he stated that, when •/=»», its value is On (z). This is incorrect (as was pointed out by Kapteyn) ; and the value for general values of v is J

{S1 ,u(-z)-pS ,y(-z)}/z, ()

when

It (v)

>

|

shall

ir.

now shew by Barnes' method §

positive integers, then

S

fll<

|

z

j

is

large

that,

when

ft

± y are not odd

y(z) admits of the asymptotic expansion

^. w ~^-[i-^-y-" + t»-

a) when

\

<

The asymptotic eocpansion of SîV (z).

10*75.

We

and arg ( - z)

i >'--'M0'-») ,

--, i-...],

and arg z < w. |

j

Let us take the integral

27rtJ_ooi- i> +i

The contour

is

to be

r(|-^/A + |i/) rd-i/A-iv) drawn by taking p

only poles of the integrand on the poles of the

Gamma

left

to

'

sins7r

be an integer so large that the

of the contour are poles of cosec sir, the

functions being on the right of the contour.

*

Gegenbauer, Wiener Sitzungsberichte, lxxiv. (2), (1877), p. 126. t Quarterly Journal, xian. (1911), pp. 161—177. % Of. (rubier, Zurich Vierteljahrsschrift, xlvii. (1902), pp. 422—428. § Proc.

London Math.

Soc. (2) v. (1907), pp.

59—118;

cf.

§§6-5, 75, 7 '51.


352

The

integral

convergent when arg z

is

j

may be shewn from

that the same integrand, centre at

that

R

<

ir,

and

may be

it

X

seen without

(a*-2?).

difficulty that it is

It

\

[CHAP.

— p — \, on

Gamma

the asymptotic expansion of the

when

function

integrated round a semicircle, of radius

R -*- oo

the right of the contour, tends to zero as

,

R with

provided

tends to infinity in such a manner that the semicircle never passes

through any of the poles of the integrand. It follows that the expression given above is equal to the

sum

of the

residues of

T(j-1ui +lv-s)r(l-l H,-$v-s) nrQir sin^TT r(i-i * + jior
2

'

i

at the points

0,-1,-2,. ..,-<_p-l), 1,2,3,...,

When we

calculate these residues

we

find that

r(|-|î + |i')sinj'7r w = oW T(l -v + m) ,

1 -

+ so that

_»-i (-yr(i-^-i-iir+m)r(i-^-|y + w)

,

.

sini>7r

x [cos i 0* -

by

1071

v)

w

.

«/_„ (*)

- cos £ (/* + v) it Jv (z)] = .

we have the formula

and

so,

and

this is equivalent to the asymptotic expansion stated in (1).

§

(2),

(^~2p),


10*8] 10*8.

353

Hemi-cylindrical functions.

Functions

Sn (z) which

satisfy the single recurrence formula

S-* (*) - Sn+1 (z) = 2S„' (z)

(1)

combined with

B.W

(2)

«•'(*)

1

have been studied in great detail by Sonine*. They

be called hemi-

will

cylindrical functions. It is evident that

S„ (z)

is


Sn (s)=/»(D).S

D = d/dz and /„ (D) is a polynomial in D of degree n

where fn (?)

(*),

satisfies

;

and the polynomial


combined with It follows

by induction

(cf. §

91 4) that

n a a /«(l)=H(-f + V(l +i)} + {-f-V(l +i)}

tt

],

and therefore

Bm {M)-ll[-D + J(I* + l)}»+[-D- + l)}*].B,(s).

(3)

If

it is

supposed that (1) holds for negative values of

S_ (*)=(-)"Sn

(4)

To obtain an

.

/

x

t Mi)-

easy to see that

(*).

tt

alternative expression to

w, it is

put £ = sinh

(3),

t,

and thenf (n even)

jcoshntf

{-sinhn*

(re

1+ 2! ? + ~4i re

]

,.

-jjf

re (re

2

-

l2)

^-31

odd)

(»even)

*

,,

..

fc £•'-•••.

(re

odd)

Hence »2 I

(5)

So <*)

Sn (^)=

8n (z), Un (z) It should be

re

2

Cre

2

— 22 )

+ \, n(n2 _ 12)

S„" (#) |j *

'

S » (,)+...

(re

n (z) Tn (z) and E (*) are hemi-cylindrical and H„(s>are not hemi-cylindrical functions.

It is to be noticed that

but

+

y

ft

even)

functions,

remarked that the single recurrence formula Z

gives rise to functions of no greater intrinsic interest than Lommel's polynomials. * Math. Ann. xvi. (1880), pp. 1—9, 71—80. t See e.g. Hobson, Plane Trigonometry (1918), §264.


354 10*81.

We

[CHAP.

X

The addition theorem for hemi-cylindrical functions.

now

shall

establish Sonine's important expansion*

Sm (z + t) = 2 J n (t)Sm _n (z);

(!)

n= — «

the expansion at the point

z,

valid when z + 1 lies inside the largest circle, whose centre is which does not contain any singularity of the hemi-cylindrical

is

function under consideration.

Take as contour a circle G with centre z such that S on the circle. Then

(f) has

no singularity

inside or

ifiM %—z —

,, ( , +0

ZlTl] c

t

=^fc smxo[ ^nO

n

a-z)jn (t)^dc

n

The

series converges uniformly

Sw (s + t)=

i

= $

%

cn

€n

on the contour, and so we have

Jn (I)

Jn (t)fn

B* (0 On (t- s)

J

%

(-^8m (z)dz

-.!,^«>/-(-a)/-(5) ••(')* But

it is

easy to verify that 2/n (- *)/„

(0 -/_ (£) + (-)"/-+« (IX

so that

»»(* + *)-./.(*)*(*) + 2 ^»(0{S«*-n(^) + (-rS M+n n=l

whence Sonine's formula

is

2)},

.(

obvious.

It should be noticed that, if

S„ (z) denotes a function of a more general type than a hemi-cylindrical function, namely one which merely satisfies the equation Sn, 1 (z)-8n+l (z)^2Snf (z), without satisfying the equation

f*

and so the formula the formulae of * Math. Ann.

xvh. (1880),

§§

S

x

(z)

=—S

V di) Sm W = Sm_n (1) is

53, 91,

still valid.

'

(z),

( z)

We

we

still

have

+ ( ~ )n &m+n ( ^'

thus have an alternative proof of

934 and 1063.

xn. (1880), pp. 4—8. See also Konig, Math. Ann. pp. 85—86.

v. (1872), pp.

310— 340;

ibid.


10-81, 10-82]

Nielsen s functional equations.

10*82.

The

355

pair of simultaneous equations

*U (*) + F

v+1 (*)

1(2)

- (2v/z) Fv (z) =

%gv

(z)/z,

where fv {z) and g¥ (z) are given arbitrary functions of the variables v and z, form an obvious generalisation of the pair of functional equations whereby cylinder functions are defined. It has been shewn by Nielsen* that the functions /„ (z)

and

it

satisfy the relation

and g v (z) must

has been proved f that,

if this relation is satisfied,

the system can be

reduced to a pair of soluble difference equations of the first order.

For brevity write

+ 9» (*) = «- (*).

f» (•*)

and the given system of equations J

is

f» (z )

~ 9* (*) - & (*)«

equivalent to

+ v) Fv (z) = zFp -x (z) - a, (z), (* - v) Fv (z) = - zFv+l (z) - ft, (z).

(^

(3)

1(4) It is

now evident <*>*

that

- tf) Fv {z) = (*-v) [zFv .

(z)

x

-

a v (z)]

= - z*F, (z) - z/3^ (z) -&-p) av (z), V V F¥ (z) = - z$ v _ (z) ~(**-v) (z).

so that

or,

x

Again <*»

We

-

F

*2 )

v (jr)

- <* + *) [- zFv+l (z) - fi m (*)] = - z*Fv (z) + za v+1 (z) -(* + p) &, (z).

are thus led to the equation

V,F,(z)

(5)

= zv,(s),

where

zm v (z) = - *£_,

|(6)

z*r y (z)

1(7)

On

(z)

= + zap+1 (z)

comparing these values of

-(*-p) «„ (z), - (* +

vr v (z),

we

v) /9„ (z).

are at once led to Nielsen's

condition (8) It

/_, (*) +/„ +1

now has

to be

(*)

- (2»-/*)/r (*) = 9.-i (*) ~ 9*+i (*) " 2S^' (*)•

shewn that Nielsen's condition

ence of a solution of the given system. *

Ann. di Mat.

To prove

(3) vi. (1901), pp.

is sufficient for

this,

we assume

51—59. 49—53.

t Watson, Messenger, xlviii. (1919), pp.

the exist(8) to

be


356

given, and, after denning ta v (z)

of variation of parameters.

F

(9)

9

(m)

= /„ (*)

by

The

(6)

and

solution

(7),

we

solve (5)

X

by the method

is

L - jy 1-k

[CHAP.

dtl

v (t) «r„ (t)

+ Yv (z) \dy + |ttJ V„ (t) vt v (t) dtl

,

where a and b are arbitrary constants; and c„ and dy may be taken to be independent of z, though they will, in general, depend on v. remains to be shewn that c„ and dv can be chosen so that the value of given by (9) satisfies (1) and (2), or (what comes to the same thing) that it satisfies (3) and (4). If (3) is satisfied, then It

F

¥

{z)

|c - £tt

*/,_, (z)

Yv (t) vf¥ (t) dtl - \ttz

¥ (z)

F„ (z) «r„ (z)

J

e

+ zYv ^ (z) \d ¥ + %tt

j

J¥ (t) «„ (t) dtl + faz Y¥ (z) J¥ (z) TB V (z)

Y^ (t) «„_, (t) dtl

= zJr-, (z) c^ - \tt -j

J

+ zY¥ _

x

(z)

<*„_,

+ |tt

l"

J_, (t) w^, (0 dtl —«„ (z),

I

that

is to say,

*/"_! (z)

r

I

+ zY^

x

- c w -WJ"{ Y* (t) v v (t) - Y¥^ (t) <*„_, {t)\ dtl dv - d_, +

(z)

*

\tt

[Jv (t)

«r„ (*)

- J^ (t) «r^ (t)}

dt\

J

But

easy to verify that

it is

Tz since (6)

{Û (z) &_, and

( jr)

and

1

(z)

T

;

by

«r„ (z)

and so (3)

<&¥ (s)

- «r r _, (z) V^ (z),

is satisfied if

F^ (z) £_, (z) - Y (z) «„ (z) ¥

|^-d-i + iw r^-

this condition,

zj¥_, (z)

- V. (z) a ¥ (jr)} -

(7) are satisfied

zJ^ (z) jc - c,_y - \tt + zY,_

+ a ¥ (z) = 0.

§ 3'63-(12),

1

(^^

1

*

I

(^)-^(^)^(^)l*|

+ a,(z) = 0,

reduces to

- c_, + \tt [ Fr _, (a) &_, (a) - F, (a) a„ (a)]} + * F_, (*) {d¥ - d ¥_i - \ir \JV_ (b) &_, (b) - J¥ (b) a, (6)]} = 0. [c ¥

X

Consequently, so far as (3) is concerned, to satisfy the difference equations f (10)

\(11)

it is sufficient

to choose c„

c - c_, - - £tt F,_ (a) &_, (a) - F, (a) a, (a)}, d, - <*_, _ |» {J^ (6) £„_, (6) - J, (b) a„ (6)} {

x

and d ¥


10*82]

and the reader

difficulty in verifying that, if these

have no

will

difference equations (with v replaced by v

the value of

F

v {z)

given by

357

+

1

throughout) are

a solution of

(9) is

same two

satisfied,

then

(4).

These difference equations are of a type whose solutions may be regarded

known*; and so the condition (8) is a sufficient, as well as a necessary, condition for the existence of a solution of the given pair of functional equaas

tions (1) If,

and

(2).

as z -* *>

/„(#)-0 where B >

0,

then we

may make a -* oo may be

so that the general solution

(12)

F. (z)

= J9 {z)

|ir,

+ where ir^v) and

5r,(*)

(**-•),

7r 2 (i/)

b -* x>

(**-'),

,

and we have

written

+ £tt

(v)

Y, (z)

,

=

L

2

(v)

(J

Yv (t) w„ (t) dt}

- |tt

rJ

v (t)

v. (0^4

,

are arbitrary periodic functions of v with period unity.

Note. Some interesting properties of functions which $ati*fy equation (2) only are to be found in Nielsen's earlier paper, Ann. di Mat. (3) v. (1901), pp. 17—31. Thus, from a set of formulae of the type

F_ v

it is

,

(*)

F

v

+

!

(z)

- (2v/«) *T («) = 2g y («)/«,

easy to deduce that

(13)

Fv+n (z)=F

v

{z)Rn

,

v

(z)-Fv _ l (z)Rn . l

,

v +i(.z)

n-l

+ (2/«) the

first

and the

two terms on the

right are the

2 SWw(«)-ffn-m-i, „ + »+i(«); m=0

complementary function of the difference equation,

series is the particular integral.

* An account of various memoirs dealing with such equations Math. Soc. (2) n. (1904), pp. 438—469.

is

given by Barnes, Proc. London

CHAPTER XI ADDITION THEOREMS The general nature of addition theorems.

It has

been proved

and

11*1.

(§ 4*73)

that Bessel functions are not algebraic functions,

obvious from the asymptotic expansions obtained in Chapter vn that they are not simply periodic functions, and, a fortiori, that they are not it is fairly

doubly periodic functions. to Weierstrass*,

of

,/„

(Z) and

it is

J

v

Consequently, in accordance with a theorem due

not possible to express

That

(z).

JV (Z + z) as an algebraic function do not possess

to say, that Bessel functions

is

addition theorems in the strict sense of the term.

There

are,

however, two classes of formulae which are commonly described

In the case of functions of order zero the two classes for functions of the first kind is

as addition theorems.

coincide

;

and the formula

J y(Z* + i» - 2Zz cos

)}

= 5 em Jm (Z) Jm (z) cos m, ro-0

which has already been

The simplest

indicated in § 4'82.

rigorous proof of this formula, which

is

due

to

Neumann f,

depends on a transformation of Parseval's integral another proof is due to Heine J, who obtained the formula as a confluent form of the addition theorem ;

Legendre functions.

for

11*2.

We

Neumann's addition theorem^.

shall

now

establish the result

J

(1)

(*0

= 2 € m Jm (Z) Jm (z) cos m, wi-0

where, for brevity,

we

write zr

and

all

*

Ch.

;

2

0),

the variables are supposed to have general complex values.

The theorem was

1893)

= V(£ + * - 2£* cos

see

stated in §§ 1

Phragmen, Acta Math. vn.

—3

of

Schwarz' edition of Weierstrass' lectures (Berlin, 33—42, and Forsyth, Theory of Functions (1918),

(1885), pp.

xiii for proofs of the theorem. t Theorie der BesseVechen Fuvctionen (Leipzig, 1867), pp. 59 70. XHandbuch der Kugclfunctionen, i. (Berlin, 1878), pp. 340—343;

—

cf.

§ 5*71

and Modern

Analysis, § 15*7. §

xvi.

In addition to Neumann's treatise cited in (1880—1881), pp. 201—202.

§ 11 -1, see Beltrami, Atti della

R. Accad. di Torino,

ADDITION THEOREMS

11-1-1 1-3]

We

which

359

take the formula (Parseval's integral)

is

and

valid for all (complex) values of vr

analytic function of 6 with period

We

2tt.

a,

the integrand being a periodic

next suppose that a

is

defined

by

the equations ts

and

it is

sin

ol

— Z—z cos

ta cos

,

a

= z sin

,

then apparent that

J

(•©)

*

= jj-

exp

f

* [* j 27T J _„ (

(£

{i

|

m=-x

- z cos

JM (Z)e» w l

cos 6)

dd

efe " in( *"' )rftf

n

I

Jm (Z)\

= J_

%

Jm (Z)

-

<£

)

—

=

+ iz sm

0) sin

"

f

mie+izain

e

<*

6)

dd

mi <•+> -» 8in<>

^

Jm {Z).Jm {z)e™*,

X m= —

e

<*>

the interchange of the order of summation and integration following from the uniformity of convergence of the series, and the next step following from the periodicity of the integrand.

we group the terms for which the immediately obtain Neumann's formula. If

values of

m

differ

only in sign,

we

for Beasel functions of order ±| were obtained by Clebsch, Math. lxi. (1863), pp. 224—227, four years before the publication of Neumann's

The corresponding formulae Journal fiir

formula; see § 11*4.

11

•

Grafs gen eralisation of Ne um arm's formula.

3.

Neumann's addition theorem has been extended to functions of arbitrary The extension which seems to be of more immediate importance in physical applications is due to Graf*, whose order v in two different ways.

formula

is

*<•)

(1)

and

•

{^-^Y - J_J'+<»

this formula is valid provided that

(

Z)Jm

{Z)

*"*'

both of the numbers

|

ze ±f*

\

are less

than \Z\. *

Math. Ann.

—

pp. 59 61. p. 241.

A

xliii. (1893),

pp. 142

—144 and Verhandlungen der Schweiz. Naturf.

special case of the result has also

Get. 1896,

been obtained by Nielsen, Math. Ann. ui. (1899),


360 Graf's proof

[CHAP. XI

based on the theory of contour integration, but, two years after it was by G. T. Walker, Messenger, xxv. (1896), pp. 76

is

published, an independent proof was given

80 this proof is applicable to functions of integral order only, and it may be obtained from Graf's proof by replacing the contour integrals by definite integrals. ;

To prove the general

so, if

argJ?=a, we have

Jv+m {Z)Jm (z)e™*

1 HI

formula, observe that the series on the right in (1)

convergent in the circumstances postulated, and

is

= — 00

= s-7

there

no special

is

exp \$Z(t-j) [r--»- *Jm {z)e">*dt

S

difficulty in

interchanging the order of summation and

integration *.

Now

write

(Z - ze-* )t where, as usual,

and

it is

-*-

•sr

all

z

admissible values of

u- contour to start

by

z,

the phase of ns\Z

This determination of

from and end at

— ao

we

and

define the angle

yfr

-*yfr

as z

relation indicated

--*-

«r

now an

is

renders

it

acute angle,

possible to take the

£ = arg m.

exp (— i/3), where

-»/r,

z sin

= vr sin

yjr,

(so that, for real values of the variables,

by Fig.

28),

then Graf 3 formula

«*./.(»)- 2

m= — oo

and, on changing the signs of (3)

taken which makes

by the equations

(2)

is

this is Graf's result.

Z — z cos = «r cos where

),

then have

§ 6*2 (2);

If

= \/(Z + z — 2Zz cos

-*~ 0.

positive or negative.

We

(Z- ze*)\t = m/u, 2

z

supposed now that that value of the square root

+ Z when

For

-st

= vru,

and

may be

obtain the

J*+m (Z)Jm {z)e™«

^r,

we have

r**J.(w)- 2 J^m (Z)Jm (z)e-mi*, * Of.

we

written

Bromwich. Theory of Infinite

Series, § 176.

r

ADDITION THEOREMS

11-8]

whence

it

361

follows that

2 Jv+ m (Z)Jm {z)™*m.

/„(«r)Ti/^r=

(4)

sin

sin

«!=— oo

Fig. 28.

in this formula,

If,

we change

the signs of v and m,

we

readily deduce from

§ 3-54 that

and

% T^m {Z)Jm (z).m, sin

r„(t!r)în/r=

(5)

sin

#j=

— ao

so

ô*)™V= sin

(6)

5

ifi=— oo

^(^^w^^ Bin

was given by Neumann in his treatise in the special case i>=0; see Ann. xlv. (1894), p. 276; ibid, xlvii. (1896), p. 356. Some physical applications of the formulae are due to Schwarzschild, Math. Ann. lv. (1902), pp. 177 247.

The formula

(5)

also Sommerfeld, Math.

—

If it is

we

replace Z, z and

«r

in these equations

by

iZ, iz

and

irsr

respectively,

apparent that I. (•)

(7)

*!"„*- 2 (-riv+m (Z)Im (z)^ 5 m, Hill sill m= — oo

K.(v)"!*v+- 5

(8)

Sill

Of these results, (7) was 201—202.

stated

K +m (Z)Im (z)™*m. BUI y

TO=— 00

by Beltrami, Atti

della R. Accad. di Torino, xvi. (1880

1881), pp.

The

following special results, obtained by taking ^—îr, should be noticed

(9)

3?,(»cos^= 2

(-y^M-W^mW,

9,(w)mnv+- I

(~r^ +wl+1 (Z)Jm+l (z),

m— — x

(10)

= «rsin^r

Z=«rcos^r,

where

and |^|<|Z|.

For the physical interpretation of these formulae the reader is referred to the papers by G. T. Walker and Schwarzschild it should be observed that, in ;

the special case in which v of the

first

an integer and the only functions involved are kind, the inequalities se*** < Z need not be in force. is

|

I

|

|


362 11 '4.

[CHAP. XI

Gegenbauer s addition theorem.

The second type of

generalisation of

Neumann's addition theorem was

obtained by Gegenbauer* nearly twenty years before the publication of Graf's paper. If cos

,

Neumann's formula we find that

Jn (m) _ K

}

is

em+n

n time? with respect to

differentiated

Jm+n(Z) Jm+n(z) dn

»

~,„t

ttr»

111

of §

Zn

zn

COS

(w +

tt)

.

d(cos<}>)n

This formula was extended by Gegenbauer to functions of non-integral order by means of the theory of partial differential equations (see § 11*42); but Soninef gave a proof by a direct transformation of series, and this proof we shall §

now reproduce;

to be noted that, in (1),

it is

sr

is

not restricted (as in

113) with reference to Z.

We

take Lommel's expansion of § 5*22, namely

«MV(C+A)} _ * (-hhyjv+P

m

)

write

O

by a further application of Lommel's expansion with f and h replaced by and z\

Z

Z2 + z

and replace f and h by in place of

Jv (vr)/™"

2

and

for brevity, it is

2

(Zz cos

•

-

(

)P

Jv+P W(Z*+z*) (Z*

;

if

we

}

+ «•)*«•+>"

-)« zp+»i cosp

Jv+p+q (Z)

^.p\ql

pro ,to

respectively

found that

pi

p-o

_ "

— 2Zz cos

Z'+*

'

%

But, by § 521,

Jy+P+g (Z) hP+g(Z)

_ S

v

q\

Z k\{q-k)\

Z«

k

+ p+2k

T( v + p + k) T(v

29

T

„

+ p + q+k + l)

and so fl

| 2 I

+ p + 2k) T (v + p + k) gP** cosP 2*p\kl(q-k)\r{ V + p + q+k+l)

(-)« (v

p -o 9 -o*-o

Jy+p+*(Z) Z»

the triple series on the right being absolutely convergent, by comparison

with

2 2 i

+2fc p-0g-0A-0 2 P+29 i>

*

i

jki

T ( v +p + A;) ^ +a« Z-+P+** _ /.)| r ( v + p + 2k) T (v +p + q + k + (g

Wiener Sitzungsberiehte, lxx.

t Math. Ann. xvi. (1880), pp.

(2),

(1875), pp.

22—23.

6—16.

1)

ADDITION THEOREMS

11*4, 11-41]

363

But, for an absolutely convergent series,

q=0k=0

k=0 n=0

and so rt

2*+™ p\kln\r(v +

*t

„ To * - o

+ p + 2k) T (v + p +

(-)*+" ( v

_ | | |

(~)k 2- +P(v+p

£ ^

+ 2k)r( V + p + k)C0SP4

~ (-)* 2"+"t 2t (y

| TO

+ m) T (y + m - k) cos*-

-o*-o

Z"

+ m) T (y + m - k) cos"*- * 2

—

J, +m (£ ) J, +m (*)

„

z"

.

Cm " (cos denotes the coefficient of am in the expansion + a )~" in ascending powers of a. We have therefore obtained

§ 3*32,

2a cos

(z)

z"

£*

<*'»(-)* 2 w»-2*r(i/+m-A;)cos", - 2*<£

where, as in

z*

^ J^^Z) J,+m

(ro-2*)!*!

xt

of (1

Z" 2

{m-2k)\k\ '(-)* 2" +w 2* (v

J,+p+2k (Z) Jr+p+oJcjz )

>

k-om-tk

J,+ p+2k (Z )

Z'

p\k\

p-oa-u

= S ^

z****™ cos*

k)

p+2k + n + l)

)

2

the expansion

îp = 2' T („)

(2)

2(v + m)

W

which

m-0 values of Z,

is valid for all

exception of

0,

— 1, —

«

This formula

is

-^^ '^±# Cw

z,

and

'

(cos

),

Z

It

and

d>,

for all values of v

with the

2,

In the special case in which v = (3)

J

= «. I (m + 4)

£,

we have

^£>

.

-M*) Pm (cos *).

due to Clebsch, Journal fur Math,

lxi. (1863), p.

227

;

it

is

also

given by Heine, Journal fiir Math. lxix. (1868), p. 133, and Neumann, Leipziger Berichte, 1886, pp. 75 82. The formula in which 2v is a positive integer has been obtained by

—

Hobson, Proc. London Math.

Soc.

—

xxv. (1894), pp. 60 + 2 dimensions.

61,

from a consideration of solutions

of Laplace's equation for space of 2v

An Phys.

extension of the expansion (2) has been given by Wendt, Monatshefte 125 131 the effect of her generalisation is to express

—

xi. (1900), pp.

fiir

Math, mid

;

»-"-»• sin 2?

Jy + ,, (or)

as a series of Bessel functions in which the coefficients are

somewhat complicated

determinants.

11*41.

The modified form of Gegenbauer's addition theorem.

The formula (1)

may be §

11 "4.

^P =

2"

r (v) £ (-)- iv + m)

established in the

-^& %^
J

(cos

)

same manner as the Gegenbauer-Sonine formula of

This formula does not seem to have been given previously explicitly,


364

though

it is

[CHAP. XI

used implicitly in obtaining some of the results given subsequently

in this section.

Unlike the formulae of formula,

it is

convenient

the formula

§ 11*4,

small that both the inequalities

ze ±i^

|

is

\<\Z\ are

true only

when

satisfied; but, in

\z\ is so

proving the

suppose that the further inequalities

first to

|2Z*cos<£|
22

!>

\z\<\Z\

are satisfied.

We

then use Lommel's expansion of

§ 5*22 (2) in

P

p=

which

is

It is

J^>(?)

when h < £ then found by making valid

= I

v v v 2* 2* 2*

%

v

j

(- Zz cos

)P

+1 {v (-)p+^ (~)p (y

(-) P+k 2 ,+p

=

^+

~ Jo

j_ v¥ y( Z , + ^2)}

^ +p + 2k) Tjv + p + k) cosp

J- V-V ^k (Z)

J-r-.p_*(Z)

+ m ) f ( y + m - k) COS "*-»*

*-o

(™ " 2*)"1 A!

= 2*i» 2 (-r (> + m) m=0 so the required result

is

J-"-w •"

*

"

piu

-Hm-2fc („

(Z)

^±^ cm

*

;

Jy+p+2k (z)

*

«/-,-„> (Z) J.+m (z)

*' (cos

Z

#,

established under the conditions

|2^cos<£!<|Z2 + *2 |,

Now

v+*+™ cosP J-^-p-a (Z) zP 2k - n) z v - p - 2A; 2fc) T (- »/ 2k) 2k+M £' p\k\nlT(l-v-p-k) 2*+"p!ib!fi!r(l-v-i>-A) 2

„?o *?o co

z, ri{v+p)

+p+

t

00

slight alterations in the analysis of § 11*4 that

2n p \k\(q - k)\V

"",-0*0.-0 = 00

•

j

|

+p + 2 k)T(-v- p- g - k) zp+* cos? {\ - v-p-k)

(-) p+1 (v

qZ« k =«

_ v t •?

|

the form

the last expression

is

\z\<\Z\.

an analytic function of z when z

lies inside

the

circle of convergence of the series*

| m. and

+ m) Z-**-™ z™ Gmv (cos "r(l-i'~TO)r(l+i' + m) (v

)

this circle is the circle of

'

convergence of the series

2

CW(cos<£).

(^J

series converges and represents an analytic function of z provided only that ze±{* \<\Z\\ and, when this pair of inequalities is satisfied, J- v (w)/-sr" is also an analytic function of z.

Hence the given

|

* Cf. § 5-22.

ADDITION THEOREMS

11-41]

365

Hence, by the theory of analytic continuation, (1) whole of the domain of values of z for which

is valid

through the

\ze±*\<\Z\.

we

If in (1)

^^-

)

(2)

Again,

now under

^P =

(3)

and

so,

find that

(-r(v-m)Z"Jv^n (Z)z"J_ v+m (z)Cnr v (eos).

2"*r(-v) 2

we combine

if

values of z

—

— v we

by

replace v

(1) with § 11*4 (2),

we

see that, for the

domain of

consideration,

2-

T („) £

(„

+ m)

%^- %^ Cm }

*

(cos *),

generally,

TO

If in (3)

we make <7 °

^

Tlt=0

(cos

i>

)

*

and use the formulae

-*-

= 1,

lim {r

(v) (v

+ w) Cm" (cos <£)} = 2 cos m$,

(m + 0)

i<-*-0

we

find that

2

F,(«r)=

(5)

6 wl

Fm (Z)Jm (^)coswi0.

«»=o

The formulae (1) and (2) have not been giveu previously; but (3) is due to Gegenbauer, and (5) was given by Neumann in his treatise (save that the functions Ym were replaced by the functions F). The formula (3) with v equal to an integer has also been examined by Heine, Handbuch der Kugelfunctionert, i. (Berlin, 1878), pp. 463 464. Some developments of (4) are due to Ignatowsky, Archiv der Math, und Phys. (3) xvm. (1911), pp. 322—

—

327.

If

we

replace Z, z and

this section

(6)

(7)

we

~

w

by

iZ, iz

= 2" r 00 5

(-)-

{v

!=£) = 2T (v)l

J^ =

2-r( y

Of these formulae, pp. 156

— 157; while

+ m)

OT =

(-.» (v 4 m) o

(8 )

and

its in

the formulae of

§ 11*4

and

find that

)J

(8) is

(6)

and

(, o

+ m)

%^- %& CV }

^W ^^

^/ ^^Cm

(cos 0),

C,» (cos

0),

Z)

-(cos0).

due to Macdonald, Proc. London Math. Soc. xxxn. were given by Neumann in the special case v=%.

(7)

(1900),


366

of § 11*4 and of this section are of special physical importance If we change the notation by writing ka, kr and for Z,

The formulae v = \.

in the case z

and

we

sin

™'

see that the formulae

become

+ a — 2ar cos 0) + a -2ar cos 0)

k V(r

[CHAP. XI

2

2

2

" »»+i \kd) J m+i (kr) n v /(m i a = tt X pm (cos $), + i) 7^ /„ \

i

.

v«

,»=o

cos

v?

+ a — 2ar cos fl) + a ~ 2ar cos 0)

k ^(r

2

8

2

V(r2

-»

5 (-r(«+j)i=^^±i^>i>,„(cos«), Va

m=0 e xp {'

A;

yXr

v (^ + «

'

2 ft

V?'

— 2ar cos 0)}

+ - 2 «r

2

2

cos ^)

'

= Z (2m +1) »»=o

7-

7-

V«

V^

^™ (cos V).

These formulae are of importance in problems in which pulsations emanate from a a from the origin, in presence of a sphere whose centre is at the origin. Cf. Carslaw, Hath. Ann. lxxv. (1914), p. 141 et seg. point on the axis of harmonics at distance

The

following special cases of (4) were pointed out

by Gegenbauer, and

are worth recording If

=

(12) If

we have

()

TZTW~~ = \ir, we

(13) If

7r,

(z»

Z= 2,

+

w -«

(

M

&

'

*

'm\P(2v)'

have

^^"

= 0,

and

2

«-o

r

df„ is

(

}

(

*'

taken to be

2 2 *r(i/)r(i/

+ i) 2

*"

'

«»i

'

JVi

.

r (2j/ +

x

,

7«> ra

/x

a formula already obtained (§5*5) by a different method; in this connexion the reader should consult Gegenbauer, Wiener Sitzungsberichte, lxxv. (2), (1877), p. 221.

More 7

Z=z,

generally, taking

y.i.W _ 2 (2s sin £9)*

Gegenbauer,

loc. cit.

.

r

$ = JV we have £ ( „ + m) j^C£)l Ci/ (C03 ^. 2 ±0,

»»=«

C

V

,

I

)

gives also special cases of this formula, obtained by taking

ADDITION THEOREMS

11*42]

Again,

can be shewn that*,

it

if

R{v)> —

sin 2"

Cm " (cos

)

Cp ' (cos

)

so,

provided that

(m±p) Y (2v + 7n) ( v + m).m\{T(v)Y

d <

-n

2i

I

and

\,

[=0 _

.„

J

367

"- 1

^

l

~ p)

R (*/)> — J,

and, more generally, (17) J.

(

y + ^-2^cos^

*> Sm "

< C0S

°"

_ 7T T (2l> + Wi)

d

^

2"-1 .mlT(v)

+W1

(g) J,+w

Z"

A

(2)

z*

w=0

simple proof of this formula t, in the special case in which and the cylinder first kind, was given by Sonine, Math. Ann. xvi. (1880), p. 37. Another direct proof for functions of the first kind is due to Kluyyer, Proc. Section of Sci., K. Acad, van Wet. te Amsterdam, xi. (1909), pp. 749—755. An indirect proof, functions are functions of the

depending on § 12-13(1), pp.

is

due to Gegenbauer, Wiener

Sitzungsberichte,

491— 502.

An

[Note.

interesting consequence of (4), which

Sitzungsberichte, lxxiv. (2), (1877), p. 127, is that, if of integration, then (cf. § 9*2) ° +)

(

(18 >

lxxxv. (2\ (1882)

2Tj

^^-^^ rf

2

was noticed by Gegenbauer, Wiener \ze±* \<\Z\ throughout the contour

^W-(^-)^9^^(cos0).

Special cases of this formula, resembling the results of § 9-2, are obtainable equal to or «r.]

11*42.

by taking

Oegenbauer's investigation of the addition theorem.

The method used by Gegenbauer, Wiener to obtain the addition

transformation.

Sitzungsberichte, lxx. (2), (1875), pp. 6—16, not quite so easy to justify as Sonine's

theorem of § 11-4

is

Q

It consists in proving that

is

a solution of the partial differential

equation 3

2

Q

dz*

+

2i>

+ 1 do

1_

+ z*

dz

z

and assuming that Q can be expanded

Q= Bm

is

independent of 0, and

dtf

in the

2 m=0

where

d*Q

+

2v cot

dQ

J~ fy + o =

-

form

Bm .Cm " (cos

tfTO "(cos0) is

),

a polynomial of degree

m

in

cos0;

it

follows that

ra

2

|a02

+ 2 " cot

tf»

ai Cm" (cos ) a^}

* Gegenbauer, Wiener Sitzungsberichte, ucx. (2), (1875), pp. 433—443, and Bateman, Proc London Math. Soc. (2) iv. (1906), p. 472 cf. also Barnes, Quarterly Journal, xxxn. (1908), p. 189; Modern Analysis, § 16-51 and Proc. London. Math. Soc. (2) xvn. (191S), pp. 241—246. t Formula (16) has been given in the special case ,-=0 by Heaviside, ;

Electromagnetic Theory, in.

(London, 1912),

p. 267, in

a somewhat disguised form.


368 is

a constant multiple of

Cmv (ooa

of am in the expansion of (1

and so )

Cmv (cos may be taken to be the coefficient And then Bm qua function of z, satisfies the )

" ".

,

equation

differential

ozl

so that

),

— 2a cos <£ + a2

[CHAP. XI

Bm is a

multiple of

vz

z

z

(

J

solution of this differential equation not

z~"Jv+m («)> the other

being analytic near the origin.

From

symmetry Gegenbauer

considerations of

a multiple of

Z~ v Jv+m (Z),

z

is

m Zm co8m

A

a function of v

in

Q and

2 bm m=0 and m only

-—

^

Q= where bm

inferred that

Bm qua

less

11*5.

is

;

in the expression

lV(cos<£),

z

£>

and bm is determined by comparing on the right.

similar process was used by Gegenbauer to establish § 11-41

seems

function of Z,

so that

convincing than in the case of functions of the

first

(3),

coefficients of

but the analysis

kind.

The degenerate form of the addition theorem.

The formula e*cos*

(1)

=

^0 |

(2 n

+ 1) in Jn+i (z)

was discovered by Bauer* as early as 1859;

who obtained

tf>)

was generalised by Gegenbauer f

the expansion e fcco8*

(2 )

it

n (cos

=2 T(i/) i

(v

m=0

Bauer's result

is

^Cm

+ m)imJ^±z

-(co8);

obviously the special case of this expansion in which v—\. -*- 0, the expansion becomes the fundamental expansion

In the limit when v of §2-1.

deducible from the expansion of § 11 41 (4) by and making Z -*. co it is then apparent from §-11*41 (9) and (10) that the physical interpretation of the expansion is that it gives the effect due to a train of plane waves coming from infinity on the axis of harmonics in a form suitable for the discussion of the disturbance produced

Gegenbauer's expansion

is

multiplying by Z" +i

;

by the introduction of a sphere with centre at the

A in

powers of z and substituting

supplied by the formula of §

• ==

*

> „=o

n cosn

t'

<£

52

for ;

each power the

we thus

• 2 v+n (v

w!

expanding ^"etecos *

series of Bessel functions

find that

+ n + 2k).r(v + n + k)

2,

4=0

T

,.

+n+sk\Z)'

TT

*•

Journal fUr Math. lvi. (1859), pp. 104, 106.

t Wiener Sitzungtberichte, lxviii. (2), (1874), pp.

lxxv.

origin.

simple analytical proof of the expansion consists in

(2),

(1877), pp.

904—905.

355—367;

lxxiv.

(2),

(1877), p. 128;

and

ADDITION THEOREMS

11-5]

we rearrange the repeated

If

369

by writing n = ra — 2k, we deduce that

series

m-*k (v + m)T (v + m-k)Jv+m (z) ^ efecos* = 2 jri_^?l_*2''+ &!(m-2fc)! m =o*=o

and

this is

Gegenbauer's

result.

Modified forms of this expansion, also due to Gegenbauer, are

(

=

e*«»*

(3)

T (v) 2

2"

+ m)-^r^-Cmr(cos4>)

(v

>

m=0 e -*cos*

4)

=

2"

T

5

(*)

(-)*»

(»>

•

cos(*cos)

(5)

sin(*cos)

(6)

=

2»I»

),

z

m=0

= 2"I» 2

Gm" (cos

*y

+ 2m)-^Ôim (cos),

(-)m .(v

2

+m

"

+ m)

m=0

"+2"? 1 ( **

2m +

1)

»»(s)

r(if

(-)" (y + .

0Wi (cos

ft),

i»=0

l

(7)

L i/ -t-

jV~*fi,(«»»)«^-

+ »o .

*

n-0

(8)

The

«^ +2 = 2-S( V + 2m) zwi;

m,i

* r(>^)r(gr(«, + »)^ j^w.

a generalisation of Poisson's integral, which was obtained by a method in § 332. It is valid only when R{v)>-\.

last is

different

These formulae are to be found on pp. 363—365 of the

first

of Gegenbauer's memoirs

to which reference has just been made.

Equation (1) was obtained by Hobson, Proc. London Math. Soc. xxv. (1891), p. 59, by a consideration of solutions of Laplace's equation in space of 2* + 2 dimensions, 2v+2 being

an

integer.

A cos

more general set of formulae may be derived from (2) by replacing -1 by cos cos + sin sin cos ifr, multiplying by sin 2* i/r, and inte-

I

Cm' (cos

cos

'

grating with respect to ^.

+ sin

The

sin

'

integral*

cos

ifr)

sin2 "

_ which /

is

valid

exp [iz (cos

when cos

yfr

dty

?^!M Cl

(cos *)

Oj (cos

n

R (v) > 0, shews that

ft'

+ sin ft sin ft' cos ^r)] sin2 *-1 ^r d^r

-2»-|r

(

„».

i to-0

• Cf.

-1

'^^)%Mc^(cos^)^ \Av-\-m) z

A

Gegenbauer, Wiener Sitzungiberichte, lxx.

(2),

(cosf),

(1874), p. 433; en. (2 a), (1893), p. 942.


370

and

[CHAP. XI

so '

Jv-\{z sin

/V sin

/Q v (9)'

(2;

sin ') 7-/ sin #)•""* 7

,.

T^ exp rL COg

V(2^T m? The

^

cog J/1J

r(2^

^

+ m)

~^— Cm"(cos^)C -(cos^). TO

integral used in the proof converges only

result is true for all values of

when

R (v) > 0, but the final

by analytic continuation.

v,

This result was given by Bauer, Miinchener Sitzungsberichie, v. (1875), p. 263 in the v=\; the general formula is due to Gegenbauer, Monatshefte fur Math, und Phys. x. 192 see also Bateman, Messenger, xxxni. (1904), p. 182 and a letter from (1899), pp. 189

case

—

;

Gegenbauer to Kapteyn, Proc. Section of pp. 584—588.

Sci.,

K. Acad, van Wet.

Interesting special cases of the formula are obtained by taking

and,

if

we put

<£'

equal to

multiply by eiZcos

£«-,

-^ I "j^^z sin

(10)

)e

iZc0ii

= 2" WK J(2n) so that the expression on the left

was given by Bauer

11 '6.

We (1 )

in the case v

>

't

I

(

m=o is

'l>

sin 2 "

and

Amsterdam, iv. (1902)

te

'

equal to

integrate,

we

or to

\n

;

find that

sm z ''d

**(*) Jy + Zm(Z)

Z"

2"

'

a symmetric function of z and Z\ this formula also

— \.

Bateman' s expansion.

shall

now

establish the general expansion

\z Jp (z cos

cos

)

Jv (z sin

(f>

sin <£) 00

= cos* x x

which

is

cos"

O sin"

.Tro> +

^ (-

n,

/u,

..

sin" 4>

2

(-)"(/*

+ i)(r(, + i))'

+v+n+ 1

valid for all values of

+v + 2n + 1) J^+H-zn+i (*)

n-0

/j,

;

v

+1

;

••

r'<-^" + -

sin2

1

—

'•

1; ' +1;

""'«

),

and v with the exception of negative integral

values.

Some

of the results of § 11*5 are special cases of this expansion, which was

discovered by

Bateman* from a

consideration of the two types of normal

wave motions examined in § 4'84. proceed to give a proof of the expansion by a direct transformation.

solutions of the generalised equation of

We

* Messenger,

xxxm.

(1904), pp.

182—188 Proc. London Math. ;

Soc. (2)

m. (1905),

pp. 111—123.

ADDITION THEOREMS

11-6]

deduce from the expansion

It is easy to

371

-

(§

5 21) of a Bessel function as a

series of Bessel functions that

\zJ

IL

{z cos

w,

(-)

'5

J¥ (z sin <£ sin

cos <$)

(^y

^ = cos* d> cos' 4> sin" .

.

2

cos*

,

•

[~(-)m cosaro tT1/

—6

"

2

•

.

cos2"1

AX <1>

,,

,

,

2^1 (- w,

= cos*

sin" 3>

r

+ v + 2m + 2n + 1) —^- n\T{v tr>/ + l)

-U/*

»-ol

x

^

.

4

x

)

+2m+1 (cos6cos>t+2m

p + v + 2ra + n + 1 sin"

<£

sin* 3>

; 1/

+1

;

«/mh-2to+»m-i(*)

,>.

sin2 sin 2 4>)i

2 m* + + 2n+l) JM+ „+2n+ i'

i

(2)

«-o L »

X

+ v + n + m + l) mS(n-w)!r(i/ + l)r(/* + m+l)

( (-)»»

«=ot x

2

cos'"*

cos2"1 &.r(fi

F (m-n,/A + i'4-m + w + l; 1

^ = cos* <6 cos*

.

.

x^

4

(-n,/*

.

sin*
i/

+ l;

sin2 <£sin2

<&)U

+ + —+ l)r(/* — + v+2» pru 1x r»/ ^ i\

"

r(/i

2

+ + w + l;

v

i'

n + l)

fi

+ 1, v + l;

cos2 <£cos2

,

/x

r

W.'+sn+i 1*7

,

sin2 <£sin2 )

,

where jp4 denotes the fourth type of Appell's* hypergeometric functions of two variables, defined by the equation

We now

have to transformf Appell's function into a product of hyper-

geometric functions in order to obtain equation (1)

we assume

formation

that

R

(fj.)

>

0,

;

in effecting the trans-

though obviously

this restriction

may

ultimately be removed by using the theory of analytic continuation.

a consequence of the following analysis, in which series are rearranged, and a free use is made of Vandermonde's theorem

The transformation

*®.$

is

+ + n + 1; fi+l, v + l; cos $ cos <>, sin sin 1 = £ "jf (~ n >*+* (/* + " + " + V+» cosM ^ cos^^« s n a s in»-
cos3

4

(-n,

2

2

/*

r -o«-o

5

i

»-

slr\(jjk+l),(v+l)r

^ (-nX^ +

"r-o.-o

2

2

i/

v + n + lU, ± r\(v+l) r t-o

(-)W^y> t\(s-t)l

- (-)«W-*»

„_

«!(/*+!).-

* Comptes Rendus, xc. (1880), pp. 296, 731.

This transformation has not been previously noticed to exist except in the special case in which #=#, see Appell, Journal de Math. (3) x. (1884), pp. 407 428 ; some associated researches are due to Tisserand, Annates {Mimoiret) de V Observatoire (Paris), xvni. (1885), in£m. C. •f-

—


372

2 (- n\ T «-r^K» +

[CHAP. XI

+ v + n + 1 ),+, (-)«+" sin* ft sin8" l)r(t-r)\(r + S- t)l (u-r)\ (fl+ 1 ),+,_„ "rÔ «-0 <- r (~ »Vh (ft + y + w + l)r+< (-)'+» sin ft sin8" = S I I (u - r) !(/* + l)r+,_ «-0 «-0 r-0 ,-t-r r !(» + !)r (t - r) \{r + s W+M 2" = I 5 i (- »)t Q* + y + n + l)t (y + < + tt + !),.< (-) sin ft sin r!(i/ + l)r («-r)!(tt-r)! + l)n _„ /-o«-0r-o ((- «), Q. + » + « + 1), + (--f « 1), a0_ ain„, - £ i 3ipM ^ w! *!(*/ + l) <-o«-o = V *2

+t (^

,

V

8'

!

tt

8f

(//,

t

= (-)w |*** ) V ^

I

+ * + n + l;

(-n,/*

v

+ 1;

sin'ft)

)

x/i(- /m — n, v + n + 1; v+1; = (~)n

(

*i\-cos^.^

1

(--n,

A*

+ + n + l;

v

+ 1;

F (-»

/i

+ » + » + l;

i/

x

Hence we \z «/M (z cos

ft

1

)

sin8 ft)

v

+1

;

sin8 <£).

at once obtain the result

cos

) «/„

= cos" ft cos* * sin" x(-)n

8

sin8 )

(# sin

ft

sin 4>)

am' *

»irfr + l)r( y + l)

r Tiv-^(- w>/* +,/+n + 1 1

x/jf-w./t + v+B+l;

j/

+1

;

'

"

+1

J

sin2

sin 8 <£),

from which Bateman's form of the expansion

is

evident.

^)

'

^^

<*>

CHAPTER

XII

DEFINITE INTEGRALS Various types of definite integrals.

12*1.

In this chapter we shall investigate various definite integrals which contain either Bessel functions or functions of a similar character under the integral sign, and which have finite limits. The methods by which the integrals are evaluated are, for the most part, of an obvious character; the only novel feature is the fairly systematic use of a method by which a double integral is regarded as a surface integral over a portion of a sphere referred to one or other of two systems of polar coordinates. The most interesting integrals are those

—

122 12*21, which are due to Kapteyn and Bateman. These no very obvious reason, seem to be of a much more recondite

discussed in §§ integrals, for

character than the other integrals discussed in this chapter

;

their real sig-

become apparent from the recent work by Hardy described in 12*22. and important types of integrals, in which the upper The numerous § limit of integration is infinite, are deferred to Chapter xm. nificance has

The reader may here be reminded of the very important integral, due to Sonine and Gegenbauer, which has already been established in § 11*41, namely f" <@v J

W(Z* + z>- 2 Zz cos )}„ w i* + *-2X»coB*r °

,

,

(C °S

x

*>

,, Sm „ *** .

a

,

= *r(2y + m)

<@v+m (Z)

Jv+m (z)

Z"

z*

'

2*- l .m\r(v)

12*11.

Sonine's first finite integral.

The formula

iWh (z) - 2

(1)

which

is

valid

,

r(y+1)

when both

R (/*)

J„ {z sin |

cos*"* 1 Odd,

6) sin"+'

o

and

R (v)

exceed

-

1,

expresses any Bessel

function in terms of an integral involving a Bessel function of lower order.

The formula was stated in a slightly different form by Sonine*, Rutgers and Schafheitlin+, and it may be proved quite simply by expanding the inte* Math. Ann. xvi. (1880), p. 36; see also Gegenbauer, Wiener Sitzungsberichtc, mcxxviii.

(2),

(1884), p. 979.

f Nietno Arehiefvoor Witkunde, (2) vi. (1905), p. 370. J Die Theorie der Bettel'tehen Funktionen (Leipzig, 1908), p. 31. Schafheitlin seems to have been unaware of previous researches on what he describes as a new integral.

V

'


374

[CHAP. XII

grand in powers of z and integrating term -by-term, thus

V* (z sin 0) sin* ^r~-^ l + 1) Jo I

+1

6 cos2 " +1

6d6

1 \V

= 2 ,^

= 2 1=0

n

2' i+, .

'

—m!r rWr

,_ +2w

is

m!r(/t +

i»

1- **

and

(2)

fi is

J*7,

A

(5 sin 6)

formula* which f

is

M (* sin 0)

*

1

cos'-"* 1

0d0

sin-- d cos2 >+> i/

= — £, we

V

M (2 sin 6)

factors

If we had taken ml. Hence, when

M0 - gH^fy

•

have

sin— 0d0 =

easily obtained

in the integrand

in the denominators.

we should have removed the unrestricted, we have

(-) f

(4)

2

effect of the factor sin'* +1

T (/t + m + 1)

In particular, by taking (3)

^ ™*

obvious.

6 as the factor,

R (v) > — 1

sin 2

I

+ w + 2)

is

be observed that the

to eliminate the factors

sin

tt

(—) m (A 2V + " +2rw+1

and the truth of the formula It will

m^

+ «» + l)T(v+l)j '

(/*

from (1)

J„ (, cos 0) tan *+i 0
-

H^

(z).

is

pî'V^ff J,

(*),

when R(v)> R (/*) > — 1. This may be proved by expanding /„ (z cos 0) and integrating term-by-term, and finally making use of Lommel's expansion given in

The Fp.

and

§ 5*21.

functional equation, obtained from (1) by substituting functions to be determined, „ + 1 in place of the Bessel functions, has been examined by Sonine, Math. Ann.

FM +

,

LIX. (1904), pp.

Some

529—552.

Lombardo Rendiconti,

p. 92.

[Scientific Papers,

I.

(2)

xin. (1880), p. 331, and Rayleigh, Phil.

Mag.

(5)

xn. (1881),

(1899), p. 528.]

It will be obvious to the reader that Poisson's integral by taking /*=-£.

For some developments of the formulae of this papers by Rutgers, Nieuw Archie/ voor Wiskunde, (2) pp.

have been given by Beltrami,

special cases of the formulae of this section

Istituto

is

the special case of

(1)

obtained

section, the reader should consult vi. (1905), pp.

368—373

;

(2)

two

vn. (1907),

88—90.

12*12.

An

The geometrical proof of Sonine's first

integral.

instructive proof of the formula of the preceding section depends on

the device (explained in § 333) of integrating over a portion of the surface of a unit sphere with various axes of polar coordinates. If

(I,

m, n) are the direction cosines of the line joining the centre of the *

Due

to Rutgers,

Nieuw Archie/ voor Wisknnde,

(2)

vn. (1907), p. 175.

1

DEFINITE INTEGRALS

12*12]

dm whose

sphere to an element of surface

375

longitude and co-latitude are

and

evident from an application of Poisson's integral that

0, it is

*

r (fi + \) V (\) (|*)"+1

f

J, (z sin 0) sin* +1

cos2* +1

0d0

Jo

= (i*y* +r+1

f

I

Jo

= (hzY + * +l

%*«»•«>• * sin2* +1

cos» +1

sin 2" <£ctyd0

.'0

e

1

w m * n ' +1 2

2

•>Jm>0,n>0

JJl>0,m>0

V

= (i*>t+, +1 ff JO '

008 '

sin*+f,,+1

2r<^+ y +t)

cos2*

sin2" +1
Jo

= K(/*+*)r(z/+i)r(j)/M+ ,+1 (^), and the truth of Sonine's formula

An

integral involving

same device*

is

obvious.

two Bessel functions which can be evaluated by the

is

fin

J

v

{z sin2 0)

J

v

(z cos2 0) sin2 "* 1

cos2 "+*0d0,

Jo

R(v) >

in which, to secure convergence, If

we

write

w 2 = sin4 + cos4 - 2 sin2 we

and use § 11*41 (16),

r( y

cos2

2*+»r(v+i)r(*)J (iff (£*)"

*

F«r^+T) io*" ^ so that finally,

1211

§

2

f*Jw (z sin

(1)

* This integral

kunde,

by

(2)

0)

(*

^^^ sin

*

sm 2
JA*
[[

f

,

equal to

is

(i-sin 2 ^cos2 ^)*"

J#

2»+*r(*+i)ra)

= 1 - sin2 20 cos2 f

sin4y+lffco84y+1 ^ sin2

/T 2^ I> + £) r(J).//, JJô,«>o

=

cos

see that the integral

fâ) /oT^ =

— \.

J„{zJ(l-n*)} (1

-n )*" 2

'

J

J-(zsinO ) s

^ sin

' +1

2

"n*v da>

^+l6cos

2 ''

'

o

sin

l

"

* cos2

<

>coa,i ''

t

dd0d

w

(1),

Jw (z cos

2

2 0) sin "+>

cos2"+' 0<*0

-

^M^lffi

'

has been evaluated by a different method by Rutgers, Nieuw Archief voor Wis-

vn. (1907), p. 400;

cf.

ateo § 12-22.

'

,

,

,


376

;

[CHAP. XII

Some integrals which resemble this, but which are much more difficult to evaluate, have been the subject of researches by Bateman, Kapteyu and Rutgers; see § 12 2. -

As a simple example of an integral which may may prove that, when R(v)> —

reader

be evaluated by the same device, the

•&,

(a* ^

- «»)*" cos t Iv U(x* - 1*)} dt = 2y+ ?j. (v+ |) -

.

by writing the integral on the 2" +

the form

left in

V

ir(v + ^)r(i) Jo Jo

This formula was given (with r=0) by Bdcher, Annals of Math.

12*13.

vm.

V (1894), p. 136.

Sonine's second finite integral.

The formula

ly^san 0)J

(1)

which

¥

is

and, in

when both R(fi) and

valid

fact,

(Zcos 6) sin^-5 cos'+'fld* =

A simple method of proving the

§

making

of the equation by Z" and then

and

-R(v) exceed

he obtained the formula of

^4^Mg+f21

- 1,

1211 from

also due to Sonine* by dividing both sides

is

it

Z -*-0.

is to expand the integral in powers p + v + 2m on the left combine to form

formula

to verify that the terms of degree

of z

and

Z

(-y**z v (Z2+z2 m ml r(p+v + m + 2) )

The proof by

We

method

this

is left

to the reader.

proceed to establish Sonine's formula by integrating over portions of

Under the hypothesis that R(/m) and R(v) we have

•the surface of a unit sphere.

exceed

— £, we

* r(

see that, with the notation of § 12 12, !

k"

+ *)

a*wfivu*Y(hzy Jo

I

J» (z sin e) J" (z cos e) sin,t+1 e cos

eizl+iZn cos *

\\

\

e dd

oJo

.

V

—

" +1

m^ n2»+i 8 Q [

i>>

yf,

'oii?n>0,n>0

da

^

J OJ J «>0,J>0

J

JO.'

-Jir

isin9

e -r/f JO m>0/

(zn+ZQ

m

*>

C0S^^ Sin2"** da> dO

J.

*»1 Qi sin

(zl+ Zm) rfv

C0S2M

s i n2^+2 ff fa &Q

Jo JJ» >0

= I

/

/

"^ 8in 8sin * ( * °°" * +Z sin w cos2

Jo Jo Jo *

Math. Ann.

^ S ^n ^ cosi" ^ sm2 1

''

xvi. (1880), pp.

35—36.

" +2

^

^

d d0.

DEFINITE INTEGRALS

12-13]

377

the exponential function involved here is a periodic analytic function of 2ir, and so, by Cauchy's theorem, the limits of integration with respect to yfr may be taken to be a and 2-rr + a, where a is denned by the

Now yfr

with period

equations vr cos

= V(£ + *"). a

and

or

i/r -f

a for

a

= Z,

a

ts sin

these limits of integration, and then write

we adopt

If

= z,

the triple integral becomes

-\fr,

i

2i.^)S i I1< cog?n0 sin** + *0dyfrdd0, eim sin 0sin*co8* coa ^

i

J

Jo Jo Jo

and this integral may also be obtained from its preceding form by replacing z by «r and Z by zero. On retracing the steps of the analysis with these

we reduce the

substitutions

o eiw sin cos * s in2M+i

triple integral to

Cos2" +1 6 sin2** <£ sin 2 " yfr d dyfr d$,

io Jo Jo

= r(y+|)r(j)

_

rr

giw?

r(j/+l)

JJ m >0,n>0

F(v +

JJl>Q,m>0

1)

£i!i±i)_£li} f'f

V^

r
CO9

m^ w2 +1 dw „

'sin2* +2 " +2

COS2*

r

f' e

sin^+2,+2

iwcos<»

<^m

2 " +1

<£<&*> c£0

^^

Jo

= ^r (/i+i)r(, + i)^±^r)

,

and we obtain Sonine's formula by a comparison of the

initial

and

final

expressions.

Sonine's

own proof

of this formula was based on the use of infinite dis-

continuous integrals, and the process of making

it

rigorous would be long and

tedious.

The formula may be extended

— \> R (v) > — 1, by

and

to the

domains in which

- \> R (n) > - 1,

analytic continuation.

In Sonine's formula, replace Z by *J(Z* + C2 - 2Zfcos0), multiply by sin <£/(£* + ^ -2££ cos 4>)* v and integrate. It follows from § 11-41 (16) that 2"

,

Jn (zain 0) J„(Zcos6)Jv (£cos 6) sin^tfcos Odd

"

(2)

[ °

«**{?

_

f

•

~*T{v + \)T{\)) 9

J^tv^ + g+r-gggcoBfl) (z^

+ Z' + ^-2Z^co8)i^"^

9tt
'

provided that JR( M )>-1, This result

is

also

due to Sonine,

R(v)>-$.

ibid. p. 45.

In connexion with the formulae of this London Math. Soc. xxxv

section the reader should consult Macdonald's memoir, Proc. (1903), pp. 442,

W.B.F.

443 13

^

^

(


378 12*14.

An

[CHAP. XII

Gegenbauer's finite integral.

somewhat resembles the

integral which

first

of Sonine's integrals,

namely fw cog I

(z cos

.

j o

cos

s(r)

sin

Jv_i (z sin

sin ty)

C/ (cos 0) sin w+ * 0d0,

has been evaluated by Gegenbauer*; we shall adopt our normal procedure of using the method of integration over a unit sphere. It is thus seen that l" e iz cos cos*

=

^tz sin

(

r

)

J

>^V

(z

—^

rm

gin

r™'

f*

sin e* s

^ C/

(cos ' °° s

( cog

* +sin ' sin * cos * )

J

/

= ^ffit?,7 1

ff

= (^smjr£-i

ff

(

e te <»«»*+*"to+)

l*)! (j) JJm>0

V") i Vi)

1 f

z

1

«?in

Airy-*

W*

(f) Jo Jo

(A 2 sin iM"~* -rr / xiT/ix

=

f 2ir

r (")*(£)

/

** sin * cos *

CS

If

we

Isin

^ cos A) cos3 ,-i Q s in 0d<6d0

cos

(

+ yfr)} cos "" 2

1

d
sin

is

a periodic analytic function of

ff J-

with

e

I (") A (i)

ia

is

equal to

Cr v (I cos yfr-m sin ^n'^dco

«>o

^Sm ff4

* (I JJwi>0

izn

0/ (n cos yfr- I sin &) m "" 2

1

da>

= (tf Sm J ?!T ("r^^^/Ccos^cos^-sin^sin^cos^sin^^sin "r

2

/

r(v)

I (£)

J

1

^^^.

,'o

Now, by the addition theorem f

Cr " (cos yfr cos — sin yjr sin

cos

for

Gegenbauer's function,

)

X *

retrace the steps of the analysis, using the last integral instead of its

*

=(

^

2ir.

i??ptC {?) (?) 1

° sin2,"_1 ^

n2l,_ ld6>

immediate predecessor, we find that the original integral (

"

Jo Jo

since the penultimate integrand

period

^

efa«in»coB(*-« (J. ( s i n

f 2w

/"*"•

^ sin2

cos

1

el>(?COSl/(+W8in(
ft"'

^

0/ (n) m2- d«

J.'»>o

= ^/VrT/ix A

0) sin „+i 0de

C£j (COS 0) C; +_ pP (COS *) 0;-» (COS +).

Wiener Sitzungsberichte, lxxv; (2), (1877), p. 221 and lxxxv. (2), (1882), pp. 491—502. was proved by Gegenbauer, Wiener Sitzungsberichte, lxx. (2), (1874), p. 433; en. (2a),

t This

(1893), p. 942.

'

DEFINITE INTEGRALS

12*14, 12'2]

When this is multiplied by sin * -1 2

of the

sum

vanish except the

r\r(2v) ( ?"l n {Zv /o + r) !

Qr

and integrated, all the terms of the integral which is

first

C/ (cos yjr)

(cos 6)

L

We

379

""1

! 'sin 2 Jo

d.

thus find that

*

eteoos«coS *

|

j

{z si

v s i n ^.) (7r ( cos #)

n

Jo

= r! and hence, by (1)

§

^ ,^ 2"^

(

p(

S1

)

(

"*

C/(cOSVr) f

)

'e^costfcos * I

j, (^ s n # i

gi

n

^ =

(2)

we equate I

V

008 '

0/(008 0)8^0^,

332,

Jo

If

sin'+i^d^

real

and imaginary

Odd

(7^ ( cos #) gin^+i

v sin*-*

(y )

parts,

co8(*cos0cos^) J,^ (2 sin

^ 0/ (cos f) Jv+r («).

i

we obtain Gegenbauer's formulae

sin >/r)C>* (cos

^sin'"^^*?

Jo

_

r

f(-)*

(

—

)*

sin""1

1 C/ (cos -f Jp+r (z), )

(r even) (r odd)

(0

and (3)

f "sin (z cos Jo

cos

<^>)

J^ (z sin

sin

yfr)

Gr " (cos 0) sin"+* 6d0 (r even)

{0,

/^V sin- f C

(_)*

12*2.

-t

r

"

(cos

(r odd)

^) /„+,.(*).

Integrals deduced from Bateman's expansion.

In Bateman's expansion of

§ 11*6,

write

=

;

and then, noting Jacobi's

formula*

{^ (-w, + i/+n+l;i/ + l; sin

2

/*

2

2

<£)}

cos2" +1

sin 2 "+1 <£cty

.'0

(/*

we deduce (1)

Jo

«/M

* Journal

1/

when R(fi) and

12

(scos8 <£) «/,(* sin2

<£)

that,

21

nir^ + n+ixr^+i)}* + + 2n + 1) T (fi + v + n + 1) T (v + n + 1)

fUr Math.

lvi. (1859), pp.

(i>)

sin

both exceed

cos <£<&£

149—175 [Werke,

—

1,

= 2 n=0

(— )n «/M+ „ +2 l+

vi. (1891),

,

i

(z),

pp. 184—202].

/


380 that

,

[CHAP. XII

to say

is

(2)

J(

V, (t) J„(z-t)dt = 2

I (-)» J^ +m+1 (z).

»=0

An important deduction from this result is that, when R (fi) > 2M

^ (0 /„

Jo

-

(,

y = J o {^-i (0 + J, +

(VM (0/,<>-of=

J0

This formula

is

due

I

Bateman*; some who considered

to

dt

-2

M+ „ (#),

fi.

special cases

independently by Kapteyn f,

had been obtained fi and v only.

integral values of

we can deduce from

It will be observed that

1)

—1

—

so that (3)

^ (* -

(01

i

and R (v) >

(2),

combined with

§ 2*22 (2),

that (4)

|

Jo

when

— 1 < R (/*) < 1,

By and

J^ (t) J_M (fc ~

interchanging

dt

= sin z,

\

J

fi

(t)J1 _ (z — t)dt = Jo(z) ll

— cosz,

Jo

and when with v

fi

— 1 < R (p) < 2 respectively. and t with # — t in (3), we see

that, if

R (/*)

both positive, then

J? (v) are

Jo

t

'

—t

Z

v)

\fi

z

It seems unnecessary to give the somewhat complicated inductions by which Kapteyn deduced (3) from the special case in which fi = v = l, or to describe the disquisition by Rutgers J on the subject of the formulae generally.

12*21.

Kapteyn 's trigonometrical integrals^.

A simpler formula

than those just considered

Pcos (z - 1) J (0 dt

(1)

= zJ

is

(z).

Jo *

To prove

this,

we put the

left-hand side equal to u, and then

it is

easily

verified that

d?u

T

,

x

and therefore

u = zJ where

A

and

B are

(z) 4-

A cos z + B sin

z,

constants of integration.

* Proc.

London Math. Soc. (2) in. (1905), p. 120. Some similar integrals occurring in the theory examined by the same writer, ibid. (2) iv. (190G), p. 484. t Proc. Section of Sci., K. Akad. van Wet. te Amsterdam, vn. (1905), p. 499; Nieuw Archie/

of integral equations are

(2) vn. (1907), pp. 20—25 ; M6m. de la Soc. R. des Sci. de Li€ge, J Nieuw Archiefvoor Witktinde, (2) vn. (1907), pp. 385 405. § M4m. de la Soc. R. des Sci. de LUge, (3) vi. (1906), no. 5.

voor Wiskunde,

—

(3) vi. (1906), no. 5.

DEFINITE INTEGRALS

12-21]

Now, when z

small,

is

u-z+

A = B = 0,

and so

and the result

by

It follows from (1)

differentiation that

(t)

.

dt

«=

zJx (z),

Jo

by a

partial integration,

(3)

(z-t). Ji (t) dt =

sin

I

am z — zJ

(z),

Jo

The formula Psin (z - 1)

(4)

which

(z>),

established.

is

P sin {z - 1) J

(2)

and,

381

Jo is

valid

when

dt

= -l

R

> 0, is

(fi)

is

write

of a

more elaborate

required to prove v

(-)" JM+an+1 (z),

/*«=<>

*

of the preceding section

We

^

character,

and the

result

it.

= V, (z - t) J* (t) dl, \

Jo

and then we have

^ + v-\yo"(z-t) +

Jo(z-t)}J»(t)dt + j;(z)

{' Jl

=

^r-Mt)dt+j;(z) z—t

Jo

= \*J»(z-t)l±P-dt + j;{z) Jo

t

by §122.

By

the method of variation of parameters

_

v

v

and, since

when z

.

= A cos z + B sin z + /*

is small, it

-

f* I

when

Hence we obtain the required

(5)

"cos (*

we deduce

,/,({)

—*-

,

dt,

t

(** +8 ),

R (fi) > 0, 0.

result.

differentiating (4) with respect to z

f

.

.

sin (z - t)

^ T(fjb + 2) +

follows that,

7*33),

Jo

A=B = By

.

(cf. §

- «>^~

dt

=- I

we

find that

(-)" 6W J"^^ (4

that


382

Hardy's method of evaluating finite integrals.

12*22.

As a typical example of a very powerful method of evaluating now give a proof of the formula (cf. § 12*12)

is

valid

when

The method it

finite integrals*,

we shall

l*W&t)J.Wt>&»0^+*Mt-$$g^y ±^,

(!)

which

[CHAP. XII

is

R (/*)>-

•£

and

R (v) > - £

more elaborate than any other method described in

this chapter, because

involves the use of infinite integrals combined with an application of Lerch's theorem t

on null-functions. iV

(

Let

J^(zri 8w*e)Jv (si*coa*6)r* + ** +3 sin» + i6coai + i0d03fl ''

By changing from $ 13*2 (5)

we

see that,

polar coordinates

|,

|

exv(-r*t).f1 (r)dr =

f"

(r,

whenever t > / («) exp

j

(r) t

6) to Cartesian coordinates (x, y)

and using

then

{-^J^zx^x^^dx

f

f*v(-y*t)Jn(zy*)y* + 1 dy

4ir(ti +z*)* + '' +1

=

j\xV (-r*t).Mr)dr,

and hence, by an obvious modification of Lerch's theorem, /2 (r) and this establishes the truth of the formula.

f

t

identically equal to

(r) is

;

12'3.

A

Chessin's integral for

Y

n («) has been obtained by Chessin, American Journal, xvi. 186—187, from the formula

curious integral for

(1894), pp.

V I

if

we

Y» (*).

+ n—+—m + 5+... 2 ;

:

,

I

Jo

i

— ,*

dt;

substitute this result in the coefficients of the ascending series for

T« («), we obtain

the formula in question, namely

T. W -»(r+*W.r.W-y »

(„

i

-/ I

must express

my

the publication of his

many

2Jn (z) - (<-*»+<*») J„ l-t

)

'""»r--

{z sit)

1)

'

d

thanks to Professor Hardy for communicating the method to me before of it. The method was used by Ramanujan to evaluate

own developments

curious integrals ; and the reader

may

use

it

to evaluate the integrals

examined

earlier in

this chapter.

t Acta Mathematics xxvn. (1903), pp. 339—352. that, if /(r) is a continuous function of r

when

r

>

0,

The form

of the theorem required here is

such that

exp(-J-2t)./(r)dr=0

/:o for all sufficiently large positive values of

t,

then/(r)

is

identically zero.

CHAPTEK

XIII

INFINITE INTEGRALS Various types of infinite integrals.

13*1.

The

subject of this chapter

is

the investigation of various classes of infinite

integrals which contain either Bessel functions or functions of a similar character

under the integral very numerous (I)

;

sign. The methods of evaluating such integrals are not they consist, for the most part, of the following devices

Expanding the Bessel function

in

powers of

argument and

its

inte-

grating term-by-term. (II) Replacing the Bessel function

by

Poisson's integral, changing the order

of the integrations, and then carrying out the integrations. (III) Replacing the Bessel function

by one of the generalisations of Bessel's changing the order of the integrations, and then carrying out the integrations; this procedure has been carried out systematically by Sonine* in his weighty memoir. integral,

(IV) When two Bessel functions of the same order occur as a product under the integral sign, they may be replaced by the integral of a single Bessel function by Gegenbauer's formula (cf. § 12*1), and the order of the integrations is then changed j\

(V)

When

two functions of different orders but of the same argument may be replaced by the integral of a single Bessel function by Neumann's formula (§ 5*43), and

occur as a product under the integral sign, the product

the order of the integrations

is

then changed.

(VI) The Bessel function under the integral sign contour integral of Barnes' type order of the integrations

is

(§ 6*5)

involving

then changed

;

may be

Gamma

replaced by the

and the method has not

functions,

this very powerful

previously been investigated in a systematic manner. Infinite integrals involving Bessel functions under the integral sign are not only of great interest to the Pure Mathematician, but they are of extreme importance in many branches of Mathematical Physics. And the various types

numerous that it is not possible to give more than a selection of the most important integrals, whose values will be worked out by the most suitable methods care has been taken to evaluate several examples by each method. In spite of the incompleteness of this chapter, its length must be contrasted unfavourably with the length of the chapter on finite integrals. are so

;

—

* Math. Ann. xvi. (1880), pp. 33 60. t This procedure has been carried out by Gegenbauer in a Wiener Sitzungsberichtc.

number

of papers published in the


384

[CHAP. XIII

The integral of Lipschitz, with HankeVs generalisations.

13'2.

was shewn by Lipschitz* that

It

lyMJ°M dt ~7(^-V

(i)

R (a) > 0, and, in

where

order to secure convergence at the upper limit of in-

numbers R(a ±ib) are positive. That value of the square taken which makes a + V(aa + &) > °

tegration, both the

root is

\

|

The

simplest method

by

coefficient

I-

I

of establishing this result

Parseval's integral (§ 2*2)

—a procedure which may be

tegrations

to replace the Bessel

is

and then change the order of the justified

without

difficulty.

It

is

in-

thus

found that ["«-•« J» (bt) dt

= - f°e- at fV"08 * dddt ~~ ir

Jo a

= l/V(a

8

and the formula

Now

is

— ib cos & +fe2 ),

proved.

consider the more general integral

re- at Jv {bt)t»-

l

dt.

Jo first investigated in all its generality by Hankelf, in a memoir published posthumously at about the same time as the appearance of two papers by Gegenbauer \. These writers proved that, if R (fi + v) > 0, to secure convergence at the origin, and the previous conditions concerning a and b are satisfied, to secure convergence at infinity, then the integral is

This integral was

equal to

(&b/q)T (/* + *) F (fi+v 8 oT(i/ + l)

M~2~

To |

b

|

|

+ v+l 2

establish this result, first suppose that b

< a

term,

i

f '

|.

we I

If

we expand the integrand

fV + .

_^\

-

1 '

a>)'

further restricted so that

is

in powers of 6

and integrate term-by-

find that

e-**J¥ {bt)tr-*dt=

2

_ ^ "~

\r/ lio.iv (-)*" (& bY +3m

m .o m r (v + m + !

* Journal JUr

Math.

lvi. (1859), pp.

1)

/

^ + ' +am_1 *~ at *

r (/x + v + 2m) a" + " +aw

191—192.

t Math. Ann. via. (1875), pp. 467—468. 433—443; ibid, lxxii. (2), (1876), pp. 343—344. J Wiener Sitzungsberichte, lxx. (2), (1875), pp. in §3-32; in the la the former, the special case M =i»+1 was investigated by the integral given integral for J,{bt). Poisson's substituting result by general the obtained Gegenbauer latter,

;

:

INFINITE INTEGRALS

13'2]

The

converges absolutely, since

final series

term-by- term integration

|

b

385

< a \

\

and

|,

so the process of

Hence

is justified*.

re-otJAtyP-'dt

(2)

(Wayr(fi + v) The

+v

fp

•M

a"I>+l)

2

/*

+ +! !/

been proved only when

result has, as yet,

.

'"

2

'

6'\

+1

a*)'

'

R (a) >

and

I

b

< a

\

\

|

but, so long as merely

R(a + ib)>0 and R(a-ib)>0, then both sides of (2) are analytic functions of b analytic continuation, (2)

;

and

so,

by the principle of

true for this more extensive range of values of

is

Again, by using transformations of the hypergeometric functions, (2)

6.

may

be written in the following forms

re-rtJrWV-'dt

(3)

Jo

V + a»)

a*l>+l)

»//* + *

(Wrfr+»)

~ (& + »>}<'+') r(v + l)* The formula

(2) has

;

Beasel functions,

r «-«*r (60^" r

=cot

1

2

\

i-/» + '

2

»

„.+

.

'

Lamb J may

by

"V + &*)*<—

/? (>*)

>

|

.ff

(v)

London Math.

|

\

aa + 6V*

it is

be regarded as

typical.

easy to deduce that

r(„+i)

aFl

l"T"»

"a-

v+1, '

-cosec^^^^^,^^^^^, —g— Proc.

'

*)

* "

provided

ft

i.

'

special cases of (2) are required in various

physical researches, of which those

(4)

l

+1 '" + 1

'2

2

been used by Gegenbauerf in expressing toroidal

functions as infinite integrals

By combining two

»M

and A (a ± io) > 0;

,

1

*+W ,,

a2 + ft2J,

due to Hobson, and Heaviside, Electromagnetic Theory, in.

special cases of this formula are

Soc. xxv. (1892), p. 75,

(London, 1912), p. 85. It is obvious that interesting special cases of the formulae so far discussed

may be

* Cf.

Bromwich, Theory of Infinite Series, § 176. t Wiener Sitzungsberichte, c. (2), (1891), pp. 745—766; Gegenbauer also expressed series, whose general terms involve toroidal functions and Bessel functions, as integrals with Bessel functions under the integral sign.

ZProc. London Math. Soc. xxxiv. (1902), pp. 276—284; (2) vn. (1909), pp. 122—141. See London Math. Soc. xxxv. (1903), pp. 428—443 and Basset, Proc. Camb. Phil. Soc. v. (1886), pp. 425—433. also Macdonald, Proc.


386

[CHAP. XIII

obtained by choosing

/* and v so that the hypergeometric functions reduce to elementary Thus, by taking p equal to v + 1 or v + 2, we obtain the results

functions.

r.~,. Wr *_.2B£fe+ii, +

(5)

(a 2

Jo

fV^,(tor^,

(6)

provided that

R(v)> —$, R(v)> — 1

+ o2 )*

*

Jir

^ ^) r +

te

,

respectively.

These formulae were obtained by Gegenbauer, Wiener Sitzungsberichte, lxx. (2), (1875), 443; they were also noticed by Sonine, Math. Ann. xvi. (1880), p. 45; and Hardy, pp. 433 Trans. Camb. Phil. Soc. xxi. (1912), p. 12 while Beltrami, Atti della R. Accad. delle Sci. di Torino, xvi. (1880—1881), p. 203, and Bologna Memorie, (4) n. (1880), pp. 461—505, has obtained various special formulae by taking /*=»1 and v to be any integer.

—

;

Other special formulae are (7)

J".

-« •/-,(*) dt 7-

y(d?+W)- ay vb ¥

w( a *+ b2 )-<*r

r

e-«
(8) (8)

[Note.

(4) viii. (1887), pp.

125

— 143,

that these integrals are derivable from the generalised form of Bessel's integrals (§ 6*2) by Laplace's transformation (cf. § 9 15). This aspect of the subject has been studied by Macdonald, Proc. London Math. Soc. xxxv. (1903), pp. 428 443, and Cailler, Mem. de la #

Soc. de Physique de Geneve, xxxiv. (1902

by

satisfied

(5)

and

integral

— J^bt)

-^—j-

/

J

o

The

differential equations

qua functions of a, have been examined by Kapteyu, Archives (1901), pp. 103—116.]

(6),

Ne'erlandaises, (2) VI.

The

— — 1905), pp. 295—368.

tdt

sinn, itt

was obtained by Neumann, Journal fur Math, lxxii.

a limit of a series of Legendre functions (cf. § 14 64). The integral does not seem to be capable of being evaluated in finite terms, though it is easy to obtain a series for it by using the expansion -

(1863), p. 46, as

COSech»T< = 2 2

5 -(2n +

A series which converges more rapidly (when Some

is large) will

be obtained in § 13*51.

same general type are given by Weber, Journal fur Math. lxxv. 92—102; and more recently the formula

integrals of the

(1873), pp.

r Jy(bt)rdt = (2b)T(

(9)

Jo which

b

l),rt

is valid

when

e«t-l

R (») >

v

+$

» )

Jn

and 1(b) 1

\

<

»r,

1

«=i(»2«-* +

6y + i'

has been obtained by Kapteyn, Mem. de la

Soc. R. des Sci. de Liege, (3) vi. (1906), no. 9.

13*21.

The Lipschitz-Havkel integrals expressed as Legendre functions.

was noticed by Hankel that the hypergeometric functions which occur in the integrals just discussed are of the special type associated with Legendre functions; subsequently Gegenbauer expressed the integrals in terms of toroidal functions (which are known to be expressible as Legendre functions), and a little later Hobson* gave the formulae in some detail. It

* Proc.

London Math. Soc. xxv.

(1893), pp.

49—75.

.

INFINITE INTEGRALS

13'21]

387

we

obtain the fundamental formulae* of this type,

To

shall

change the

notation by writing

a

6

is

we thus

obtain the formula

= T (fi + v +

f°V
(1)

provided that

The

R

(fi

+

v)

>—

(2)

1)

PM-' (cosh a),

1.

special case of this formula in

Math.

= i sinh a,

a complex number such that

where a

Set.

— cosh a,

which i»=0 had been given by Callandreau, Bull, des

xv. (1891), pp. 121—124, two years before

Hobson published the general

formula.

It folio ws at once from (1) that

f%-<»^,g,(*8inha)P'
(2)' v

Jo

provided that

The

R (p + 1) > R (v) j

sin

^^ + {ft

T f> - v +

v) it

1)

Q/ (cosh a),

j

modification of (1) which has to be used is positive and less than 1 is

when

the argument of the

Legendre function f

[%-<«»*

(3)

jy (j s n £) p. dt = T (fi + v + 1) Pf (cos £), i

Jo

and hence we

find that

[V ""gy, » T^

(4)

Jo

.

(tain V

~x

t

= £)*»(& '

-

sin u7r

7 , Sin(/A+1>)7T •

,

•

r (^ — v + 1)-^

x [Q/ ( cos # + °») eivni

7T

+ &" (cos £ - Ot) «-*"**].

Some special cases of this formula have been given by Hobsou, Heaviside, Electromagnetic Theory, in. (London, 1912), p. 85.

An

and by

loc. cit. p. 75,

apparently different formula, namely

[**'*«** I () IA)

(5) (5)

Jo

cfr

Vt

= &-j(cosh«) V^tt)

'

has been studied by Steinthal}. This formula is connected with formulae of the previous type by Whipple's § transformation of Legendre functions, which * Since,

by a change of variable, the integrals are expressible in terms of the ratio of b to a, no The various expressions for Legendre functions as hypergeometric series which

generality is lost.

—

97 204. are required in this analysis are given by Barnes, Quarterly Journal, xxxix. (1908), pp. The reader will remember that it is customary to give a different detiuition for the Legendre

f

function in such circumstances p.

471

;

and Modern Analysis,

;

cf.

Hobson, Phil. Trans, of the Royal Soc. clxxxvii. A,

§§ 15'5, 15*6.

X Quarterly Journal, xvm. (1882), pp. 337—340. § Proc. London Math. Soc. (2) xvi. (1917), pp. 301—314.

(1896),


388

[CHAP. XIII

expresses a function of cosh a in terms of a function of coth

general formula of the same type /"•

,

^*IT .

e- tt

(6)

9

(t)V- i

Jo

1-

Q*

cosvtt

,

*

(cosh a)

dt^-r—/ r- ,'"\. sm (m + y) 7T V(i7r) smh"-*1

R (/* + v) > — v in

replacing v by

(6),

and

22 (cosh a)

we

find that

>

1.

sinn*-* a

Jo

and

.

,

•

In these formulae,

On

The more

a.

is

when

this formula is valid

If we take cosh

a— 0, we

R (fi) > R (v) |

|

and

R (cosh a) > — 1.

deduce that

p KAt)^dt^^r(^y(t^y

(8)

o

a result given by Heaviside* in the case v—0.

When

ft=l,

becomes

(7)

*^¥-a r ••*«+* K ®dt—^~ vn smh a

(9)

9

sin

Jo

and hence,

if

,

v=0,

—a±

v

„\

j,_ arcsinh

s/(a 2 - 1)

arc sin ^(1

- a 2)

arccosa

/, If

we

a by + i'6, we

replace

/'"

and

so,

when / (b) |

|

<

find that

io*-

(io)

The former

of these

is

j^.

due to Basset, Hydrodynamics, n. (Cambridge,

1889), p. 32.

[Note. Various writers have studied the Lipschitz-Hankel integrals from the aspect of potential theory to take the simplest case, if (p, , z) are cylindrical coordinates, we have ;

e-Pt J
/;o

1 2 V(p +*8)'

J

It is suggested that, since e~# (zt) is a potential function, the integral on the left is a potential function finite at all points of real space except the origin and that ont the plane

3«=0

it is equal to 1/p, and so it is inferred that it must be the potential of a unit charge at the origin. But such an argument does not seem to preclude the possibility of the integral being a potential function with a complicated essential singularity at the origin, and so

this reasoning

must be regarded as

suggestive rather than convincing.

* Electromagnetic Theory, in.

t

On

the axis of

z,

the integral

(London, 1912), p. 269. is equal to a constant divided by z |

\

INFINITE INTEGRALS

13-22]

389

For various researches on potential theory with the aid of the integrals of this section, may consult Hafen, Math. Ann. lxix. (1910), pp. 517—537. For some develop-

the reader

ments based on the potential function e-*>

j~Jo[k J{(x- *)2 +y2 }]/(0 dt,

see Bateman, Messenger, xli. (1912), p. 94.]

13*22. Applications of the addition formula to the Lipschitz-Hankel integrals.

deduce from the results of the preceding sections combined with § 11-41 (16) that, if all four of the numbers R(a± ib ±ic) are positive and R (/* + 2v) > 0, while tsr is written in place of V(&2 + c9 — 26c cos ), then It is easy to

re-otJ, (bt) Jv (ct) P-

(1)

1

<**»

7ra^+4

dt

\'r~W>p«~ sin- +
["

, wr(/i+2>>

1

r

T(2i/+l)J

// 2

X

i±

V

The hypergeometric function reduces and so we have

/;^<^,
(2)

The

case

/i

= 2 may

^ /+ 2, +

2

i

_^\

T

2

'

aV

'

'

to an elementary function if

sin8

fi

=1

y

# ^

or 2

^^p2^)-

be derived from this by differentiation with respect to

a.

These formulae, or special cases of them, have been examined by the following writers: Beltrami, Bologna Memorie, (4) n. (1880), pp. 461 505 ; Atti delta R. Aecad. delle Sci. di

—1881),

—

201—205; Sommerfeld, Konigsberg Dissertation, 1891; Gegenbauer, Monatshefte fur Math, und Phys. v. (1894), p. 55 and Macdonald, Proc. London Math. Soc. xxvi. (1895), pp. 257—260. Torino,

xvi. (1880

pp.

;

By taking

/x=

— 1, v = l

in (1),

(" - a Jll?±dt = e t Jo

*

t

so that the integral on the (1896), p.

An (3)

integral which

cos at

j

I

[\/( a 2 + 2-2cos0)-a}(l+cos0)cty,

which was encountered by Rayleigh, Phil. Mag. iv. (1904), p. 260], is

may be (bt)

find that

2ir J o

left,

195 [Scientific Papers,

we

K

associated with (1)

(ct)

dt

= jo

o

expressible as

v{a2

an

(5) xlii.

elliptic integral.

is

+ (6 + c)2 _ 46csin2

-

}

This was discovered by Kirchhoff * as early as 1853 the reader should have no difticulty in deducing it from § 1321 (10) combined with § 11-41 (16); it is valid if all the numbers ;

R(c±b± id) are positive. * Journal

fUr Math. xLvm. (1854), p. 364.


390

A somewhat similar result,

namely

v e- at f- J^{bt)Jv {ct)dt

(

(4)

= W($c)T(2;i + l f' r(i/ + £)r(|) Jo

sin 2 "

)

which

valid

is

[CHAP. XIH

R (a±ib±ic)>0

when

(a

2

d

+ 2mccos<^-c2 cos2 ^>+6 2

and R(fi)> —\,

due

is

)'

t

+i'

to Gegenbauer,

Wiener

Sitzungsberichte, lxxxviii. (2), (1884), p. 995. It is most easily proved integrals of Poisson's type for the Bessel functions. In the memoir cited

also given a list of cases in

functions

is

§ 1323).

(cf.

Gegenbauer s deductions from the integrals of Lipschitz and Hankel.

13*23.

A

which the integral on the right

by substituting Gegenbauer has expressible by elemeutary

formula due to Gegenbauer, Monatshefte fur Math, und Phys. IV. (1893), 401, is obtained by combining the results of § 13*2 with the integral

pp. 397

—

formula of § 5*43

two Bessel functions;

for the product, of

thus possible to

it is

express certain exponential integrals which involve two Bessel functions by

means of integrals of trigonometrical functions*. The general by Gegenbauer is deduced by taking the formula «/M (bt)

R (a)

r e~

2at

v

(bt)

=-

tJo

J^+ v

("2bt

cos

(f>)

cos

by e~'lat VkJrv and integrating from > 1(b) and R (p + v) > - h then

multiplying if

J

(/a

— v) d,

to ao

it

result obtained

it is

;

thus found

that,,

|

i

J» (bt) Jv (bt) t^"

dt=-

Jo

rJof "e-^Jn+r (2bt cos<£)

0*-+"cos

(fx

-v) .ddt

"""Jo

= -[

tJo

The

\

Jo

e-^Jw (2bt cos W+" cos (fj,-v). dtd

inversion of the order of the integrations presents no great theoretical

difficulties

;

hence

re-*at J»(bt)Jy (bt)P' +v dt

(1)

Jo

_ r (fi + V + f) 6»+» This

result, in

f**

Jo

7T*

the special case in which

/i

= v = 0,

COS*-*-''

(a«

(2)

(3)

particular cases of (1) take

/." e

/*

~ iatJ >

=1

and v equal to

an

£) M

— 1881),

and to — 1.

earlier note

t

(2« 2 +6')*-2("2 +&2 )^

by Gegenbauer,

"

r+ *

,

*'

p. 204.

It is

M J*M *~%«£j&+W

f" c-~J*(b£)3t

* See also

1

had been obtained previously by

Beltrami, Atti della R. Accad. delle Sci. di Torino, xvi. (1880

As

COS (fl-v)(f>

+ 6«coB

ibid. pp.

379

— 380.

found that

INFINITE INTEGRALS

13-23, 13*24]

where the modulus of the complete corresponding formula

elliptic integrals

/''-^W-aift,

K and E

is &/>/(«*

we deduce from

).

Beltrami's

wtW

that

(2)

/V"-/l(6l)/.<*)«*- a^-*)

(5)

where R(a)>\R(b)\, arid the modulus k t of the and (4) may be modified in a similar manner.

>

The formulae (3)

elliptic integrals is b/a.

It was stated by Gegenbauer that the integrals in means of elliptic integrals, but he did not give the

(2), (3)

and

by some formulae Kiel Programm, 1890. (5)

are expressible

results in detail;

deducible from the results of this section were given by Meissel, [Jahrbuch liber die Fortschritte der Math. 1890, pp. 521 522.]

—

13*24.

+&2

is

( 4>

Replacing 6 by

391

Weber's infinite integral, after Schafheitlin.

The formula

J

v

in which

T(^)

(t)dt

p-*+i

Jo

2"-* +i

< R (u.) < R (v) + \, was

r (v - $n + i )

obtained by

Weber*

for integral values of

was extended to general values of v by SoniDef pletely general result was also proved by Schafheitlin \. v.

The

result

The formula

is

of a

more recondite type than the exponential

in § 13*2 ; it may be established as

of convergence are satisfied,

a limiting case of these formulae, we have by § 13 2 (3)

r±m

lim

A

direct

r

a— Wo 2v r

is

integral formulae given for,

since the conditions §

(u

fV-H + l

+iy iFl \2>

—

2

v+1 '

>

V'

at once obtained.

method of evaluating the integral is to substitute Poisson's integral for the and then change the order of the integrations this is the method used by

Bessel function,

;

Schafheitlin, but the analysis

limited range of values of

and

and the com-

-

Jo t"-^ 1

whence the formula

;

/*

is

and

intricate because the result is established first for v

a

and then extended by the use of recurrence formulae

partial integrations.

Analytical difficulties are, to a large extent, avoided by using contour integrals instead of the definite integrals of Schafheitlin. * Journal filr

Math. lxix. (1868),

p. 230.

a Smith's Prize question, Jan. 29, 1867. f Math. Ann. xvi. (1880), p. 39.

The

special case in

which

[Math, and Phys. Papers,

X Math. Ann. xxx. (1887), pp. 157—161. § Cf. Bromwich, Theory of Infinite Series, § 172.

If v

we suppose that

= was

set

by Stokes as

v. (1905), p. 347.]


392

R (fi) <

R (v) > — \,

and

we then have

[CHAP.

TTTTT

(the integrals being absolutely con-

vergent) r(o+)jr

(—t)dt

L Fo^

/"(°

1

=

2-r

(,

+>

+4)

/"*"

<"*~ J. «»<««»«)«*•''«*

ra) j + .

— 2l Sin /47T. r (ft) COS £/A7T

/"*" /

sin2 "

cos-'1

0d0

Jo

_ — 2i sin/x7r. T (^fi)

"^-^r^-i/i + i)' By

the theory of analytic continuation, this result

When

R(fi)

when p and

valid

is

v are

R (fi) < R {v + §).

subjected to the single restriction

>

0, we deform the contour ir.to the positive half of the and we at once obtain the Weber-Schafheitlin formula.

axis taken twice,

real

The integral* H,(-<)cfc

/•("+>

may be

treated in exactly the

analysis

is

same manner; the only

that cos(tfcos0) has to be replaced by

difference in the

— sin (2 cos 0),

and

so,

by

Euler's formula (adapted for contour integrals), the factor cos \ fitr has to be replaced by — sin \pnr. It is thus found that f

provided- that

When

(o

+)

H

r

R (fi) < R (v + f

R (/*) > —

1

_ 2t sin /iir. T (|/t) tan \fitr

(-Qcft

(fi)

^ 0.

may be deformed

the contour

,

R

and

)

into the positive half of

the real axis taken twice, so that /"•

K

'

provided that If

M

v

(t)dt

we take

2"-»+ i

Jo

This

R (u) < R (v) + f that

-

(3) result,

—

T

1

/•"

Hi(2Q
_2_

2

/'J.

und a>

1^

and

1 \ + l&ry _

~ir«^**'

used by Struve, Ann. der Physik

for both small

iTT.

combined with the asymptotic formula

W* was

T (}/Q ta n (j /nr) r K v-Sfi + l)'

=

t'-^ — 1 < R (fi) ^ and /* = 0, v — 1, we see Jo

large values of x.

f

The

cos(2j?+^7t) 27r 8 ' 8

^'2

'

Chemie, (3) xvn. (1882), p. 1014, to tabulate

H

1

(2t)dt

last integral is of

importance in the Theory of

Diffraction. * Generalisations obtained

by replacing Bessel functions by Lommel's functions

the integrals of this section and in

Videmkabernet SeUkabs Skrifter,

many

(§ 10*7)

in

other integrals are discussed by Nielsen, K. Dantke

(7) v. (1910),

pp. 1—37.

INFINITE INTEGRALS

13'3]

By

[Note.

differentiating (1)

393

under the integral sign we obtain Weber's result

j / (01og«ft--y-log2;

(4)

been investigated by Lerch, MonaUhefte

this formula has also (1890), pp.

The formula

Math, und Phys.

I.

for functions of the second kind, corresponding to (1), is °°

f provided that

|

#(i>)|

Y

v it)

r (fr) r (fr - v) cos (|M - y ) w

dt _

This result has been given by Heaviside, Electro-

magnetic Theory, in. (London, 1912),

p. 273,

when

i>=0.]

Weber's first exponential integral and

13*3.

The

filr

105—112.

its

generalisations.

integral formula

|" J

(1)

(at)

exp (-p^) tdt = .

was deduced by Weber* from

i

exp (-

^) which

his double integral formula

will

be

discussed in § 14r2. This integral differs from those considered earlier in the chapter by containing the square of the variable in the exponential function. It is supposed that

|

argp

j

< £tt to secure convergence, but a is an unrestricted

complex number. It is equally easy to prove Hankel'sf

r J (at) exp (-pH*)

(2)

by a

v

.

more general formula,

0*-1 dt

Jo

To

direct method.

secure convergence at the origin,

it

must now be

supposed that J

R((i + v)>0.

To obtain the

result,

Joo is

we observe

723)

I^(\a\t).\ex V (-^)\.\t^\dt

/Jo

convergent, it is permissible § to evaluate the given integral

J, (at) in powers of

t

by expanding

and integrating term-by-term.

* Journal filr Math. lxiz. (1868), p. 227.

an

that, since (by §

Weber also evaluated

(2)

in the case fi=y + 2, v being

integer.

t Math. Ann. vni. (1875), p. 469. See also Gegenbaaer, Wiener Sitzungsberichte, lxzii.

(2),

(1876), p. 346.

J This restriction

may

be disregarded

if

we

replace the definite integral

(0+)

§ Cf.

Bromwich, Theory of Infinite

Series, § 176.

/

by the contour


394 It

Xm

thus found that

is

\m ( 1 f]\v+2m

/

eo

f<*>

m= orn\r(v

Jo

«=o

and this If

[CHAP.

is

m

rx r

+ m + l)J

r (i/ + m + 1

!

ran

)

'

2/>" +'l + 2m

"

equivalent to the result stated.

we apply Rummer's we find that

transformation

first

(§

4*42) to the function on the

right in (2),

("jriatJexyi-pH^.t^dt

(3)

Jo

=

ex

Vr(,ti)

and so the integral

is

{i'-to+H'+iiij,)-

P("v)- fl

expressible in finite terms whenever

fi

—v

is

an even

positive integer.


j~Jw (at) exp (-pH>)

(4)

provided that R(v)

>—

.

= fêx?

t>« dt

This integral

1.

by Sonine, Math. Ann. xvi. (1880), pp. 35

is

(-

^)

,

the basis of several investigations

—38;

some of these applications are

discussed in § 1347.

In order that the hypergeometric function on the right in (2) may be Rummer's second transformation (§4*42), we take fi=l and, we replace v by 2i>, we then find that

susceptible to if

;

JVôOexpt-j^).*- êxp (-§£).!, (g£),

(5)

a result given by Weber in the case v = \. If

we

replace v by

* (6)

f

—

easy to see that

v, it is

F„ (a*) exp (- pH*) dt

j o

-" when

|

fi

(i»)

|

< '£

;

W

exp

and,

if

("

¥

2

tan ) I/"

we make p 00 Z

(7)

„ Y

/

2v

x

-* 0, (a 7

(at)dt

=

(8)

|

R (v)

\

<

%,

by using

+

* Kw ($)

—

vir ,

u>

§ 7'23

;

and, in particular,

rY (t)dt =

Jo

sec

w]

being now positive), we find that

tan

Jo

when

V7r

(*?)

0.

INFINITE INTEGRALS

13*31]

Formulae

and

(5)

(London, 1912),

(6)

395

were given (when i/=0) by Heaviside, Electromagnetic Theory,

III.

p. 271.

Another method of evaluating the integral on the Proc. Camb. Phil. Soc.

vm.

(1895), pp. 122

— 128

;

left of (3) is

suggested by Basset,

the integrals have also been evaluated

with the help of Laplace's transformation by Macdonald, Proc. London Math. Soc. xxxv. (1903), pp. 428—443; see also Curzon, Proc. London Math. Soc. (2) xm. (1914), pp. 417 440 and Hardy, Trans. Camb. Phil. Soc. xxi. (1912), pp. 10, 27, for formulae obtained by making p2 a pure imaginary. ;

For some applications of the integrals of this section to the Theory of Conduction of Heat, see Rayleigh, Phil. Mag. (6) xxn. (1911), pp. 381—396 [Scientific Papers, vi. (1920), pp. 51—64].

Weber s second exponential

13*31.

The cussed

integral.

result of applying the formula §11*41(16) to the integral just disis

to

modify

it

by replacing the Bessel function under the integral same order.

sign by a product of two Bessel functions of the

= V(a + 6»- 2ab cos 2

If «r

we thus deduce

exp (- p*t?) Jv (at) Jy

f

)

and

if

R (p) > - \ R (2v + fi) > 0, ,

(&**)

P-

1

dt

J o

"

rVSW)

/."

il

exp ( "

^

^

The hypergeometric function reduces

({ab/p*)"

If

(1)

|

arg|>

|

<

\ir,

that

(

to unity

a2 +b 2 \

we expand the exponential under the

j\xV (-p*r)J

v

This formula

is

(at)Mbt)tdt

valid if

R (v) > — 1

=

" +-' **

^ =

when

/*

[*

fab cos

integral sign,

2

we

;

so that

d>\

.

,

,

?J

find that

±ex V (-^±?)/„(^).

and arg p < |

|

|tt.

Like the result of § 13*3, this equation is due to Weber, Journal fUr Math. lxix. (1868), Weber gave a different proof of it, as also did Hankel, Math. Ann. vin. (1875), 470. The proof given here is due to Gegenbauer, Wiener Sitzungsberichte, lxxii. pp. 469 p.

228;

—

347. Other investigations are due to Sonine, Math. Ann. xvi. (1880), p. 40; Sommerfeld, Konigsberg Dissertation, 1891 Macdonald, Proc. London Math. Soc. xxxv. (1903), p. 438; and Cailler, Mem. de la Soc. Phys. de Geneve, xxxiv. (1902 1905), p. 331. Some physical applications are due to Carslaw, Proc. London Math. Soc. (2), vm. (1910), (2), (1876), p.

;

—

pp.

365—374.


396 13*32.

[CHAP.

XHI

Generalisations of Weber's second exponential integral.

When the Bessel functions in integrals of the type just considered are not of the same order, it is usually impossible to express the result in any simple form. The only method of dealing with the most general integral °°

\

Jo is

J"M (at)

2

1

dt

to substitute the series of § 11*6 for the product of Bessel functions

integrate term-by-term, but

In the special case in which X is

Jv (bt) exp (- pH ) t*-

it

—

and

seems unnecessary to give the result here. v — /x, Macdonald* has shewn that the integral

equal to

by a transformation based on the

An

exceptional case occurs

results of §§ 12*11, 13*7.

when a = b

;

if

R (\ + + v) > 0, we then have fi

A.

J.(at)JAat)e^(-pH2 )t^dt^ J

o

„ffi 8

+ v+l 2

3

\

—+

fi '

v

2

2

^ p^+vr(fi+1)r(v+1)

+ 2 \ + n+v 2

'

+ 1* + iA

by using the expansion of § 541. Some been investigated by Gegenbauerf.

P+1

,

'

a*\

>

V

+ 1 >P + P + 1 ~p)> >

special cases of this formula have

13*33. Struve's integral involving products of Bessel functions.

now be shewn

It will

m K)

that,

JMMt) at ^

f-

Jo

t»+>

when

R (ji + v) > 0, then

rfr + iQr«3) -2»+T(fi + v+$)r(vL + $)r(v + $y

This result was obtained by Struve, Mem. de V Acad. Imp. des Set. de St Petersbourg, (7) p. 91, in the special case p = v = l ; the expression on the right is then equal

xxx. (1882), to 4/(3ir).

R

In evaluating the integral it is first convenient to suppose that both exceed £. It then follows from § 33 (7) that

R (ft) and

(v)

r JMJ +

v (t)

J

t»

x

f

J

>

m _~2^'-(2ji~1)(2i--1) irr^ + h)r(v-rh) .

2

• r*- ri' mn{tmn0)an(ttdn

^

j) coa

J

J o

_ Q cos2v 2

^ gin

$ sin

t"

London Math. Soc. xxxv. (1903), p. 440. 999—1000. t Wiener Sitzungsberichte, lxxxviii. (1884), pp. * Proc.

^ d0d4>dt

INFINITE INTEGRALS

13*32, 13-33]

397

fact that tr 2 sin (t sin 6) sin (t sin <£) does not

In view of the

cally the smaller of

and

1/tf-

sin

sin

exceed numeri-

the repeated integral converges

,

and the order of the integrations may be changed.

absolutely,

Since * 00

sin

j*

sin 0) sin

(t

we

sin

(t

<£)

\\ir sin 0,

(0

,

{0

_ ~ I^tt sin

,

t*

J

^ >

) )

find that the triple integral is equal to fin

\tr

ri> I

cos8*"8

cos8

""8

Jo Jo

+ $ir\

I

J

J

cos8*

-

1)

cos8

j

= - cos2""

"-8

sin

<£

cos8*" 8

1

<•&

cos8"

-8

<-6

sin d0d

sin

sin8

dd0.

we have

But, by a partial integration, (2v

-8

sin8

cos*4

"8

sin8 0rf0[ cty

j J

*

d0

sin"

J

cos8*-1

+

.

cos8*"8

J

+ y -i)r(p 2IV+i> + *)

sin8 <-W<£

r(j*

The other /••

*

integral is evaluated in the

J.

so

we have

jAt) dt = r(^ + y -i)r(i) {(2At-i)+(2 y - 1)}

jt)

2*+T(/* + v + j)r(/*+i)r(i; + i)

**+'

Jo

same manner, and

'

whence the result stated is evident. The extension over the range of values of fi and v for which merely R (/* + v) > is obtained by the theory of analytic continuation. It

may be shewn

in

a similar manner that, when

r HM (QH,(Q Jf

(2 )

r+>

Jo

r(M +*)r(j) 2* + "

This result was also obtained by Strove

By

using § 10*45

we

find that,

R (ji+v) is positive, then also

when

R

ro*+v+£)r (/*+£) r(*+$)' (ibid. p. 104) in

(/i)

and

the case

/*

« = 1. i»

R (v) exceed J,

**+"

jo

/"•

(2M -l)(2v-l)

/"**

"2'*+"-*«-r( M +i)r(i/+i) Jo Jo

f|* {l-oos

(<

if

a and

j8

are positive,

it

- cos (< sin )}

<*

x cos2* -2 Now,

sin 0)} {1

Jo cos2

"" 2

<£

sin

appears from a consideration of

* This result is easily proved by contour integration.

sin d$ddt.


398

[CHAP. XIII

round a contour consisting of the real axis and a large semicircle above

r }

Hence the proving

(1),

triple integral

will

this

may

is

(2) is established in

prove in like manner that,

/;^^=

3)

and

under consideration

and consequently

The reader

(

" C am (1 - cos (at)} {1 - cos ($t)} d _ P Jo

that

(at) sin (pt)

d{

t%

equal to the triple integral evaluated in

way

the same

R (p) and R (v)

if

it,

as

both exceed \ then ,

2*+" 0* + v - 1) r 0* + £) r

be extended over the range of values of

p and

(1).

(v

+ £)

v for

which

R (v) > \

and

R(n + v)>l. The

integrals

may

^+„+ i

<*'»

^ +|>+2

J^

]

*

be evaluated in a similar manner, but the results are of no great interest*.

13*4.

The

The discontinuous integral of Weber and Schafheitlin. integral

Jo

t

and 6 are supposed to be positive to secure convergence at the upper limit, was investigated by Weber, Journal fur Math. lxxv. (1873), in which a

pp. 75

—80,

in several special cases,

(i)

x=n=

0,

i/

=

namely, (ii)

l,

X = -£,

/*

= 0, v=±$.

The integral was evaluated, for all values of X, p. and v for which it is convergent, by Soninet, Math. Ann. xvi. (1880), pp. 51 52; but he did not examine the integral in very great detail, nor did he lay any stress on the discontinuities which occur when a and 6 become equal. Some years later the integral was investigated very thoroughly by Schafbut his preliminary analysis rests to a somewhat undue extent on the theory of heitlin

—

J,

linear differential equations.

The special case in which X=0 was discussed in 1895 by Gubler§ who used a very elegant transformation of contour integrals unfortunately, however, it seems impossible The analysis in the 0. to adapt Gubler's analysis to the more general case in which X ;

#

special case will be given subsequently (§ 13'44).

* Some related integrals have been evaluated by Siemon, Programm, Luisemchule, Berlin, 1890 [Jahrbuch ilber die Fortschritte der Math. 1890, p. 341] Wiener Sitzungsberichte, t See also § 13 43 in connexion with the researches of Gegenbaaer,

lxxxviii. (2), (1884), pp. 990—991. X Math. Ann. xxx. (1887), pp.

Math. Ann. xxx. § Math. Ann.

161—178. The question of priority is discussed by Sonine, 582—583, and by Schafheitlin, Math. Ann. xxxi. (1888), p. 156. in die Theorie (1897), pp. 37—48. See also Graf and Gubler, Einleitung

(1887), pp. xlviii.

der BesseVschen Funktionen, n. (Bern, 1900), pp. 136

—148.

INFINITE INTEGRALS

13*4]

The first investigation which we shall give The conditions for convergence are* (R(p.

is

399

based on the results of §

+ v + l)>R(\)>-l,

(a±b)

[R(p+v + l)>R(X)>0, it

13*2.

(a

= b)

being supposed, as already stated, that a and b are positive.

We

shall first suppose that the former conditions are satisfied,

<

shall also take b

constants

o, fi,

7

The

a.

defined

by the equations

+ 1,

)2a =/a + j/-\

ry=V+l,

|\

=7-a-

[V

=7— 1.

It will be supposed that these relations hold It is

known

that

r-J>(°t)M>>t)

P

Jo

dt=

positive value, the integral

replace b

now

£,

end of § 13*41.

to the

^W.(H^

lim

on the right

by z and the resulting

R(z)>0&nd\I(z)\<

down

C—+0J0

t

since the integral on the left is convergent

we

and we by choosing new

analysis is greatly shortened

is

;

now, when c has any assigned

convergent for complex values of b

integral

is

;

an analytic function of z when

c.

_M J«-fi(at)Jy-i(zt) dt

fy-^-'^zr^' =

e

PV

Jo

w (j Y+2W (-)W (~) (^)

. 2I

'

T

w!Ti V(y + m) Jo

So

m\T(y + m)

U=o e

-ot

J

(crf )

J

^p+sm-1

dt>

provided thatf P

m =o wi!r( 7 + m) Jo is

absolutely convergent; and

it is

'

easy to shew that this

is

the case

when

z\
J/

*^0

|

<

c,

mfo

w!T(7 + m)

x ,FX (0 * It follows

+ m,

$

•(a2

+c ) a+M, r(a- 9+l) 2

y

- £- m

;

a

-£+1

:

^p)

from the asymptotic expansions of the Bessel functions that the conditions

R(p + v + l)>R(\)>-l are sufficient to secure

convergence when a = b, provided that

t Cf. Bromwich, Theory of Infinite Series, % 176.

/*

-v

is

an odd

integer.


400

[CHAP.

XHI

and the hypergeometric function on the right may be replaced by*

r(a-/8 + i)r(j)

F

i_o_ m

/

c r(a-^+l)r(-j) + r(i-y3-m)r(a + m)V(a8 +

Now

__*__)

i.

.

s. _±\ m++ il!_ Po m ™''*'<* ^ (ra + m + *)'

j,

,

,

.

i

c»)

the moduli of the terms in the expansion of

sFiia

+ m^-p-m;

J; x)

do not exceed in absolute value the alternate terms in the expansion of — */%)~ A ~ 2m where A is the greater of 2a and 2/S — 1 1; and, similarly, the moduli of the terms in the expansion of

(1

,

|

2

F

X

(a

+m+h

1

1

j

m

—fi

$

>

*

x)

do not exceed in absolute value the alternate terms in the expansion of t »- l -

V«)-^-

(1

Hence the terms

/V*.

in the infinite series which has been obtained do not

exceed in absolute value the terms of the series

• (-.y»(|g)T+2m-i

w =o ml

QaY-t r(2a+2m)

T (7 + in)

(a'

+ c )-** 2

V (j) (1 - V«)-«-»» - 0- to) r(a + m + £)j

V

l\T(l

+

ir(-|)|(i-^)-^-«-n

,

= (^/(a + c But this last series is absolutely convergent when < V(a* + c ) — c, and it represents an analytic function of z in this domain.

where |

z

|r(|-i9-»)r(a + m)|J

2

9

a?

).

2

|

Hence, by the general theory of analytic continuation,

y

^(«m-i(*o A

v

/,

_ | "

(-)* (j*)** 8™- 1

(ja)*-*

T (2a + 2to)

2 a m ,o m!r(7 + m) (a + c )*+T(a-/3 + l)

provided that £

satisfies

the three conditions

R(z)>0,

Now

take

C to

\I(z)\
|*|< v (as + /

)-c

c3

be a positive number so small that

6
and take

< c < G,

a

s

so that also

6<

V(a"

+ d*)-c

* Cf. Forsyth, Treatise on Differential Equations, (1914), § 127.

INFINITE INTEGRALS

13-4]

we may take z = b, and when this has been we use fonctions majorantes just as before, we find that the resulting

Then

in the last integral formula

done,

if

series has its

terms

than the terms of an absolutely convergent series

less

* (_-y»(ft)y+8«-i (| q)"-^r(2«H-2m) m\T(y + m) " (a* + C2 a+m

m -o

)

T(j )(l - JX)-* '** l\t(l - fi -m)T(a + m + [~

+ ,

X =&l(a* + &).

where

401

iJ\

ir (-|)i(i-yx)-^-^-n

|ra-y8-m)r(o + m)|

J'

Hence, by the test of Weierstrass, the original series converges uniformly %c^C, and therefore the limit of the series when c when

with respect to c -*

is

We

the same as the value of the series

when

c

= 0.

have therefore proved that

&

c-»-oJo

and therefore

Ja -p(at)Jy -i(bt) d Jo

"

(-)m iy+zm-i p (2a + 2m) T (\) +2m»- 1 a»+*+*» r (1 2«-^ m) m! r £ - w) f + i=o (7 It has therefore

(1)

that

is

+ »» +

rf

*-2r— â^r^ra-^)-

2

*

1

^' 7 '^'

to say

provided that < b < a, and that the integral obtained by Sonine and Schafheitlin.

is

convergent. This

is

the result

we interchange a and b, and also fi and v, throughout the work, we when < a < b and the integral is convergent, then

If that,

(3)

£)

been shewn that

^^

io

(a

[" JO

Ja .p(at)Jy -

pT^

X

q-gr(q) r (7 - a) T (q - fi +

(bt)

at

~

27-«-0 &2»-y+i

2

find

1)

î(a,a-7 + l;a-/3 + l;|2 ).

.

[CHAP. XHI


402

Now

so happens that the expressions on the right in (1)

it

the analytic continuations of the

this

phenomenon The

13*41.

in

some

same

(3) are not

There is consequently a be necessary to examine

function.

when a = 6; and

discontinuity in the formula

and

it will

detail.

critical case

of the Weber- Schafheitlin integral.

In the case of the integral now under consideration, when a =

b,

we

have,

as before,

fiW*!^*. C—+0J0 iim {"îiMLMjt, K

Jo

R

assuming that

Now

t

t

(//,

+ v + 1) > R (X) > 0,

to secure convergence.

consider e

is

***

P=**

/.

where z

K

a complex variable with

R

(z) positive.

R (z) > la we may expand

the integrand in ascending powers of a and integrate term-by- term, this procedure being justified by the fact that

When

the resulting series

We

is

convergent.

thus get, by using § 5*41,

/

l-lt

J*-fi{at )Jy -i(at)

dt

Jo )a^+y+2m-i ~zt (_)m (£ tt f« e

â-Hm-1

p (g -

+ 7 + 2m) - £ +7 + »») m-OJ m! r (a - /3 + m + 1) )a-^+y+2m-i m 2m) r (a p (2a + + 7 + 2m) (_) (| a _ ~~ l *""+""• m)V(a-/3 + y + m)' mir(a-fi + m + l)T(y + m =o .

F (7+ m) T

Now and so

the integral on the

when

its value,

left is

an analytic function of

z has the small positive value

ft

,

(a

c, is

z

when

R (z) > 0,

the analytic con-

tinuation of the series on the right.

But, by Barnes' theory*, the series on the right and may be represented by the integral

its

analytic continua-

tions

xi

J_ (

(ia)--*-*-*

27rJ_ aci

and

^

T (2a + 2s) T (a - /3 + 7 + 2«) p + s + l)r(7 + s)r(a-/3 + 7 + *)

41 "- 1

+ «r(a-/tf

this integral represents a function of z

which

is

analytic

^

when arg z < |

\

tr.

It is supposed that the contour consists of the imaginary axis with loops to ensure that the poles of T (- s) lie on the right of the contour, while the poles of T (2a + 2s) and of T (a - /9 + 7 + 2s) lie on the left of the contour.

evaluate the integral by modifying the contour so as to enclose the poles on the left of the contour and evaluating the residues

When

\z

* Proc.

\

< 2a we may

London Math.

Soc.

(2) v.

(1907), pp.

59—118. See

also §§ 6'5, 6-51 supra.

INFINITE INTEGRALS

13*41]

The sum

at them.

of these residues forms two convergent series proceeding

in ascending powers of z

Ja-

/"

fi

403

(at)Jy^{at)

R (z) >

when

hence,

;

and \z\< 2a,

,

Jo

(_)m(^ a

_1 • ~2 wl

rn\

Now c

^ r (7 - a - <8 - m) T (a + \m) -£-£m) F( 7 -a - iw) T( 7 - £ - *m)

Y -a-g-m-i

(-)™ (â)- w

*

1

)

T(l

22 (7

— a — ft) > 0,

and then make f-

n (1) .

-1 .gY-»-*+™

c

-*

0,

and

so,

1

(at)

«F=^

provided that

when we make

we deduce

Ja-,(at)Jy.

J

r (a + ft - 7 - «t) T (â + £7 ~ iff + 4w)

aC

> 0,

i2 (a)

2 assume the positive value

that

_

(|a r--g-ir(7- a -/3)r( a)

"2r(l-/3)r(7-a)r( 7 -^)

R (7 - a - ft) > 0.

From the Gaussian formula for ^(a, ft;

7;

in the value of the integral, though there

1),

is

there

is

therefore no discontinuity

a discontinuity in the formula which

expresses that value as b increases through the value

The

result

f

(2)

may be

a.

written in the alternative form

^^dt

J

qq)*- 1 r (\) r (\n + \v - $\ + ^)

R (a* + v + 1) >R(\) > 0.

provided that If

/a

—v

is

an odd integer the integral converges when

> i2 (X) > —

1

this case next demands attention.

We place of

shall ft

make a change

and

i>

__]_[-* "2wt.L.

^

if

-1

,

^

"1 w *

T (X - m) T (a - |X + \m) £X+1)T (- p - \m + \\)T (a + \\ - $m) 2 mt w!r (;>- £w +

_1

"

i

*- A r(o

;

r (2a + 2s) T (2a + 2s - X) p( + p + * + i)r(o-/) + «)r(2o + «)

(|o ) to+M ,+

a+p and a — p—l R (X) > 0, we then find that

by writing

in notation

in the preceding analysis

•

(-)**

(la^-"

1

•

1

(_)m(â)- TO - 1 Â+ "

,

r(-X-«i)r(a + im)

in


404

[CHAP. XIII

and hence

f Ja+P (at) J^-x (at)

w

Jo

2^r(J5 +

*

~

T (X) T (a - j\) i\+i)r(a+^\)r(i\- i>)'

aK

_

}

A

1

\ = 0. This should be compared with the more obtained from § 13*4, namely that, when b < a,

general formulae

unless

Ja+P (at) /q-p-i (bt ) dt

f"

f

(4)

Jo

U+l) a*-^r(a-p)r(p T (p + 1\ + 1)

2* a*-r-K r(a-p)

and,

when

\"- hX ' P ~ hX] '

a

~P

'

a>)

'

> a,

b

(aQ Ja-p-i (bt )

«/«+P

f

(5)

* Fl

dt

Jo

a«+JT(a-iX)

= 2>6^-^r (a Since X

when a

On

*

4 + i)r(ix- P

' )

yp-(/

-^ j.

tt

j. tt+y + +1n - iX; ,

,

i'

,

,

1;

as \

p)-

the functions on the right in (4) and (5) do not tend to limits

^ 0,

b.

the other hand, 1

when X (| a

[**

and the residue at

*

=—a

is

is

zero, the contour integral

)«.+«-i

jr (2a + 2s)p r (-

becomes

s)

(— )*/(2a).

It follows that

(6)

Jo

J. +p (a*) J.^-x

(bt) dt

-

-I

(- W(2 aX

U according as 6

<

a,

6

= a, 6 > a.

Since

^(a.-i?; a-?; l)-(-)p p T(a-p)/V(a), \

it is

evident that the value of the integral when b

when

b

The

—a—

and

is the

fy-coM^- ^j:-/' is also easily

mean of its

limits

Amsterdam,

.

obtained by inserting limits in § 5*11 (13); this formula has been

discussed in great detail by Kapteyn, 116.

=a

= a + 0.

result of taking X»»l in (2) is

( 7)

which

b

iv. (1902),

pp.

Proc.

102—103; Archives

Section of Sei.,

K. Akad. van Wet.

Nderlandaises, (2) vi. (1901), pp.

te

103—

INFINITE INTEGRALS

13-42] 13*42.

405

Special cases of the discontinuous integral.

Numerous

special cases of interest are obtained

to the constants X,

three values are given for an integral, the

— a, and

>a

by giving

special values

To

v in the preceding analysis.

ft,

save repetition, when value for b < a, the second

first is its

when two values only are given, the first is the value for b ^ a, the second for b > a and the values are correct for all values of the constants which make the integrals convergent. The following are the most important special cases * Jât)j^bt) a (b/ay/fj,, ai (1) [R(ri>0] t Jo (Ha/W/*for b

the third for 6

;

;

:

r

Jp. (at)

(2)

dt

F Jo

=

a* sin \iltt 2

i> _1 cos dt

=

{fi

arc sin (b/a)},

a cos 11

Jo

[R(fj,)>-1]

+ V(6 2 - a )}"

[fi {b

JM (at) cos bt

(3)

sin{/*arcsin(6/a)},

sin bt

[R(ri>0]

IfjLir

I

[fi{b+^(b*~a2)}»' sin

\fjj

v(« 2

,«,

(4)

/

Jo

Jn (at) sin bt .dt=

\

x

[B( H.)>-2] aM COS

cos

|

b°-)

or 0,

{/j,

|/47T

arc sin (b/a)}

V(a2 -62)

,» (5)

arc sin (b/a)}

Jât) cos &£ dt = .

-'o

-(

[«(m)>-i]

or 0,

1

sin £ai7t

a**

v

V(6

-a

3

2

).{6+ v (&2 -a2 )}'i /

Special cases of preceding results are

r Jo (at) sin

(6,

/

0, I

fa

.

dt

= » -j

,

/V(6

(?)

I

«7 (a£) cos

bt.dt

2

fiM« = « -:

Jo

2

-a

2 ).

-n

,

la

These two formulae, which were given by Webert, Journal p. 77,

are

known

as Weber's discontinuous factors

determining the potential of an

;

fiir

Math. lxxv. (1873),

they are associated with the problem of

electrified circular disc J.

—

*

Numerous other special cases are given by Nielsen, Ann. di Mat. (3) xiv. (1908), pp. 82 90. and (5) diverge for certain values of n when a = b. t The former was known to Stokes many years earlier, and was, in fact, set by him as a Smith's prize examination question in Feb. 1853. [Math, and Phys. Papers, v. (1905), p. 319.]

The

integrals in (4)

t Cf. Gallop, Quarterly Journal, xxi. (1886), pp. 230—231.


406

Another

[CHAP.

XHI

special formula is

(8)

f

\

(at)

JM_, (bt) dt =

J

[R (fi) >

1/(26),

0]

(0;

and

we put

if

fi

= 1, we

(9)

obtain Weber's result

The

result of putting

/*

= £ in

1

(8) is

by Voss, Encyclopadie der Math.

article

Some

=

(bt)dt

known

]

1/(26),

as Dirichlet's discontinuous factor; see the

Wisa. n. (1), (1916), p. 109.

other special formulae have been found useful in the theory of Fourier series by

W. H. Young,

Leipziger Berichte, lxiii. (1911), pp. 369

Another method of evaluating Math,

J

Jo (at)

f

(ibid., p. 80),

und

(5)

—387.

has been given by Hopf and Sommerfeld, Archiv der

Phys. (3) xvin. (1911), pp. 1—16.

A consequence of formula (1) 2

must be noted, When

J\ + n {x) = 2v

/

v

> 0,

we

have, by § 5*51

(5),

J^(t)~

*

.'o

= 1, and so (10)

I

.W |<1,

-

(§ 2 5)

|^ +

1

(*)j
an interesting generalisation of Hansen's inequality which was discovered by Lommel, Miinchener Abh. xv. (1886), pp. 548 549.

provided only that v be positive; this

The reader may

is

—

find it interesting to

deduce Bateman's integral *

jV (aO{l-/oW)Y =

(11)

{J [log «./«),

from the Weber-Schafheitlin theorem. 13*43.

Gegenbauer's investigation of the Weber-Schafheitlin integral.

In the special case in which the Bessel functions are of the same order, Gegenbauerf found that by his method Weber's integral could be evaluated in a simple manner. If

R (2v + 1) > R (\) > R (v + \) we

f"

have

dt

\^Jv (at)Jv (bt)fK

(\oby

"T

f"

/»

Jv

[t

V(q«

+ 6 - 2ab cos 2

Sm „„.,.,, Wdt .

+ 1) r (J) J o Jo t— (a* + fc*-2a& cos <*>)*" (ab)" T (v - \\ + £) [" f* sin " d 2 -22a& cos 0)"-*A +i 2* T (v + \) F (I) r (§\ + £) J (a +6 Jo(a* o "

(v

2"

2

* Messenger, xli. (1912), p. 101

;

for a proof of the

Messenger, xlii. (1913), pp. 92—93. t Wiener Sitzungsberichte, lxzxviii.

(2),

formula by another method, see Hardy,

(1884), p. 991.

INFINITE INTEGRALS

13-43]

by

§ 13"22.

When b < a, the r (v — AX. + 1)

expression on the right

b"

2

we

/b\ n C w

°°

v-^r(» + i)rft)raui)

Now

407

.?.

U)

is

tv-™ («•«*«-*«

J.

from the recurrence formulae

2/x- 1)(1 -^)-*"" 1

= (n +

J* {(i

-*)-*»(V(,)}

^

- *p+» Cs(t)} - - (» + 1)

{(1

(1

C^

1

(5),

- *}*+*-! C* n+1 (z),

see that 1

(n

+

(1

l)!" .'

-*)*-*

CW*)***

=-

(1

-l *

-

*»)'-*-*»+*

J

^|(l _ *«)*»+* Cmm (*)l dz

= (n + 2/*-2v-l)P z{l-z -y-lCn»{z) dz n

n+2fi-2v-l1 C _ 1 ~ 2, 2v+n + l /'/ + ^TT J_ 1

(n + 2p-2v-l)(n

2v

-^ s| 1

+ 2n-l)

+n+ l

(1

M

-°'^ G

r1

>J

dz

ii-*y-kC*n-i(s)dz,

so that

0«+i (cos r Jo

)

sin 2" d

(n

Hence

it

^

+ 2fi-2v-l)(n + 2u-l)[ n c „ 9JJJ -' (cos * s,n (to + .+i)(.+i? '. ..

.

.

follows that

f Jo and

this agrees

with the result of § 13'4.

The method given here is substantially the same as Gegenbauer's but he used slightly more complicated analysis in order to avoid the necessity of appealing to the theory of analytic continuation to establish the result over ;

the more extended range jK (2v

By expanding (I)

+

1)

> R (X.) > —

1.

the finite integral in powers of cos

Cj(a£\J

dt (bt)

,

we obtain the formula

wrfr-fo+j) „ (2v + 41 - A.

*°M—

— 2v

'

+ i3 - \

-

;

v+1

4a 262

\

>@TFf)-

which is valid whether a > b or a < b. This result was given by Gegenbauer, and with this form of the result the discontinuity is masked.

The reader will find it interesting by putting b = a in the finite integral.

to

examine the

critical case

obtained


408 13*44.

The

[CHAP. XIII

Gublers investigation of the Weber- Schafheitlin integral.

integral

J^(at)Jv

I

(bt)dt

Jo

method due

investigated by the

will

now be

first

to consider the more general integral

[«

J.(at)Jv (bt) t

.'o

K

to Gubler*.

It

is

convenient

dt

even though this integral cannot be evaluated in a simple manner by Gubler's (v) > 0, R (X) > \, R (/* - X) > - 1 and, methods. It is first supposed that

R

as usual,

a and

From

b are positive,

;

b.

the generalisation of Bessel's integral, given by § 6*2

that the integral

We

and a >

is

(2), it is

evident

equal to

take the contour as shewn in Fig. 29 to meet the circle

j

z

j

=

1

and the

Fig. 29.

line R(z),

=

only at z

=±

i;

and then,

for all the values of z

and

t

under

consideration,

R\hbt(z-l/z)}^0; and the repeated integral converges absolutely, since

Jo is

The order

convergent.

t

K

J-»

I

of the integrations

may

therefore be changed,

we have

r J„(at)JAb t)^_ dt Jo If

we

?

1

f<°

+

>

~2frij. a>

f"

write

b(z-l/s) *

Math. Ann.

Jy. (at)

*

]

—

exp

H-

a(e-l/Q,

xlviii. (1897), pp.

37—48.

1

;)}

z-*-1 dtdz.

and

)

INFINITE INTEGRALS

13-44]

and suppose that that value of £ r- J»(at)jv {bt ) J

"

^

is

taken

for

_ j_ ~ 2*i j _.

|

x

[\a (i

x

^ 1, we

+ 1/0}*-"-'

rv + i)

2™

£j

have,

by

§

132,

Qay r{p-\ + i)

/«>+>

dt

which

409

-ji^

V(ji

.'-.({T+i/cr^

- A. + 1, /* + X;

f/x,

+ l)

/it

4,1

+ 1; i^r«) d*>

by Kuramer's transformation*.

Next consider the path described by £, when z describes its contour. Since the value of £ with the greater modulus is chosen, the path is the curve on the right of the circle in Fig. 29; and the curve is irreducible because different branches of z, qua function of £, are taken on the different parts of it. The curve meets the unit circle only at e ±iu> where ,

b

a> is

the acute angle for which

— a sin m.

Now

both the original integral and the final contour integral are analytic \ when (X) > — 1, so long as a ± b. Hence we may takef X =0,

functions of

provided that

R R (fi) > — 1

;

and then we have

Next write £ = zt and then Z2

_ br + a ~r(aT+6)'

+ a) +6

t (6t

~

6

ttr

'

and the r contour is that shewn in Fig. 30; it starts from - b/a, encircles the origin clockwise, and returns to — b/a) where the contour crosses the positive half of the real axis, we have arg t = 0.

we

adr 2r(bT + a)'

dz

_.

Since

1

+ ?)

z(l

find (on reversing the direction of the contour) that

[ ^ J,, (at) J o

Jv (bt) dt = ^-. *""* =a

(0+ '

fi°+)

„,1

Ima**

* Journal fur

(1908), pp.

t If

Math. 115—119.

X^fcO,

n. f.

W *(-H-i)

l

(1 \

] _x

xv. (1836), p. 78,

>

(ar

+

&)*<"*-» dr

formula

(57).

b2

Wc+^+d

+-,2 «, a J

+ W )*C+»*-i) ,

(1

,

dfO.

See also Barnes, Quarterly Journal, xxxix.

the hypergeometric function does not in general reduce to an elementary function,

and the analysis becomes w.

+ a)-it"+*+l

—b/a

h" °.

t**"-*-1 (6t

'

( J

intractable.

14


410

[chap,

xm

If we expand in ascending powers of b*/a? and substitute the values of the Euler-Pochhammer integrals, then Gubler's result

J^

(at)

Jv (bt) dt

I

yrUM + iv + i) is

F

Lfi+ v

+1

v

—n+l

manifest.

Fig. 30.

A

13*45.

The

modification of the Weber- Schafheitlin integral.

£ *'<&*<& *.

integral

which converges if R (a) > I (b) and R (v + 1 — X) > R (fi) is expressible in terms of hypergeometric functions, like the Weber-Schafheitlin integral, but unlike that integral it has no discontinuity when a = b. j

\

,

|

\

To I

b

j

evaluate

< a j

j

it,

J

expand

¥

(bt) in

powers of

b,

assuming temporarily that may be a

in order that the result of term-by-term integration

convergent

series.

Jo

t

By

using § 13 21 (8) -

it is

found that

MV ' + n+l)J b"Y(\v-\X + \it + \)T(\v-\\-\». + M+l nv— A+l r> + i> V — \4-/4+l V — \ — fJb+1 V+l ^^

K

«-ow!l (v

.

\)

'

a*)

and, in particular, (2)

/;

provided that

Formula

K^MHÛ^^L^lR,

R (v + 1) > R (/*)

(1)

|

|

and

R (a) >

\

I (b)

was given by Heaviside* when

* Electromagnetic

/j,

|

=v=

and \

Theory, in. (London, 1912), pp. 249, 268, 275.

is

and

— 1.

INFINITE INTEGRALS

13-46, 13*46] 13*46.

411

Generalisations of the Weber-Schafheitlin integral.

To obtain the

values of integrals containing three Bessel functions under

the integral sign, take the integral

Mat)J

r-

(ot)

v

dtt

Jo

t

replace 6 by VX& 2 + c2 — 26c cos <£), where b and c are positive, multiply sin 2 " /(b2 + c2 - 26c cos ) iv and integrate. It is thus found that f

-

,/M (a«)

^

J, (bt)Jv (ct)

l

o

where

-Er

_

+ i)r(i)Jo

(,

= V(6 + c2 — 26c cos 0);

—

J„ ^ ^"^~ sin io (at)

f•

( «rt)

.

and the integral on the right

2

convergent

f-

(\bc)"

^-r

by

****

is

absolutely

if

R(v)>-%,R(/jl + p+2)>R(\ + 1)>0. Change the order of the integrations on the right integration with respect to

t is

;

then the result of the

an elementary function of -or

if

\ -r v + 1 = ± /*,

by the formulae

1-x"^-- 1

J,o

.:.-i,.,fw..

j

.a

( a>tsr )

-

It follows that

/•

JM (at) Jv (6<) J, (c«) t '* dt 1

(i)

Jo

(

V

*6

4W-TTrp7n

in which the value of

A

f

62

is less

*

arc cos

+ c -a

than, between, or greater than the two

a

;

^^

R (/*) and R (v)

numbers

+ c) exceed — |. 2

,

In particular

/J

sinS

t

>

(6-c) ,(6

(2)

1

2

2

^r Zoc

2

provided that both

,

*

is

0,

according as a2

'" " h ~ c + 26c cos ^"

J„ (at) J, (M) * (ct) f = ^, r

%

)rW J." 8in" **

Multiply by a" +1 and differentiate under the integral sign with respect to and we then obtain the interesting result that, if R (v) > — \, f

(3)

°°

J

v

|

(at)

Jv (bt) J

(ct)

o

when

a, 6, c

—=

A " -1 r(l/ + i)r(i)

2" -1

dt

v

(a6c)

,

2

are the sides of a triangle of area A; but

of a triangle, the integral

>

if a, 6, c

are not sides

is zero.

This formula is due to Sonine, Math. Ann. xvi. (1880), p. 46 other aspects of it have been investigated by Dougall, Proc. Edinburgh Math. Soc. xxxvu. (1919), pp. 33—47. ;


412

[CHAP. XIII

It has been observed by Macdonald* that the integral on the left in (1) is always expressible in terms of Legendre functions. The expression may be derived from the integral on the right in the following manner:

When

are the sides of a triangle, by the substitution

a, b, c

sin

= sin \A

$

sin 6

we have (a2 .'

- 62 - c + 2 be cos 0)*-"2

1

sin2 "

d

o

= (26c)*-"-

1

(cos

- cos Ay-"-

1

sin2"

(j>d

Jo

==2 2*- 2 "- 1 (bcy-"-> sin 2*" 1 £

'r

A

(l-sin2 J/lsin2 ^)"-isin2"^cos2'*- 2 "- 1

t

.

^^

Jo

= and

4*-'

(6c)*—

1

therefore, if iJ

triangle,

(4)

If,

T{V

sin*-* jj. (/a)

and

v)

rfJ+%~

R

(v)

— £,

exceed

î (* +"> i-»; /* + *; sin2 M),

•

and

J.

J"

(at)

however, a2

Jv (bt) Jv (ct) *w
>

(6

+ cf,

are the sides of a

a, b, c

we have

M P ^(

cos

A

)-

and we write

a?

- 6 - c = 26c cosh ^, 2

2

we have 2 [ (a

Jo

- 6 2 - c2 +

26c cos

)*-"

_1 sin2 " d>d<£

= (26c)*-"- 1

£4 + cos <£)*-"- x sin2 " <}>d

(cosh J

Ii^î>

= (26ccosh^-"- 1

x.J^^^ + ^|î;* + l;sech l,

so that, /-\ (o)

In /n\ (6)

when a > 2

(b

+ cf, we

(6c)* / *\ r /J^ r / .wix Tj J»(at)J v (bt)Jv (ct)t -*dt=±—t

manner, we deduce from

r /i^ r .v (bt) J, <«) *„ (at) J.

f°° rr

Jo

where 26c

/

-x

X=a

/

2

-f b-

+c

2 ;

j-i <»«

..

§

*

j^

and in

-1

COS

I>7T

sinh*

.

-

*^ — ~i-M Q

—

0* COS

Z>7T

.

- l) ** *

(X2 —

;+9)w

(brf(hey^ An{

this formula a,

6,

c

may

R (a ± ib ± ic) ;

this result is also * Proc.

due

London Math.

.

.

.

,(cosh<#).

13*45 (2) that

provided only that the four numbers

are positive

^),

have

C* t

like

2

to Macdonald.

Soc. (2) vn. (1909), pp.

142—149.

^* + *

,

irs

0,., (A )•

be complex,

INFINITE INTEGRALS

13-46]

413

The apparent discrepancy between these formulae and the formulae

[Note.

donald's paper

is

of Mac-

a consequence of the different definitions adopted for the function

Qn m

;

-

see § 5 71.]

Other formulae involving three Bessel functions may be obtained by taking § 11*6 (1), replacing z by x, multiplying by

formula

2Jp (xcos$)/xk and integrating. It

thus found that

is

0)

J» ( X C0S

\

^ C0S ^) J» i X S * n

cos*1

u==

X

(f>

cos*1

<3>

sin"

*£

S^n

^) Jp ( X C0S

^) Â^l

cosp 6

<£ sin"

nir(/* + n

oL

+

l)

r(â+^ + |/j-^\ + w+l ) +lv-ip + \\ + n + l) u+ v + pP — X + n + 1, ^p — X — u — v p ^ .^,(

l\lfi

~

^

(— n,

p.

+v+n+1

x-^-w,

/a

+ + n + l;

x

2 J*,

;

*>

+1

sin 2

;

n

;

p

+ ,1

„

;

„\

cos2 6\

)

i

and cos 6 Some

is

not equal to + cos (<& +

I

,

Some a2

have been given by Gegenbauer in a letter to Kapteyn, van Wet. te Amsterdam, iv. (1902), pp. 584 588.

special cases of this result

—

Proc. Section of Sci., K. Acad,

it is

i/+l; sin 2 3>)

R (p. + v + p + 2) > R (X) > - £

when

If cti,

i/

,

extensions of formula (3) have been given recently by Nicholson*. am are positive numbers arranged in descending order of magnitude

. . .

easy to shew that,

if ctj

> 2 am

,

R(v)>—1,

then /

oo

m II

(8)

J J

n=l

ft* dt

Jv {an t)-^^0; »

the simplest method of establishing this result

Gegenbauer's formula of

§ 11*41

is

by induction, by substituting

[on the assumption that

R(v)> — |]

for

Jv (a m^ t) Jv (am t), and then changing the order of the integrations. When a a ... am are such that they can be the lengths of the sides of a polygon, the integral is intractable unless ra = 3 (the case already considered), or m = 4. 1

lt

2

,

* Quarterly Journal, xlviii. (1920), pp. 321

in § 13-48.

—329.

Some

associated integrals will be discussed

w

r

^


414

When

Ox, a.2 ,

[CHAP. XIII

a 3 a4 can form the sides of a quadrilateral, we write ,

4

16 A 8 so that

A

is

=

+ aa + a + ai -2a n

IT (a1

s

),

the area of the cyclic quadrilateral with sides a Xi a z a % a 4 ,

,

to deduce its value,

it is

simplest

i>

to obtain

first

when R(v) >\, and deduce the value

.

= 0;

when but an expression for the integral by analytic continuation; the

The integral can be evaluated in a simple form only* for v

=

value of the integral assumes different forms according asf «i i.e.

+ a4

A ^ 2

according as

>

Oj

fla+Gts.

as a 8 a 4

We write or = of + a3 — 2ag a 3 cos Gegenbauer's formula, so that

Jb^

a rr JJ/'^^-r^T^W.

ra

dt

t in

.

and

2

2

,

4

J

(a 2 t)

v

™»> * *

^**

—a

where the lower limit is given by «r =a x + a,, whichever is the smaller.

v

Jv ((M) J» (
i.

J

J

replace

and the upper limit by

by

{a 3 t)

^ #

m—

+a

Gtj

4

or a2

We

write

- (g - a y __ (a! + a y - (a - a ) w - (a — a„) (a, + a ) — (a 2 — a sr

2

4

4

;

4

2

so that the

upper limit

for

a;

or

is 1

3

A/^(a

1

a.i

2 ,

4

t

2

2

2

a3 a i ):

2

)

this expression will be

called 1/k.

We now carry out the process of analytic continuation (unless a, -f a4 =a 2 a3 when the integrals diverge at the upper limit if v — 0), and we get -!-

f"

=

nJ

,

(a n t)tdt

n=l

+ a*) - «'} 2

-..\

[ [{(«!

-—

-1 or

{**

- («i - «*) [^ - («« - «3> 2

1

2 }

{(a,

+a

2

3)

-

-st

2

}]-*

Wsr

^.

1/fc

V{(1

- # ) (1 - k2 a?)\ 2

Hence f«o

(9) .'0

4

nJ

(a n t) tdt

=

/_1_ ^.

/

A

V

7T

2

V(ai«2 0sO\ /' A

—v #

n=l

l7rV(G-i«a«»a4)

W(<

where If denotes the complete elliptic integral of the whose modulus is less than unity is to be taken. * For other values of v f

We

still

it is

first

kind,

and that one

expressible as a hypergeometric function of three variables,

suppose that a 1 'ââ 3 â 4

.

;

INFINITE INTEGRALS

13-47]

415

Nicholson has also evaluated

when

R (v) >

and a > 0. The simplest procedure

a special case of the

last,

is to

regard the integral as

so that it is equal to

1

,-, /"\o„ {2a9* /

^Wi^Wm-h

(1

° S * )] "' + C_^x>,-*

sin 2 " <6 do

,

f2a'(l -cos.fr) )*'

d hence*

r \j (atv*

no)

*- -

a2

""sr

< 2f/ >

r ^>

The discontinuous integrals of Sonine and Gegenbauer.

13*47.

Several discontinuous integrals, of a more general character than the WeberSchafheitlin type, have been investigated by Sonine f and Gegenbauer J; modifications of these integrals are of importance in physical problems.

The

example § which we

first

due to Sonine, namely

shall take is

(a
f0,

=

some

|£!^>p-v_iW(a! _

(o>J)

ls))

secure convergence, a and b are taken to be positive and R(v) >R (/*)>— 1 a = 6, then we take R (v) > R (fi + 1) > 0. The number z is an unrestricted complex number, and the integral reduces to a case of the Weber-Schafheitlin

To

if

integral

The that, if

Jo

when

z

is zero.

integrals involved being absolutely convergent ||,

c>

MV

J

2

(<

+ *-)*"

Too /YM-ooi

1

b-

An

see from § 6'2 (8)

then

0,

- ssJ.

*

we

L,

J- (bt) (' +

c+coi

f

arithmetical error in Nicholson's

'

""""' exp

/

f"

—

- t* + z-\~\ dudt

L*° l"

r(a?—b*)u

az*~]

work has been

)J

,

corrected.

The

result for values of

R (v)

between and £ is obtained by analytic continuation. t Math. Ann. xvi. (1880), p. 38 et teq. X Wiener Sitzungsberichte, lxxxviii. (1884), pp. 990 1003. phys. de Geneve, xxxiv. § This formula is also investigated by Cailler, Mem. de la Soc. de

—

(1902—1905), pp. 348—349. The convergence is absolute only when R (y)>R (m + 1)>0; for values of this condition, the formula is to be established by analytic continuation. ||

v

not covered by


416

[CHAP. XIII

When a b, we have to apply

zero; but,

axis,

and the and

§ 6*2 (8),

then we obtain^ the formula stated*.

A related

integral

/V,(

(2)

may be

îfflU a ^|^J!)pw M^>)i

evaluated in a similar manner.

We suppose that a and b are positive f, and

R

that in evaluating (/*) > — 1 convenient to suppose that jarg.2 <^7r, though we may subsequently extend the range of values of z to |arg^|<^7r by analytic

the integral

it is

;

1

continuation.

From 2J

by

§ 6'22 (8) it follows that

the integral on the

- £ a ( u + -^^-)

/, (bt) P+1 u-"- 1 exp I

§ 6*22 (8);

and

Now make

V(tf

£

interpretation

is

\ir.

If

we put

+ ») p»-'

equal to

dudt

<

z

= iy,

where y >

^_^ {y V(a +&2)} _ ,

j

0,

we

^^

find that

[y

v(a2+ 6a)|]

and it is supposed that the path of integration avoids an indentation above the singular point, and that given to V(<2 — y2 ) which makes the expression positive when

provided that i£ (v) the singularity

is

this is the result stated.

arg z-*-±

« )rrre-i<«-»-»

of (2)

left

t

1

;

= y by

t>y.

we had put z — - iy, we should have had the indentation below the and the sign of i would have been changed throughout (3).

If axis

real

In particular Jo
(4)

Jo

(?=59

where the upper or lower sign above or below the axis of y. *

For physical applications of

V(a8

With certain

limitations,

this integral, see

a and b

S

taken according as the indentation passes

is

Lamb,

pp. 122—141. t

+6)

may

be complex.

Proc.

London Math.

Soc. (2) vn. (1909),

INFINITE INTEGRALS

13-47]

The

last

417

formula (with the lower sign*) has been used in physical investigations by

682—683;

Somnierfeld, Ann. der Physik ttnd Chemie, (4) xxviii. (1909), pp.

Bateman, Electrical and Optical Wave-Motion (Cambridge, 1915), If in (1)

we

divide by

provided that a

>

&**

if

a

>

and

>—

i2 (ji)

In (5) replace

by

i/

this

;

might have been

by the same method.

established independently Similarly, from (2)

and make 6—0, we obtain Sonine's formula

R (| v — £) > R (p) > — 1

and

see also

p. 72.

we have

1.

a by 2

2i>,

and integrate from 6 =

sin

= §7r.

to 6

It

follows that

m {)

r ^MV(<

3

this is valid

The

+ ^)

~W+W~

Jo

when

22 (v

—

}

^ +1 ** _ r(^+i) f ^-^^sinfl)^ sin" +1 m*"-"'o

|)

> R (/*) > —


case of interest

when

is

2p

is easily

= 2/a + 3,

1.

expansible in powers of z; but the only

and we then have

f;^^^=^H,<

(8)

.

1

2 *),

so that

j;3g(*--r**-J&$*.
(9)

and these are valid if R (v) > \. The last formula was established in a different manner (when i>=l) by Struvef and from it we deduce the important theorem that J, when v > $ and x>0, H„(#) is positive. Struve's integral is of considerable value in the Theory of Diffraction. ;

Some

variations of Sonine's discontinuous integral are obtainable

multiplying by b»

+1

and then integrating with respect to b from

to

by

b.


^m+i f Jo

(bt)

—^ + ^—

t*

dt

the upper limit in the last integral being b or *

My thanks are

a,

due to Professor Love for pointing out to

whichever

me

t Cf. § 10-45.

(3)

the smaller.

the desirability of emphasizing

the ambiguity of sign.

t Ann. der Physik und Chemie,

is

xvn. (1882), pp. 1010—1011.


418

<

If b

the integral on the right seems intractable, but,

a,

— a sin

put u

[CHAP.

when

b

>

XIH a,

we

and deduce that

R (v + 1) > R (/*) > —

provided that

1

this is

;

one of Sonine's integrals.

If we replace a by u in (1) and then take a ^ b and integrate with respect u from a to oo after dividing by u v ~ we find that, when z is restricted to be l

to

,

positive,

_ a"- b* /"* ~ ^-"-'Jo 1

9«l»-l

fl""* «/y _ M -i (re) cfo

**

^"^•'-'1 ^-M-i(^)exp{-<2 /•»

V(v) -

§

13 3

(4),

(^+62 )}rft;^

Jo

Jo

2n—+i a >- i fr*

by

2 )''

foo

Too

Tjji

= r 7w y^-x 1 (v) z

+6

(w2

/

aia

z*

jo

and thence we see that

W

provided that a

M+1 « = 6 2> -! r (y) *m ( *)> + ^ )iy+1 * and i£ (v + 2) > R (/a) > - 1 the restriction

positive

removed.

J,

(11) Jo

Formula

(10),

(ff

;

is

which may be written in the form

J M («)

(12)

that z

Jo

(

^+

^)iv

t

at-

^

gv

,

where i2 (v + 2) > .R (/*) > and 6 > a, has been generalised in two ways by Gegenbauer*, by the usual methods of substituting Neumann's integral and Gegenbauer's integral

The

first

§ 13-1) for

(cf.

the second Bessel function.

method gives

2*- i r(/*)

J

6"

provided that b *

> 2a and

v

(az)Ji(az) zK+v

R (v + X + f ) > R (/*)

Wiener Sitzungsberichte, lxxxviii.

> 0.

(2), (1884),

pp. 1002—1003.

'

INFINITE INTEGRALS

13*48] If

= V(«-2 + c — 2oc cos 2

-or

P J*W

(14)

),

the second method gives at

t

(?Tfy

Jo

419

î: r.™'^g& ~***«* t

_2*- T(ji) 1

> a + c and

if 6

By

i2 (2i/

induction

(15)

/o

it

J.M

J

v

(az)

J

v

(cz)

+ f ) > i2 (» > 0.

follows that, if b

•

> 2a,

~*

v+ ,r.

where the product applies to n values of

a,

M = -5^-' 7 * L

J

and

R(nv + ln + %)>R(n)>0. If the induction of the second

find

still

The

special case of (15)

when

this has

is

used after applying the

first

method

once,

we

z-»-0 is

iy,,.{bt)u[JA«t)-\r-

(16)

dam,

method

further generalisations.

nv - x

^

*- 2

k)

been pointed out by Kluyver, Proc. Section of 749—755.

Sci.,

?[f^Tj]

;

K. Akad. van Wet.

te

Amster-

xi. (1909), pp.

13 '48. The problem of random flights.

A

problem which was propounded by Pearson*

sional displacements)

is

as follows

(in the case of

two-dimen-

:

" A man starts from a point and walks a distance a in a straight line he then turns through any angle whatever and walks a distance a in a second straight line. He repeats this process n times. ;

"I require the probability that after these n stretches he r+Sr from his starting point, 0."

is

at a distance

between r and

generalised form of the problem, in which the stretches may be taken the help of to be unequal, say (h, « 2 , -.an, has been solved by Kluyverf with subsequently the discontinuous integrals which were discussed in § 13*42; and

The

RayleighJ gave the full details of the analysis of the problem (which had been examined somewhat briefly by Kluyver), and then obtained the solution of the corresponding problem for flights in three dimensions. If sm

is

the resultant of a1} a 2

,

. . . ,

am (m =

1, 2,

. . . ,

n - 1), and

if

6m

is

the

* Nature, lxxh. (1905), pp. 294, 342 (see also p. 318); Drapers' Company Research Memoirs, Biometric Series, in. (1906). Wet. te Amsterdam, vm. (1906), pp. 341—350. t Proc. Section of Sci., K. Akad. van 321—347. [Scientific Papers, vi. (1920), pp. 604—626.] xxxvn. (1919), pp. % Phil. Mag. (6)

J


420

[CHAP. XIII

angle between sm and a m+1 then, in the two-dimensional problem, the angle m between — ir and it are equally probable. ,

Now

Pn

let

{f",

a lt

all

values of

an ) denote the probability that after n stretches less than r, so that the probability between r and r + Br is «2, ...,

the distance from the starting point shall be that the distance

lies

dP n (r;

a 1} a2 ...,an ) dr ,

~

It is then evident that

Pn (r;

a,,

a2)

. .

,

.

a„)

=

... /

J

J

J

— it and

where lf 2 ...,0n -z assume all values between assume only such values as make* ,

each set of values of

for

Now

(§

lt

2

,

...

,

0n_z

d0n_t d0 n _* ...d02 d0u tt,

while

.

1342)

£<'>

rf^(r«)j;(«.«)*=n' and

so,

— iy tuple

i/ iîs discontinuous factor is inserted in the (n

may be taken

range of values of n_x

We

n_ 1 is to

to be

(—

integral, the

ir, tr).

change the order of the integrations with respect to

n_1

and

t,

and,

remembering that s2,,

= s2n_i + a\ - 2*n _i an cos

n_lt

we get

r\ J

by

J

I

1

—vJ

(rt)Jo

(s n

= 2irrf

t)dtd0n . 1

We

§ 11-41 (16).

next 8

S „_i

make

J

-Pn (^

J

= s n _2 + a2„_j - 2s„_2 an_j cos

a1}

Og,

. .

.

(sn -x f)

(an t) dt

the substitution

2

and perform the integration with respect to cess we deduce ultimately that ,

an )

=r

r» Jo

and

J, (rt)

Jo

n _2

.

By

n_2

,

repetitions of this pro-

n J, (rt) II J" (am <) dt,

w-l

this is Kluyver's result.

We shall now consider the corresponding problem for space of p dimensions. In

this

problem

it is

no longer the case that all values of m are equally likely. which m is regarded as a co-latitude) are

If generalised polar coordinates (in

used, the element of generalised solid angle contains sin p

~2

m d0m and m ,

varies from

to

tr.

m only by the The symmetry with respect

factor to the

polar axis enables us to disregard the factor depending on the longitudes. * It is to be

remembered that tm

is a

function of the variables 6\,

2

*

•

•

»

#m-i

•

If less

421

INFINITE INTEGRALS

13-49]

P„(r

than

we

r,

P

a„ \p) denotes the probability that the final distance deduce, as before, that

ay, 02,

;

...

n (v, a1 ,a2 ...,an \p) ,

.

rtteL-pf '('...

1

I" VVP

is

,

~ 2) r ($))

H

n\in*->em .de„de^...d0,de

l,

JoJnt-l

.'c.'o

where the integration with respect to 6n- extends over the values of which make sn < r. The discontinuous factor which we now introduce is x

^J.^^-SS^Îo,

(sn

n-i

>r)

and then, since by § 11*41 (16), 1

we

^p— wr+e^de^ - r

(*p

- *) r <*) -^—-^r-

•

(

W)b~

>

infer that

P.(r;« ,a if 1

...,a m

|p)-r{r
a„ are all equal to a, and w is large, we may When the displacements aâ^ approximate to the value of the integral by Laplace's* process. The important part of the integrand is the part for which t is small, and, for such values of t,

Hat)*'-1

*

aH\

(

so that (§13-3)

Pn (r; a, a,

....

a \p)

~

f^/'Ô^Ôexp (- ^) dt

^ 'TV^)&'^w,WT

1

''^

2naV + l)V2«aV -»-»V been carried much further by Rayleigh in 1

'(ip

'

This process of approximation has the cases p = 2, p = 3, while Pearson has published various arithmetical tables connected with the problem. 13*49.

The

The discontinuous integrals of Gallop and Hardy.

integral

f-

J»{a(« + «)lfr,lft(C+«)

J -co is

convergent

if

a and

(*

+ *)*

6 are positive

condition must be replaced by

(? +

and

i2

(ji

i

rff

*)"

+ v) > — 1 when ;

a = b the

last

R (/a + v) > 0.

The special case of the integral in which /*=0, v=$ has been investigated by Gallop, Quarterly Journal, xxi. (1886), pp. 232—234; and the case in which a=b has been investigated by Hardy, Proc. London Math. Soc. (2) vn. (1909), pp. 469. The integral is obviously to be associated with the discontinuous integrals of Weber and Schafheitlin. * La thiorie analytique de$ ProbabiliUs (Paris, 1812), chapter in. The process recognised as a somewhat disguised form of the method of steepest descents.

may

be


422

To evaluate the integral in the general Hardy is effective; suppose that a^b, and -ft

(/*)>£» so

Write

t

method discovered by

at first let us take

R(v)> — £,

may be substituted for the second the integral's which will be used are absolutely conin place of t + f, and let z - £= Z, so that the integral to be

tna

Poisson's integral

t;

Bessel function and vergent.

case, the

[CHAP. XIII

all

evaluated becomes

J»{a(Z+t)}Jp (bt)

f-

(Z+ty

i_co

(46)'

= '

v

(kby

=

f

L

00

r

r( y +i)r(f)J—

= r(y + 4)ra) J

—z^ty

f'J^[a(Z + t)\

f«

r ( ,+W(4j ,L

n

J o

J

(at)

~*^ cos

2 (4&y

r*

- (2»>»ro.7i)i>+i) J. by a

{6 ( *

^ btc08 ^ sm'^ d^ dt

"^

cos

cos ^ ht cos *^ cos

~V

I

co

(

(a*

*

sin2v ]

^

cos

*****

^

sin2v

* d * dt

^

***•

=\

a case

- * cos' **"' cos <**« «

special case of § 13'4 (2).

This integral

is

expressible in a simple

considered by Gallop, or

We

when a = b, the

easily obtain Gallop's

£

(1)

f"

/ON

manner only when

/*

t

case considered by Hardy.

two results

sma(z + t)

sina(^

+ «)

Jo {H) dt

= wj

r ,,,.,.

9

f

a

(bz)>

cosw^.dM

(6

^ a)

.,

and Hardy's formula

w

r»

J—

^{o^+qi/ô^+o} (r+0" u+o*

The reader

r(^ + y )r(j)

r^+^r^ + i)

will find it interesting to obtain (1) eai i* +

/

z+t

t)

J

by integrating

(bt)

dt

round the contour formed by the real axis and an indefinitely great semicircle above has to be supposed that there is an indentation at — z when z is real.

The

integral

has also been considered by Gallop.

Z

+t

To

evaluate

z+t'

it,

we observe that

it

;

it

INFINITE INTEGRALS

13-5]

and

may be

so the integral

P

f- sin a

(z

+ t)} J

f-

sin

written in the form

dt+ f JO

(bt)

J -oo

.'

sin

. (« + 1) ?

-oo

1)

J

(bt) dt

sino(*-M)

f

Z

.'O ~

^+«

sin ai Jo (&0

JO ra

a (z +

_2

+£

r oo

= 2 cos as

423

a

/"

oo

I

•'

.'

rf<

+t

cos

w (* +

sin

ttz

ô (&0 «ftdw

- oo

roo

— 2s

cos

m (z +

Jo (&0 dtdu

J

+ 2z

'o .'o

ra

/•oo

= 2 cos az

sin at

{bt) dt

I

/•«> /

sin ut

J

(bt)

dtdu.

Jo Jo

Jo

Hence, when a>b, ...

f°°

\t\sma(z + l

(4)

t)

,,^-n eft

T

Jo (&0

J

,

=

2cosaz -77^

—

rrt

=—2 COS—a^ V(a

but,

when a <

2

,

+

f

a

2*

sinw^ -—

,

j--

du

Tare cosh a/6 .

+ Zz

sin (zb cosh 0) d#,

I

Jo

b-)

6,

^

!

(o)

-

J_

'-J (bt)dt

= 0.

13 *5. Definite integrals evaluated by contour integration.

A

large

by considering

of definite integrals can be evaluated

number

integrals of the forms

^(z)H^(az)dz, 2^./*(«) ®*(bz)H» taken round suitable contours; function,

and that a

it

is

supposed that

(az)dz,

(z) is

an algebraic

is positive.

The appropriate contours

are of two types.

We

take the

first

type

when

upper half-plane; the contour is (z) has no with its centre at the origin, axis real above the semicircle taken to be a large at the origin) which joins (indented axis real together with that part of the singularities except poles in the

the ends of the semicircle.

We

take the second type

when

(z)

has branch points in the upper half-

type by inserting loops starting from and ending at the indentation, one loop passing round each branch point, so that the integrand has no singularity inside the contour. plane; the contour

A

is

derived from the

more powerful method

(cf.

§

first

131) which

is effective

integrals with Bessel functions under the integral sign

is

in evaluating

to substitute for the

Bessel function one of the integrals discussed in § 65, and change the order of

*


424

[CHAP. XIII

the integrations since the integrand in §6'5 (7)

is (a?-*), qua function of #, number, the double integral usually

;

where 8

an

is

arbitrarily small positive

when the

converges absolutely

produces no theoretical

and the interchange

original integral does so,

difficulties.

Hankel's integrals involving one Bessel function,

13*51.

Before Hankel investigated the more abstruse integrals which will be discussed in Chapter xiv, he evaluated a large class of definite integrals* by considering 1

taken round the is positive,

m

first

f

zo^H^jaz)

type of contour described in § 13*5. In this integral, a is a complex number with

a positive integer (zero included), r

is

and

positive imaginary part,

\R(v)\
The only singularity of the integrand

inside the contour is the point

r.

It

follows that 1

1 [« of-

{Hv u

(ax)

- e*™ JT,w

j_

,

p tiwwes. (£_ r2)»»+i

47riJ

It follows from §

(1)

(axe**)}

(x'-^)m^

ZiriJo

["[(1

362

X_

J_

f<"+>

^HV(az)dz (^_ r2^m+i

2m, J

r ,„-, B

_j__ (A 2.m!\rfW

l

„, "

(ar)] Kar»'

(5) that

+ e 0>-*)*») Jv (cue) + % (1 - e (p-")«)

Y. (ax)]

J o

-^ —

xf>~*dx

(tfr

This result can be expressed in a neater form by writing r = ik, so that (k) > 0. It is thus found thatf

R

(2)

J"

[cos

\(p-v)tt.Jv (ax) + sin

(p

- v) tt Yv (ax)] .

{

(

\»»M-i

/

££^

J

\tn

-«£n(ra) * Hankel's

^

'.(«*»•

—

work was published posthumously, Math. Ann. vm. (1875), pp. 458 461. A partial v=n, p=2n + 2, m=2n was given bj Neumann, Theorie der

investigation of the integral with Bessel' schen

Functionen (Leipzig, 1867), p. 58. t The evaluation of integrals of this character which contain only one of the two Bessel

functions

is effected

in § 13*6.

A

!

^

,

:

INFINITE INTEGRALS

13'5l]

The reader should J^

425

notice the following special cases of this formula

vn.Jv (ax) - sin vtr

{cos

(3)

.

f

°°

.

J „ (or)}

^-^^ = ——l-î-jZ

m #"v + «/„ (a.r) <£r x JAflx)dx _ a it"- '» l »

A'„ _ m (ak)

2™ »» ~(^+i^ + i~~ The former is valid when — 2m - f < 72 < and the latter when - 1 < R (v < 2m + 1 For an extension of (4) to the case when m is not an integer, see $ 13*6 (2). (4)

J

.

I

(i/)

The

special formula

—

xJ (ax)dx

/""

...

~^^f

(5)

Jo

has been pointed out by Mehler, Math. Ann.

)

,

„

.

(a*),

xvm.

(1881), p. 194,

and Basset, Hydro-

dynamics, n. (Cambridge, 1889), p. 19; while Nicholson, Quarterly Journal, xlii. (1911), p. 220, has obtained another special formula } o ( ax ) (6)

Some

(ak) '

integrals.

resembling those just given may be established here, to prove them without using Cauchy's theorem.

integrals

it is

Q

k

by a complicated transformation of repeated

though

K

dx

x* + i:*

/:

most convenient

Thus, Nicholson has observed that f*

J

_

(ax) dx x*

j

2 f" fl" cos (ax cos 0)

~Wo

+ k*

,

- ,

aF+T*

Jo

1 [i*

fCJo

= ~{I by

§ 10*4 (11),

provided that a and

,».

More

r°°

Jo(ax)dx

generally, if i? (v)

since,

by a

R (k) are IT

,

T

(ak)\

>

both positive; so that ,.

T

/

;xi

> — £, we have

Iv (ak) - L„ (ak) = and

(ak)--L

T ( v + i)Y(l) ll"

~ e akC0S6 sin2 "

W

special form of (2),

_

.

J

J

provided that 22 (*>)

— oo

*£

< 2 and a

~r A?

is

6~

(2fc) ft*

positive, it follows that

AW-Ma*)- irr(y + i3r(i) io

j_.

~^¥

H~" f °° J, (aa?) t'H„ (ax) 2 ar' + fc ""tTJ-oo -f-

__

= ^r

Jo

[(1 H-

*-)

^ (a*) +

»

*""

,

(1

- *"*) H„ (a*)]

^—

s

-

a


426

and

[CHAP. XIII

we have the formula

so

(8)

[cos \vtt

I

where a >

0,

.

J

(ax)

v

R (k) >

+

sin \vir

.

H„ (ax)]

-\
and

,

2

= ^v \IV (ak) L„ (ak)},

The change

in the order of the

integrations presents no great theoretical difficulties.

A

somewhat

similar integral

is

f

°°

xv

If

we choose

(ax)

v

+

x?

Jo

which converges

K

dx '

k*

R (v) > — \ and R (a) > 0. k so that R (k) > 0, we have, if

rx K v

v

«»

Jo

{ax)dx

_ Y(v + l)

+#

~

r(i)

p

by

§

616

(1),

f" (2a) cos xu.dudx v

Jo io

+

(a?

#)*>«

+

8 )-**

_ 7rrQ + i) f" ( 2aye- vk du ~ 2kT(i) Jo (M + aT+* s

'~ 1

7r2A;,

4 COS

when we use

and

§

therefore,

1041

when f" Jo

(10)' v

(3).

Hence, when

•

The

,

Kv

R(v)> — ^,

}

— •_£

a

= v+1 2 x2 + kf> xv cos 4k

— 0)

v

vtt

and that a and

r <**(**) dx _

R (k) 2

v

Y

v (ak)}. v Ji

(ax) dx

To evaluate it, we suppose that we then have

gay

are both positive;

r

C* cos (ax cos 6)

Jo^+^"^"T(i» + i)r(i)Jo./o =

{

J -^

has been investigated by Gegenbauer*. |

{H„ (ak)' -

are due to Nicholson, and the last has also

^f°

i

integral

R (v) > —

- F_, (a&)},

.R(i>)<£,

These formulae (when been given by Heaviside. m,

{H_„ (ak) V7T

«?

+ k*

[*%-«*«>•• w T(v +ûln, \)Y($)kU (

sin

ViWd*

sin2 " Odd,

and so

* Wiener Sitzungsberichte, lxxii. Gegenbauer's result is incorrect because (2), (1876), p. 349. he omitted to insert the term - X, v (ak) and consequently the results which he deduced from his formula are also incorrect. A similar error was made by Basset, Proc. Camb. Phil. Soc. vi. (1889), ;

p.

11.

pp. 422

The

—

424.

correct

result

was given by Gubler, Zurich

Vierteljahnschrift,

xlvii.

(1902),

INFINITE INTEGRALS

13-52]

The condition R{v)> —

R (v) > — f

condition

An

integral

which

,

may now be

\

may

be evaluated in the form of a series by this method

psinha^ is

y

+

1

+ \I(b)\<7r and R(»)>

;

it is

supposed that

-1.

-L f^^ z HW (bz) z v+i dz 2m J sinh nz

taking

round the contour used in this section, we find that the definite integral

sum

is iri

times the

of the residues of

sinh at the points

i,

2i, 3i, ....

J o

The

sinh irx

tt2

It follows that

r*^^J (bx)a?*

(12)

¥

v

'

x

sin 2 (-)«-»»»+i '

dx=it

na.Kv {nb).

„=i

series converges rapidly if b is at all large.

An integral und

is

sinh irx

a generalisation of Neumann's integral described in § 13*2

\R(a)\

By

replaced by the less stringent

by analytic continuation.

J

which

427

expressible as a similar series

was investigated by Riemann, Ann. der Physik

Chemie, (2) xcv. (1855), pp. 132—135.

The generalisation of Hankel's

13*52.

integral.

Let us next consider the integral

27TIJ

This differs from Hankel's integral in containing the (complex) number ft The conditions for convergence (with the second

in place of the integer m.

type of contour specified in

§

135) are*

\R(v)\
a>0,

is chosen with a loop to exclude the point ik, as shewn in and then there are no poles inside the contour; and the integral round

The contour Fig. 31,

the large semicircle tends to zero as the radius tends to infinity. 1

——

;"°° I

\H

to

(ax)

— «(p-*W H

w (axe ni W

——dr—— tik+) Z9-iH

-JLf ~~

Now

I

2m iviJo

2

(*

+#r +1 As

27riJo

= — lg(Jp -(»)"» —

in § 13-51,

* we

Hence

re-1

(z*

v

w(az)dz

+ k*)*+1 \irj_

r^p-^r^ + i)-

take R(k)>0.

1


428

Hence, when we expand [(l

H

(1)

v

(az) in ascending

powers of z, we find that

+ 6 (P-r-U)H )t 7; (ax) +i (i _ ,o-r-*M) Yv (ax)]

J^

(

W

w-eKp-r-a^w

r f,,

r(/*+ +

V2

sinv7r.

x

" /'

„ _,. .,„_„.. -2M-2 '"

J+ ^+

1

r (\p + %v 4- m) (fafr) ™ m\r(v + vi+l)r(^p + ^v- fi + m) 2

2

.

m=0

1) L

W

•

[CHAP. XIII

^«m!r(- + m + l)r(^-^- At + m)J'

'

I/

and therefore

x '~ dx

00

f

[cos (£/>

(!)

- \v - fi) Tr.Jv {ax) 4- sin (\p -\v-fi) tt.

Jo

l

1

F„ (ax)] /^ w-n +jfc2

TT^rP-aM-a

2 sin vrr

x

_

.

F (/* 4-

1)

1 lf( v +i)r(is p+isv- H,y**{~ 2~>

(ja*)*-

r (j p - \v)

2

"^

fp-yp-y

.

w

+

1 »

„.

TV

a'JrVI

Fig. 31.

whether the integral of this type which contains a single Bessel function cannot be evaluated; it seems that the only effective method of evaluating it is the method which will be explained in § 136. It is natural to enquire

13*53.

Hankel's integrals involving a product of Bessel functions.

Integrals resembling those of § 13*51, except that they contain a product of Bessel functions instead of a single Bessel function, have been investigated

by Hankel* by applying Cauchy's theorem

*rij* in which a

^ b > 0,

m

is

K{bz)

R O) •

(z*-r*T*

'

is a complex number with a any cylinder function of order /i, and

a positive integer, r

positive imaginary part, 9B^ denotes |

to tjie integral

|

4-

R

(/jl) I

< R (p) < 2m 4- 4.

Math. Ann. vm. (1875), pp. 461—467.

C

-

INFINITE INTEGRALS

13-53]

[When £>

is

a Bessel function of the

first

kind,

It (p)

j

429

may

i

be replaced by —It(fi) in

this inequality.]

When a = b, the presence of a non-oscillatory term* in the asymptotic expansion of the integrand shews that we must replace 2m + 4 by 2m + 3 in the inequality in order to make the integral, when taken round a large semicircle above the real axis, tend to zero as the radius tends to infinity. The contour to be taken is that of manner of that section, we find that

^ (M

[r

(1)

ds

Hvil)

(ax)

" *" r

^

§

13*52

Numerous It

id \ m

strata?)

(f-lr^

[•^"»-o)ff.» oh.

when.p = 2wi + 3 and a =6, the

that,

round the large

integral

semicircle tends to a non-zero limit as the radius tends to infinity ; and,

^

in the

by Hankel.

special cases of this result are given

must be pointed out

we proceed

if

{hxelH) Ht,{l) (axeni)] 1

-

and

;

if

= c, Hfif) (az) + c2 H^) (az),

(az)

we then obtain the new formula (2)

The

)

(«) +

^(«^^K«^](^^|-,

particular case of (1) in which p

of the first kind deserves special (3)

= 2, m =

mention

;

and

^

is

a Bessel function

it is

f J» (bx) [cos \
tt

F„

.

(oar)]

—

= $Trie* '""^ •* JM (6r) #>> (or), provided that a If

we take

jt

>6 >

=v

i2 (/a) >\R(v)\ — 2.

and

and

.R

(»>)

>—

P° J (ax) J (bx)

(4>)

according as a $

1,

we

-^L. -

see that i

*«* J* ( br)

H

(l)

»

(

ar)>

6.

The existence of the discontinuity in the expression for this integral was pointed out by Hankel. If

we modify formula f"

(5) J

(3)

x

^rp

-^m

(&*) [cos

we

see that, if a

H/* -

v) if

•

«A.

>b >

(ax)

and

R (k) > 0,

+ 8in$(fjk-v)Tr. Yv (ax)] dx

= * Since

HM

<2>

(az)

H m (az) ~ — «*G* -•>** when ¥

then

|

«

f

'

I»(bk)Kv (ak).

is large.


430

More

generally, taking equation (1) with

r

(6)

,,

Jo

rcosiCo+ a— ^— [cos£(p+

m=

/i— z>) v) tt.J„ 7r. Jv (ax)

,_

^=

and

+ sin

[CHAP. XIII J^,

we have

F„v (ax)] dx + fi n — v) it. Y

i J (p

= -I (bk)K (ak) !
ll

In this result replace p by p + v, a by V( a' + c + c2 — 2accos 0)*", and integrate with respect to 6 from to 7J-; we find from Gegenbauer's formula, § 11*41 (16), 2

multiply by sin 2 "^/(a2

0)

+ p) 7r J v (ax) + sin | (p + p) it Y

[cos \ (p

j.

s

J\

v

(ax)]

dx

<

^-I^bfyl^cfyKriak)*?-*. This process

repeated as often as we please

may be

and we

;

find that, if

a>b+1c, ( fi )

r JO

then ~ *P I

X

to)

/+ KL ,t

" J ( c^) -> s i {p + + (^ - 1) *} t ^ (a*) /*

*

M=i

•

+ sin \ [p + p. + (N- 1) v) ir Y .

= -!„. (bk)

UI

v

(c n k)

.

K

v

v

(ax)]

dx

(ak) k»-*. .

n=\

Again, by considering

SjJi^[n-W] *"<»>* round the contour previously used, where both

and

b

p.

differ in the different

factors of the product, we obtain the slightly more general result

(9)

I" x^~k*

[n

+ provided that a

>S

Jflibx)]

sin \ (p |

£ (6)

|

[C0S '

* (/°

+ Xfl " v) W

'

Jv {ax)

+ 2/i - v) 7T F„ (cm-)] dx = - [II 7M (&&)] #„ (a&) A?" and £ {p + X (/*)} > -R (*>)

2

.

I

|.

+ 2/i — v

is an even integer, the integral on the left involves functions kind only; a result involving the integrals of products of functions of the first kind of this type was given by Gegenbauer, who overlooked the necessity for this restriction (cf. §1351).

If p

of the

An

first

extension of Hankel's results 1

fy.

round the contour, where a ^

is

obtained by considering

ÎM^+ni b

>

0,

HV(az)

m is a positive integer, and

\R(v)\)<2m + 4, + R(ji). It follows that

=

_J_

(

Ar r,^ j.i^<-r+»! *,»

"

(or)

:

INFINITE INTEGRALS

13-54]

431

and, in particular,

a result obtained in a much more elaborate manner by Sonine, Math. Ann. xvi. (1880), pp. 13*54.

An

56—60.

Generalisations of Nicholson's integral.

interesting consequence of Mehler's integral of §

Nicholson*, namely that,

The method by which

when a and k

1351

(5) is

this result is obtained is as follows

=\ This repeated integral

j" /*

-r^-p J>

may be

+ ** -

when p

(p~*)

of origin of the polar coordinates

p cos

{» V(p*

2pi cos *)) dfdp.

regarded as an absolutely convergent double

integral, since the integrand is

is large.

Now make

by writing

= k + r cos 6,

p sin

= r sin 0,

and we have l

K (ak\ T9 {ak)--] (ah\t K.[a*)J

X

9

and


To

due to

are positive,

J*( ar ) rd0dr

('

J or + .

2jfer

-

[* Jo(ar)rdr

O oB0+2Jf~J o v'(^

+ 4^)'

be .established.

generalise the result consider 'z"- 1

f

H ^{az)dz v

(z*

+ bk y +i i

taken round the contour shewn in Fig. 32.

Pig. 32.

It is supposed that

a

is

positive, and, to ensure convergence,

\R(v)\R(fi)

+ ^-.

* Quarterly Journal, xlii. (1911), p.

'2 ,

24.

a change

,


432

It is also supposed that

arg k

|

<

j

[CHAP. XIII

and the loops in the contour surround

\tt,

the points ei«ArV2,

By (2)

analysis resembling that of § 13*52, the reader will find that [cos

j

e^ks/2.

(\p- ^v-

-4 - w k V 2 )""* (

f

2/x) 7T

.

J

v

(ax) + sin

(\p- \v-

2/x) tr

.

(akW 2Y + m T(±p + \ v + £m )

|

Y

(ax)]

COS Qg

p+y-4/n+2m "*

v

^ +4jk,^

+1

4

2sini/7r.r(/t+l)Lm=o»i!r(i/+m+l)r(i 3+ii/-/A+i?») /

(akl^2)- v+2m T(ip-jv + jro) p - - 4/z + 2m 1 _ v COS 4 ~mt w!r(-i/+m + l)r(i o-ii/-/A + iw) ""J i/

'

/

If the series on the right are

that the former or if p or if p

is

compared with those given in

§ 5-41, it is

seen

— 2 = i/= /* + £ Up — 2 = — v— p. + \

expressible as a product of Bessel functions if p

— 4 = v = p. + \, while — 4 = — = /4 + £.

the latter

is

so expressible

i>

integral which contains a single Bessel function will be

The corresponding considered in § 13*6.

13*55.

A

Sonine's integrals.

number

of definite integrals, of which special forms were given

Sonine, Math. Ann. xvi. (1880), pp. 63

—

66,

by

can be evaluated by the method

of contour integration.

The most

general contour integral to be taken •l

r

is

eUz+k)

2tti

round a contour consisting of the parts of the |s|

terminated by the lines arg (— z)

=

8,

\z\

=

circles

R,

= ± ir, and the lines which join the extremities

of these circular arcs*.

m is an integer and k is not a negative real number. = z & tends to zero as 8-^0, provided that R (p) > R (v) round The integral = R tends to zero as R -* oo provided that round \z\ integral and the It is supposed that |

\

j

,

R(p)<

m + $.

By Cauchy's theorem we have 2iriJ(^ +

^M+lV

w!

^

* Cf. 3/odern Analysis, § 6-2; or § 7*4 supra.

|

1

INFINITE INTEGRALS

13*55]

433

and thus we have (!)

^T^^-^-^M*)} m! dkm /• of= -L, ^_ [ep*HJ» (# + &) ~a—\1

Too „\(x+k\ i{x+k)

1

1

e *

=~

i„. Mm+i

In particular, taking ra

kr !!^ (k) = we consider the

.'

= 0, we

dx

H

^

+ ") *" + 2 * COS COS + 1 F„ (#) sin (p — v) 7r] dx.

get

?—

[ J„

(*) {sin (p

+

+ 2i cos i/tt cos pir]

v)ir

A;

+ iF„ (x) sin (/o — v) 7r]

cfa-.

integral 1

r

—

p-iiz+h)

r (" zY-'H® (- z) dz,

-

cr--

we

(xe-« l " )\

»p—

+

«c

o

'

(*) l sin
"

1

7r

If

J

fao gi(X+k)

1

(2)

l

- €T**H¥ m

(#e'"') v

l

2ttiJ

find that rco

1

(3)

hr*H® (k) = -

p-i(x+k) -p-i

f— [J, (a) {sin (p +

-

*>)

tt

- 2i cos i/ir cos pw\

— iYv (x) sin (p — If

we take p =

v

1,

«/„(&)

/e\

More

two

we

i

f»sin(a?+&) v ' X -r K

* 1— x+k

(a?

+ k)

,

.

,

(x)dx,

r/

v

,

to Sonine*.

=v+

and

l

— \ < R (v) <

£,

we get

o±vri

/•» ,,±1(2:+*:) ~if

ir J o

Jr

,

.

2 f* cos

generally, taking p 1

2 T

=

due

results are

7r] c£r.

get

//\

T7-

last

(6)

0,

r/7x

//i\

(4)

The

=

z>)

J, (x)

dx=

+

^

± iYv (k)},

A> [Jv (A)

2i cos vir

a result also due to Sonine.

—^

cos(x

... o Bywntmg

and using the formulae

Y

(7)

and hence from (8)

§§

(*)

+

(6)

= --

k)

and

1

* I

TJO

0„

1

-

f

.,

.

P ^?

efo

,vj.

sm « (. + *) A. ,

j§

Sonine deduced from (5) that

(7) of § 13*42,

irJox + k

356(2), 9-11

Yn (k) = - -

^

+-

**

sin

I

(fr

cos

6?)

dd,

tt.'o

(2),

O + A) Jo (*) dx + -T

* See alsoLercb, Monatshefte filr

' I

sin (k cos

- \ mr) cos n0f/0.

.'

Math, und Phys.

i.

(1890), pp.

105—112.

*


434

A new method*

13*6.

We

shall

[CHAP. XIII

of evaluating definite integrals.

now evaluate

various definite integrals by substituting for the

Bessel function, under the integral sign, the definite integral of § reversing the order of the integrations.

As a

first

6*5,

and

example consider the integral of Hankel's type

J

af~ x

/'

(aw)

v

dx,

Jo in

which

it is

at first supposed that

R(v)>0, and a

is

The i

+ 2)>R(p + v)>0;

R(2fi

a real (positive) number, in order that the integral integral

is

/"<

r°°

f(-s)

af- i

{

\axyâ

_A_r

r <**» +

y

this is evaluated (by

,-v

f

af-

}

^

J

v

(a?

!o

(ax)

_ _Ti_~:

s)

4-

(//,

we

find that

dx

+ krr +1

_ q"fr+»-»^r(ip + i y )r(/i+i-ip-iF) l2 (p + v.p + v +I 2*+i r(p + l)T(p+l) 2 \ 2 a^-pr(^ + £p-M-l) *-p. u+2-""^/ '*'

'

The hypergeometric

=

2

By

+

'

4;

a '*^

functions on the right are reducible to Bessel functions

in certain circumstances

p

o^-\

,'

'

if

&.

1)

swinging round the contour so as to enclose the

poles on the right of the contour) 1

i

+ «> r <" + \~ **»-*»-«> 1

°°

converge.

equal tof

x (jar*. jfr+*-»-»

When

may

;

the former

p

if

=v+

2 or p

= v + 2fi +

2,

the latter

i/.

the principle of analytic continuation (1)

is

valid

when

-R(p)
Jo

we

find that

(x*+kr +l

>ro.+i)^"

(

}'

a formula obtained by another method by Sonine, J/a*A. Jrm. xvi. (1880), valid

p. 50;

it is

when

-KR(v)<2R(lt) + $. * This method is due to Lerch, Rozpravy, Math. (1896), p. 233]; he shewed that

f /

but no other use has been t

The change

_

,

fit

v. (1896),

made

/*""

1 y

2at

— ooi

no. 23 [Jahrbuch iiber die Fortschritte der

T(-s)x2'd8 ]

«r(«

+ i)

'

of the. method.

of the order of the integrations

may

be justified without difficulty.

"

.

INFINITE INTEGRALS

13-6]

435

A formula of some interest is obtained by making p = l, p> = — \, the hypergeometric functions then reducing to squares or products of Bessel functions

and another such formula

;

deduced that

It is thus

(3)

Jo

(cf.

R (p) > — 1

provided that

R (p) > — \

found by making p

= hv {hak) Ki

7*T*F

provided that

is

=l—

v, p,

=v—

\.

§ 5 '41)

"

{l
and that

;

Next consider stf~x

Jv (ax) dx

Jo f in which a

>

and arg k |

J

< \ir.

R(p)>0, The

It is first to

J2(4^

be supposed that

+ 4)>i2(p + v)>0.

integral is equal to

Iff"

r<-«)

^-'(i^^j.^ r(i;+* + i)r(M +

87n'J_ wi

~ sin (£p + £i/ - 2fi) 7T r (/a +

1)

i)

x {\ay** (k v/2>'+'' +2»-^- 4 ck

Lm=o »»! r(v + m +

1)

T (i/» + i v - 1* + |m)

x cos (±p

^ laid

on

p.,

v

is

m! T (2/* - }p + \v +

a representation of the integral when the conditions to be

and p are 4i*( M )

Now

+ JgL >£(,>)>-£(*).

take the cases in which the

functions,

By

reduces to a product of Bessel

or

p

— 4 = = /i + £. i/

we then obtain the formulae

(5)

(6)

first series

namely

p-2 — v — fi + ^ § 5*41

-'>+4wl r (fi + m + 1)_ 2m + 3) T (2/* - 1/3 - \v + 2»8m + 3)J

(-)m (a/fc/V2)^ +4

*

This expansion

+ i v — fi + \ m) ir

+

4*rr ~ (2*r r (» + i) J

J

(*•

Jo

(^ + 4*T + *

-

w w *

M2^=il^HT) J ^

The former of these is valid when in both, a > and arg k\<\rr. |

'

72 (v)

> — £, the

(a *)

latter

'

^^ ^>

when

(

R (p)

> £, and,


436

an example suggested by

Finally, as

x of'1

f

in which a

>

and arg k < |

\

integral /•» root

j :

is

2tti ITl.'o .'<,./_.* J _ooi

R(p + l)>R(p + v )>0.

equal to 1

'

T(-s)T(p-{-v +

2s)r(fjL

+ l-p-v-2s)

irk*-*-1

T

~ sin(p + v — ft) tt T (p, + 1) mI=o w! .

M=0 first series

f

provided that

00

(laky*™ T(p + y + 2m) m + l)r(p + v- fi + 2m)

r(v +

r (fi + m + 1) sin $ Q + y - - m) ir 1 m!^(^ + ^-|/> + Jm + f)^(|/A -^-i >^-^m + |)J• fi

/!

reduces to

J

v

(ak)

by Lommel's functions

/(_.

when

= 0, and the second series

§ 107).

(cf.

x v Jv (ax)dx

/*

irk"

r __

is

then

In particular we have .

,.

„

,

....

— ^ < i2 (v) < §.

The reader will chapter

\_

(-)"»

'

(jafc>rH ~'>+w

_ y

expressible

V2

r(v+«+i)r(/* + i)

27rij_ocf

The

'

+2s of- QaxY ^ T(-s) V V2 ^^ / / 5+l) ^-T\ ~, TT +1 ,/ <&<&» T(i/ + (x + ky ^.

I

+ ky+i

It is first to be supposed that

ir.

R(v)>0, The

shall consider

Jv (ax) dx

(x

Jo

we

§ 13'55,

[CHAP. XTII

may be

find that a large

number of the

integrals discussed in this

evaluated by this method.

13 '61. Integrals involving products of Bessel functions. If an integral involves the product of two Bessel functions of the same argument (but not necessarily of the same order), it is likely that the integral is capable of being evaluated either by replacing the product by Neumann's integral (§5*43) and using the method just described, or else by replacing the product J^ (x) Jv (x) by i

r xi

__

r (- s)V (/a+i>+

2s

+ l)(^y* + " +M

2Tri] -aoiVi

in

which the poles of V (— s) are on the right of the contour while those of 1) are on the left this expression is easily derived from § 5*41

r (p, + v + 2,5 +

;

by using the method of obtaining The reader may

find it interesting to evaluate '

xp

/: by these methods. final

§ 6'5.

The

~l

J,,,

(.r

(ax) 2

Jv (ax) dx

+F)x + i

result is a combination of

element in each function

is

a 2 >P.

two functions of the type

3

F

t

,

and the

INFINITE INTEGRALS

13*61]

Another integral formula, obtainable by replacing p-i

f°°

(1)

x

/

Jv (b/x)

integral, is

+ 2^+P+ir(

^ +1

3

ft+ i)r(^+^+ip+i)-°

when a and

6 are positive

-T- +1 —a~ - +1 ~2— +

+1 V"

is valid

by an

Jv (b/x) dx

Jn {ax)

-2^-p+ir(, + i)raM+ i,-ip+l)-° /3

This

437

"

'

'

'

1 '

1;

~W) l6";

and

-Rfa + %)
Messenger, xtvn. (1918), pp. 134

had been given previously by Cailler, Mem. de la Soc. de Phys. de Geneve, xxxiv. (1902—1905), p. 352; Bateman, Trans. Camb. Phil. Soc. xxi. (1912), pp. 185, 186; and Hardy, who discussed the case of functions of orders + £, (see § 6*23). 137; special cases

An

interesting example of an integral* which contains a product f

lim

°°

is

due da

(a) J {x) -^ J xexp (-p*a?) (— p'a?) J _ Vv (w) «/„ v (a?) &

+

4^r^ {1+21o^^-^

(,/+1)

-^( 2 --

,,)

i]'

which may be written in the form

&X SIT!

°°

f

«

/

r

„\ f

/

\

r

\

/

+ 4^"l"-

f°°

)

d%

{» (2) - * (' +

1)

-* (2 - ') - 2 log 2p).

proved that

It is easily

jo

y)

V7T

d# i# exp^^)^^)^^)-^^^/

.

«x f

r

_ j_

/

x

t

,

sin

\

r- rt+.<

r(-«)r(* + 2),ft«)-»

*+«< s(s

_1

this series is

To

r(- s )r(2« + 2) rfs + s+i)r(2-i/+s)(2^),M

+

•

i>r(i/

(2n

4 „=i w.(n+

and

i/tt)

1)

!

!

+ l)!(2p)-«*

T(v + n +

1)

an integral function of

T(2 -

*/

+ w)'

1/p.

when p is argp \
obtain an asymptotic representation of the integral, valid

j

|

small and at

and

—

|

,

,

.

* This integral was brought to my notice by Mr G. G. Darwin, who encountered problem of Diffusion of Salts in a circular cylinder of liquid.

it

in a

—

-


438

Hence we

find

V(-s)T(2s + 2)

f*+**

J_

SiriJiî

s (s

ds_

+ 1) T (v + s + 1) T (2 - v + *) (2p)M

^-^^T)

{f(2)-yfr(v +

_ (l-2v)^nv7r ~2ttv{1-v)

and

[CHAP. XIII

l)-ir(2- V )-2\og2p}{l + 2v(l-v)^}

(-)» (2p)n T (%n) 2 | P + Z 2n(n-2)r(v + l-ln)r(2-v-ln).(n-2)\ n 3

so

(— p2 a?) J J - vV (x) exp (-p*a?) J v (a?) (%) J,_ (%)

(2) lira

x

—

dx

«-*-+o

+

4^=l0 {1+2logi8 -^

(,,

+ 1) ~^ (2 " I,)} ]

l)-yjr(2-v) ^^(2)-^(, + l)-t(2-.)-21og2p}^^; ->^^ 2

(X>

v

(-)"(2j>)T(in)

+ 2 2H(n-2)r(i/+l-^)r( 2 -''-£»).(™-2)! „t 3 In the special case v

iiioM) +

lim

(3)

= 0, we f

find that

exp ( _

p^) /t

w w £]

~ _i2^ + Mffo7r ,

13'7.

j,

|r(m + t)l'(2p)-*' + l) 2 (2m + 3).(2m +

.

2

(2m

l)!'

Integral representations of products of Bessel functions.

From Gegenbauer's formula

of §

1141

(16) an interesting result

is

obtain-

able by taking the cylinder function to be of the first kind and substituting the result of § 62 (8) for the function under the integral sign.


"ski. L.i and

if

sm *•'

we change the order 1

l

of the integrations, /"«+•'*

R (v) > - \

»

exp i(

we

Z* +

f,

<

dtd *' \

find that z*\

T

(zZ\ dt

but the former restriction may obviously be replaced by R(v)>- 1, and the latter may be removed on account of the symmetry in z and Z. It is also permissible to proceed to the This result

limit

is

proved when

by making \z\-*\Z\.

and

\z\

\Z\;

INFINITE INTEGRALS

13-7, 13-71]

By

using the results of

R

provided that

The formula

(i>)

§ 6*21 (4)

> — 1 and

|

z

<

j

and

Z

j

(5),

we

439

same way that

find in the

\.

was obtained by Macdonald, Proc. London Math. Soc. xxxn. (1900), differential equations, and he deduced Gegenbauer's integral by reversing the steps of the analysis which we have given. The formulae (2) and (3) were given by Macdonald, though they are also to be found in a modified form in Sonine's memoir, Math. Ann. xvi. (1880), p. 61.

pp. 152

(1)

— 155, frpm the theory of linear

A further modification

on the right in

of the integrals

(2)

and

(3)

was given by Sonine,

the object of the change being to remove the exponential functions.

For physical applications of these 410—427.

integrals, see

Macdonald, Proc. London Math. Soc.

(2) xiv. (1915), pp.

We product

K

v,

shall

K

v

(Z)

K

v

(z)

as an integral.

next obtain a formula, due to Macdonald*, which represents the v (z) as an integral involving a single function of the type

v (Z)

K

namely

ww-trM-*-*?]*'®*

(1)

This formula

is

valid for all values of v

\ArgZ\<7r,

but

K

The expression of

13*71.

it is

|

arg £

convenient to prove

to extend it

it

j

<

7T

when

when

and \a,rg(Z + z)\ <

\ir\

Z, z have positive values

X,

x,

and

by the theory of analytic continuation; the formula, which

is

obviously to be associated with § 11*41 (16), is of some importance in dealing with the zeros of functions of the type v {z). It is possible to prove the formula

K

without the rather elaborate transformations used in proving §1141(16); the following proof, which differs from Macdonald's, is on the lines of § 26.

By

§6-22(7) we have

K

v

(X)

K

v

(x)

=\ **

— ~| £t

If

*

°°

\ J —00

J

e f

J

-r«+«)-Xeodi *-*eoSh« dtdu

—oo

|

e-ivT-Xoo&MT+U)-xco*h(T-U)

(Xe T +xe~ T)e u be taken as a new variable

/ * Proc.

dlJdT

—oo J -oo v,

in the integral

e -Xco&(T+U)-xco»h(T-U) flJJ -oe

Lon'Um Math.

Soc. xxx. (1899), pp.

169—171.


440

r it

becomes

I

f

exp

X

If

— a\ v +

2

[CHAP. XIII

+ a? + 2Xocco8h2T\l ~

dv ~^~>

*f

and so we have

K. (X) K, (.) = I,

£ [ -~-«M«"* e

exp

-

[

(|

-

^}] £

AT.

and, on performing the integration with respect to T, we at once obtain and x are positive. Macdonald's theorem when the variables

X

Nicholsons integral representations of products.

13"72.

We shall now discuss a series of integral

representations of Bessel functions

which are to be associated with Neumann's integral of §

The formulae

5-43.

of this type have been developed by Nicholson*, and the

two which are most easily proved are

K

(1)

IIL

K^tfz cosh

(z)Kv (z) = 2f

i)

cosh

(p-v)tdt

t)

cosh

(/*

Jo ,00

K^

=2

v

(2z cosh

+ v) t dt,

.'o

when arg z\<\ir, while To obtain

K

IL

these formulae

{z)K v (z) = -\ T*

The repeated

and v are unrestricted.

fi

|

J

0-s

(cosh

K

But

+ u = 2T,

H-coshM) cosn

^ cos h vudtdu.

t-u = 2U,

v

(z)

= \\ "J

I

e-«*«>* r cosh u cos h

/i(T+U) cosh v (T - U)

2 cosh

(ft

+

v)

T cmh\iL — v)

U+coshjjtr—

v)

T cosh

U + sinh

-

v)

v)

integrals corresponding to the last

vanish; and, if

^(T+U) cosh v(T-U)dTdU.

—00 J —00

+ sinh (ft + v) T sinh O we interchange

K„ (z) Kv (z) = \\ we

\

~ e %z cosh

integrate with respect to

integrate with respect to

(/j,

(/a

T sinh

+

v)

(/a

U

+ v)

U.

two of these four terms obviously Tand 6r in the integral

the parametric variables

corresponding to the second of the four terms,

If

which shews that

is absolutely convergent, and it may be regarded as a In the double integral make the transformation

srcosh

The

(5)

— 00 JJ — 00

t

h (z)

622

apparent that

it is

K

§

integral

double integral.

and

we use

T we

T C09h u cosh

U we

we

(/a

+

obtain the formula v)

obtain the

Tcosh (/a

first

form of

obtain the second form of

* Quarterly Journal, xlii. (1911), pp.

-v)UdTdU.

(1).

220—223.

(1),

and

if

we

INFINITE INTEGRALS

13-72, 13-73]

441

The formula 7M (z) I v (z)=- j" /M+ „ (2s

(2)

T Jo

cos 9) cos

which is valid when R(fi+v) exceeds —1, mann's formula.

we take

If

fi

=

and change the sign of (**

I (z)K v (z) = ~

(3)

7T

More

we

generally, if

find that, if

R (v

Im (s)

(4)

K

v

we combine

If

,

,

r

„

provided that

<

;

.

we take — m) 0

.

,

}

(z)

+ Yv

2

§ 13*6,

positive

formula

(3).

(z).

integral, corresponding to those just discussed, (z) is

— v,

and in particular

Nicholsons integral * /or J,2

2

by m and

find that

a result of which a more general form has been given in

J* (z) + F„

v

v

()

13*73.

and

/x

K _m {1z cos 0) cos (m + v)

/„(s)/f (*)=|

5>

we

Neu-

(2zcos0)co$v0d0.

1,

+ |)

— \ < R (i>) <

y

= — m and

§ 6'16 (1)

T

K

at once deducible from

is

o

ft

-—

with

(3) ,

=

.

v,

(ti-v)0 d0,

rigorously.

which represents

It is first necessary to

(= x), and it is also necessary to appeal some such principle, in the course

to Hardy's theory of generalised integrals, or

of the proof.

Take the formula

(§ 6*2 1)

H® (x) = — 7TI

ex sinh w .'

-" w

dw.

_c

which the integrand tends to zero as w -*• oc on the contour, it is clear that when an exponential factor exp {— \w-} is inserted, the resulting integral converges uniformly with regard to \, and so it is a

From

the

manner

continuous function of

in

\.

HJ$ ( X ) = *

Phil.

Mag.

]

j

Hence f lim

(6) xix. (1910), p.

—

exp {- Xw2

}

^

8inhtt,

-

w dw.

234; Quarterly Journal, xlii. (1911), p. 221. 22—66 ; Trans. Camb. Phil. Soc. xxi. (1912),

t Hardy, Quarterly Journal, xxxv. (1904), pp. pp. 1 tive

—

48.

In this integral (as distinguished from those which follow) the sign lim

is

commuta-

with the integral sign.

w.

B. F.

15


442

[CHAP. XIII

By Cauchy's theorem, the contour may be deformed into the line / (w) = \tr, so long as X has an assigned positive value writing t + \iri for w, we get :

H ® (x) = ¥

lim X-^+0

In

(t

-x

J

+

\-rri)')

- vt

eixcoaht

dt

rx

a—\v-ai

in Hardy's notation.

exp {- X

r-

W*

like

jy„W (#)

manner

=

e-'jcch «-,«

G

with an implied exponential factor exp {— \(u

— \tnf\-

Since the requisite convergence conditions are

fulfilled

when X > 0, we may

regard the product of the two integrals e l (t)ei * co* ht

|

(in

-

vt

dt x I

—X

.

ei

{u)e- ixco *Ylu

- vu

du

J —CD

which e^t) and

e2 (u)

stand for the exponential factors) as a double

integral

" / I

.

We

-X

—X

J

^ (t)

%

(u) e «(cosh <-co«hii)-v«+«) (dtdu).

thus find that Tr

2

^'

1'

=G

#„< 2 (x)

(a?)

>

T—X

J

[*

e <*(coBh*-corii «)-„(*+«)

—»

./

(^^

with the implied exponential factor

exp [— X (t

Make

the substitution ^tt2 [J* (x)

t

+

^rriy

— X(u — \irif].

+ u = 2T, t — u — 2U and

+ F„

2

(x)}

=gT— x r— J

.'

« 8te "

then

inh

r " il,hr- 8,,r (dZW).

co

with an implied exponential factor

exp {- 2XT 2

- 2X ( U + Jui*)

8 }.

In view of the absolute convergence of the integral,

it

may be

the repeated integral in which the integration with respect to first,

replaced by

U is

performed

so that |tt2 {J* (as)

+ F„

2

(x)}

=G

r I*-x e

2iesinh Tsinh

l

'-lvT

d UdT

L-'o J

X

+

r ! - x e-- ixsmUTsmhU+2 T dUdT ''

J

J

with an implied exponential factor in each case equal to

exp - 2XT2 - 2\ ( U + \ nrif) {

(

INFINITE INTEGRALS

13-73]

We

u

(

first

443


U + 1 irif] e% * sinh Tsmh

[" exp {- 2\ J — CO

dU,

T is positive. When T is positive, the tf-path of integration may be deformed into the contour I (U) = %ir; if we then write U =v + \iri, where v is real, the integral which

in

becomes exp {- 2\ (v +

[

2

7Ti') ]

e" 2* sinh Tcosh v dv

J -cc

= 2 exp (2\tt )

exp (- 2\v*) cos 47r\v e~ 2xsinh rcosht

2

=

'

.

2 exp (2XTT2 )

J

e-torinhroodi«dv

I

-'o

- 2 exp (2\tt ) 2

r

{1

- exp (-

2\v 2 ) cos 4tt\v}

e

- ixsinh Tcoshv dv.

Jo

To approximate

when X

to the latter integral

is

small,

we use the

inequalities

0^1— exp (— 2\v ) cos 47rXv 2

= 1 - exp (- 2\v ) + 2

so that, for

2 exp (- 2\v2 ) sin2 27r\t> £ 2Xv2

some value of 6 between

and

+ 8tt

2

X 2 v2

,

1,

exv{-2\(U + $Triy}eii** inhTainhu dU

f" J -en

=2

|"[ "

exp (2\7T2 )

e-2*«nhreo8h«

jv

_ 2 \ + 8tt \ ) 6

= 2 exp (2\w«) [ #„ (2a? sinh T) - (2\ + 8tt \ 2

If

we

2

2

2

)

6

f" w2 e-2*Mnfa rcosh* ^ w

i~ K^ (2x sinh T)\

1

.

treat the integral

exp{-2X(i7+^7n")2 }e- 2 a!sinh:rsinh ^rf£/' «

f*

J —CO

in a similar

manner, we find

2K <6 ^

where

X

On

1,

it

(2x sinh T)

provided that

collecting the results

equal to

- 2X0, \£- K„ (2x sinh

T is

T)\

,

positive.

means the same thing

and remembering that Jo

as

lim s-^+oJs I

,

we

find that

\Tf*\j?{x)+Yvnx))

=

lim

lim [* [exp (2\tt 2) iT {2x sinh T)

- (2\ + 8tt \ ) 2

+

lim A

lim

I"

2

exp (2\7ra )

j^^^sinh T)\ (o/x

1 e- 2 " r dr

J/*-oJ

lô(^smhT)-2X6 \^ K (2xsmhT)\ ]e*> T dT. Jm-oJ

++06-*+0J6 L

1

l

I

*/*

ll

—

T


444

Now, qua function

of T,

\~ K» (2x sinh T)\ T is

when

small,

convergent

is

X

\~K

(

ll

Jo

2 2 1 7T {J, (x)

Jh-o

making \

we

replace x

by

we have the

"

K

is

(2x sinh T) (e~ ivT

+

e

2" r

) rfT.

2,

2

when x > 0,

=—

(a?)

I*

t

K (2x sinh T) cosh IvTdT.

both sides of this equation become analytic functions

E (z) > 0.

provided that

f

-*•

Jo

+ F„

J, 2 (a?)

e^ T dT

(2xsinhT)\

+ Y* (x)} =

It is therefore proved that,

z,

[(log sinh T)*},

and, since the integrals

;

(cy*

are convergent, the result of

If

=

and so we may proceed at once to the limit by making 8 -* 0,

since the integral

of

[CHAP. XIII

Hence, by the theory of analytic continuation,

result

J* (z) +

(1)

r„ 2

^—TKo (2* sinh T JO

0)

t)

cosh 2vtdt,

R (z) > 0.

provided that

Another integral formula which can be established by the same method* (2)

v (z )

To prove

we

replace

I^- Y

d

J

v

OV

we

this formula,

by

x),

d (z)

-^4^=-OV

first

I yt, $H

2>

<

(x) {x)

suppose that z

dH"{1)(x) -

2i

=

5-5-.

IT

=-

dy

G\

= - 4? G

For the

o

is

a positive variable (which

OV

-

*

rK,(2zsm\it)e-^dL .!

and then

OV

202.

7T

is

I

full details of

G

v

(u-t-Tri) (?x (ooriU-cosh «) °°

H

f~ J

J

—QD

^G rr -T--

H « (x) dH'*W a dv

Cr

i

e-y(t+u)

_

^tdu)

(2U+iH) e2« 8inh rsinh u e~ iv .(dTd U)

(2tr +7ri) e2ix8inhTeinh£r e-2,T d

^^

(2U+iri)e-2i* aiahT »inhU ei r dUdT. '''

the analysis, see Watson, Proc. Royal Soc. xciv.

a,

(1918), pp. 197

((

INFINITE INTEGRALS

13-73]

T being

Now,

positive,

445

we have

8

U + \ irif) e

2i** inh

(2U + iri) exp {- 2\

I" —00

T>inh

u dU

.'

=2 and, since

i>e-2*

sinh

Tcosh »

(

"

(t>

an

js

+ 7n) exp {- 2\

(i>

function of

0(jd

+ in)*} e-'2assinh Tcosh *dv

v, it

may

be proved that the

last integral is 2iri

t* e-** sinh Tcosh • dv J

where the constant implied in the symbol to x its integral with respect to T from In

(A),

(\) is a function of is

T such

that

convergent.

manner,

like

("" J

+

— 00

(2

U + Trt) exp {- 2X U + $ irif] e" 2te sinh Tsinh * dU

—00

(" 2v exp (- 2XV2 ) e~2«8inh Tcosh* fo = Q.

=

J —00

Hence

it

follows that

W dY ^ - F, W ^ -^ = - - fT

/" fa)

"

fa)

dv

9l>

e

-^sinhTcosh*-2,T rf

^r

TTJo J -ao

= -- f "jfo^sinh T)e~» T dT. 7T.'o

The extension

to the case in

complex with a positive

which the argument of the Bessel functions

real part is

made

is

as in (1).

It should be mentioned that formula (2)

is

of importance in the discussion

of descriptive properties of zeros of Bessel functions.

The reader may

find it interesting to prove that

=tf [,r,, )

?W. r

; (, )

i

^].,-,,[,, (t

,

4|i»-r. W

^],

and hence that

w ^(^Lf^,)^.. (8 )

v v

'

Cv

cv

Other formulae which (4)

may

X -*

(o

'(zicoshZT-v^K^vBrnhê-^dT.

nz* J

be established by the methods of this section are

y.w/rW+FdWfrW = -„ f K _n(2z8inht) .{eC+W + e-<*+")t cos(p-v) n}dt, v

IT

(5)

J^z)

JO

Yv {z)-Jv {z) FM (z) =

these are valid when previously published.

* ain( *~ v)

n

R(z)>0 and R (ji- v) |< 1 |

j~ *,-» (2înh t) «-»+>)« dt; ;

they do not appear to have been


446

Deductions from Nicholson's integrals.

13'74.

Since

[CHAP. XIII

K (^)

is

a decreasing* function of

£, it is

clear from §13*73(1) that

Jf{x)+Y*{x) is

a decreasing function of x for any real fixed value of

Since this function

is

approximately equal to

v,

when x

is positive.

when x

2/(irx),

is large,

we

shall investigate

x{J*{x)+Y*{x)} and prove that

a decreasing function of x when

it is

increasing function oi x

when v<

v>\, and

that

it is

an

\.

It is clear that

^[x[J/(x) + Y*(x)}-]

= — f ™[K .

K

8^ 2

(2x sinh T)

+ 2x sinh TKQ

(2x sinh T) tanh

'

(2x sinh T)}cosh

2vTdT

T cosh 2vT

7T

+

d „ (2x n mJ « m 1S C*°°if sinh T) [cosh 2vT - -^ {tanh Tcosh 2vT\ ^ Jo .

,

2

,

,

[

on integrating the second term in the integral by

=~

" f

t' Jo

Now X tanh XT integrand

is

parts.

dT,

Hence

K (2x sinh T) tanh Tcosh 2vT {tanh T - 2v tanh 2i»T} dT. is

an increasing function of X when X >

negative or positive according as 2v

>

0,

and so the and 1

< 2v <

1 or

;

last

this

establishes the result.

Next we prove

that,

when x ^ v ^ 0, (a?

is

an increasing function of If

we omit the

-i*y[JS(x) +¥,*{*)}

x.

positive factor 8 (x*

pression under consideration

[xK (2x sinh t) +

—

2 i'

)

-i /7r2

from the derivate of the ex-

we get 2 (x*

-

v2) sinh

t

.

K

'

(2x sinh

t)}

cosh 2vt dt,

Joo

and is

to establish the

theorem stated

it is sufficient

positive. * This is obvious

from the formula

K ^)=r e -^ coaht dt. J

o

to prove that this integral

K

INFINITE INTEGRALS

13-74]

We

447

twice integrate by parts the last portion of the second term in the

integral thus 2j>

2 °°

sinh

K

t

J

(2x sinh

'

t)

cosh 2vt dt

Jo

-

v sinh

t

sinh 2vt

K

'

-v

(2x sinh t)\

-y-

(sinh

tK

'

(2x sinh

t)}

sinh 2vtdt

J

=—

r°°

v

d {sinh

-j-

\

Jo dt

=—

v

2a;

I

sinh

tK

t

'

(2% sinh

cosh £iT

t)}

sinh 2vtdt

sinh

(2a;

t)

sinh 2i/£d£

.'o

= — x sinh £ cosh tK

£)

cosh 2vt

,

+ =

(2% sinh

Jo

L

x

j-

\

[sinh

t

cosh £ iT

(2a;

sinh

Jo dt

( [x cosh 2£lf (2x sinh Jo

«)

£)]

cosh

2itf eft

+ 2a? sinh t cosh

the simplification after the second step

2

* J5T

'

(2# sinh

t)]

cosh

2i/i eft;

produced by using the differential

is

equation

zK " (z) + K: (z) - zK The

(z)

= 0.

integral under discussion consequently reduces to 2 [- 2x sinh tK (2x sinh

| •

t)

- 2x* sinh

3

'

1

(2x sinh*)] cosh 2vt dt

°

=

r—

+x = a;

j

K (2x sinh T

r°° jfiT

I

jfiT

(2# sinh

(2a?

sinh

t)

«)

t)

cosh 2v£

- 2 sinh

2

[tanh 2 1 cosh

£

cosh

2i>«

+

rf ~=-

+ 2v sinh

2i>t

3

—r-

)~1

(sinh't -j

£

sech

t

cosh sinh

2irt

eft

2i/£] oft,

Jo

and

this is positive

because the integrand

positive

is

;

hence the differential

coefficient of

and the

is positive,

result is established.

Since the limits of both the functions

are

2/tt, it

x \JV* (x) + F„2 (x)}, follows from the last two

(a?

-

i^)*

{Jy* (x)

results that

+ rr

«

(x)}

when x^.v^\,

An elementary proof of the last inequality (with various related inequalities) was deduced by Schafheitlin, Berliner Sitzung*beric1de, v. (1906), p. 86, from the formula (cf.

§5-14)

(4r«-l)JVr»(0* = i( a 2 + 62 where <$

(x)

)

_^Qf^ + ^;

= aJv (x) + b Yv (x).

2 .

(A )

|

+ 2(l-^)^2 (a;) + ^; 2

(. t .-)],


448

is

The next consequence which we that, when v is positive, r

To obtain

this result,

,

shall

dYv (v)

^

we observe

[CHAP. XIII

deduce from the integrals of § 13'73

dJv {v)

v

that the expression on the

left

may

be

written in the form

-^T K (2v sinh T)e-'" T dT

=1 irv

7T J o

-s[ But,

for

1

-/.Vjr-( 2 " inh s)*]-

t, 2v sinh (^tjv) is a decreasing function of v, a positive decreasing function of its argument, we see that

each positive value of

and so, since

K (x)

is

- °V< K f is

a decreasing function of

v,

(2v sinh |-) dt

and therefore

d Yv {v)

[MpY

dJv ( v) v 1AP) .

dv

dv

> lim

=

lim

by using the asymptotic expansions of

§ 8'42

;

and

this establishes the result

stated.

The asymptotic expansion of Jv2 (z) + F„2 (z).

13*75. It

is

easy to deduce the asymptotic expansion of

Jv

*

(z)

+ F„2 (z) from

Nicholson's formula obtained in § 1373, namely

J* (z) + F„8 (z) = —2 (" if

T

for

we

t)

cosh 2vt dt

;

have, by § 7*4 (4),

cosh*

where and,

(2* sinh

.'0

when

|

i2p

v is real

|

m=0

T "

(2m)!

<

and

cosiC(i>7r)|

p

is

(2^)!

so large that

(2p)l

p + %>v,

Mp lies

between

and

'

INFINITE INTEGRALS

13*75, 13-8]

We

449

at once deduce the asymptotic expansion

7r

that

is

by

to say,

§

1321

JS(*) +

(1)

2

(2w)!

m=0

Jo

(8),

YS{s)~X {1.3...(2m-l)}^; TTZ * rn =o

<&

R (z) > 0, but

it may be extended over the wider range and p exceeds v — \, the remainder positive, and z is arg z < 7r and, if v is real less than, the (p + l)th numerically after p terms is of the same sign as, and

this

is

j

proved when ;

|

term.

13*8.

Ramanujan s

Some

extraordinary integrals have been obtained by

integrals.

Ramanujan* from an

application of Fourier's integral theorem f to Cauchy's well-known formula

_^cos

av

ff.e

2M+ „- 2r(/A+

j

which

is

valid if

R (p + v) > 1.

r.

e ittd£

The

application shews that r

..ro. +B r(.-»

=

,

J

(2cos^)^+- 2 eitY

s

r 0* +

t is

any

real

By expanding formula,

a)

if

it is

v

- x)

,

y . >t , >

( jf

j

<7r ),

.

(

0;

(

where

f)T(v-ft

<[>r)j

number. in ascending powers of

x and

y,

and then applying

this

seen that

/1%^^-W

— 7r < < <

7r

;

for other real values of

t,

the integral

is zero.

In particular

I" J„t(x)J».s{x)dS

(2)

= J^{Zx).

J —oo

In view of the researches of March, Ann. der Physik und Chemie, (4) xxxvn. (1912), 29—50 and Rybczyriski, Ann. der Physik und Chemie, (4) xli. (1913), pp. 191—208, it seems quite likely that, in spite of the erroneous character of the analysis of these writers J, these integrals evaluated by Ramanujan may prove to be of the highest importance in the

pp.

theory of the transmission of Electric Waves. * Quarterly Journal, xlviii. (1920), pp. | Cf. Modern Analysis, §§ 9 7, 11-1.

t

Cf. Love, Phil. Trans, of the

294—310.

Royal Socccxv.

a, (1915),

pp.

123—124.

CHAPTER XIV MULTIPLE INTEGRALS Problems connected with multiple

14*1.

The

difference

integrals.

between the subjects of

this chapter

and the

last is

more

than one of mere degree produced by the insertion of an additional integral sign. In Chapter xiii we were concerned with the discussion of integrals of perfectly definite functions of the variable and of a number of auxiliary parameters; in the integrals which are now to be discussed the functions under the integral Thus, in the first problem which

sign are to a greater or less extent arbitrary. will

be discussed, the integral involves a function which has merely to

satisfy the

conditions of being a solution of a partial differential equation, and of having

continuous differential coefficients at

all

points of real three-dimensional space.

In subsequent problems, which are generalisations of Fourier's integral formula, the arbitrary element has to satisfy even more general restrictions

such as having an absolutely convergent integral, and having limited total fluctuation. 14'2.

The

Weber's infinite integrals.

now be considered involve Bessel functions only seems desirable to investigate them somewhat fully because many of the formulae of Chapter xiii may easily be derived from them, and were, in fact, discovered by Weber as special cases of the results of integrals which will

but

incidentally;

it

this section.

Weber's researches* are based upon a result discovered by Fourierf to the equation of Conduction of Heat

effect that a solution of the

du

_ d*u + d*u + &u hy dz

dt~dx*

2

is

u

= if"

I"

r

®(x + 2X^t,y+2Y
z

+ 2ZJt)

x exp {- (X 2 + where

<£ is

Weber

an arbitrary function of first proved that, if

its

F + Z )} dXdYdZ, 2

2

three variables.

{x, y, z) is restricted to

be a solution of the

equation 2

<£>

32

3 2 4>

3? + ?£ + !?—**•

?)

<»

—

*

Journal fur Math. lxix. (1868), pp. 222 237. La TMorie Analytique de la Chaleur (Paris, 1822), § 372. The simpler equation with only one term on the right had previously been solved by Laplace, Journal de VEcole poly technique, t

vni. (1809>, pp.

235—244.

_

;

MULTIPLE INTEGRALS

14*1, 14*2]

451

then

u

(2)

provided that

= exp (- kH) $ (x, y, z),

has continuous

the integral converges in such a

and second

first

way*

differential coefficients,

and

that transformations to polar coordinates

are permissible.

The method by which a more general If

this result is established is successful in expressing

we change

to polar coordinates

2X \/t = r sin 6 cos we

by writing

2Y»Jt

,

equation (4) below].

[cf.

= rsintfsin

2Z^/t

,

= rcos0,

get

=-—

r ao r

1

w

a single integral

triple integral as

v

fir

3>Oc

(4tt01 Jo Jo J-*

+ rsin0cos,

y

+ r sin

sin

<£,

x exp

Now

consider the function of

^(x + r 8in0 cos

iff(r)=j

It is a continuous function of coefficients

r, vr (r),

when r has any

r,

,

y

(

z + rcosd)

— j-\

r2 sin

ddddr.


+ rsindsin^), z + rcoad) sin 0dd0.

with continuous first and second differential and the result of applying the

positive value

;

operator (

d\

dr\

drj

d

1

r2

,.

to vf (r) is

shew that the last integral is zero. If we make use of the equation (1), which satisfies, we find that

We proceed differential

to

the difficulty! caused by the apparent singularity of the last integrand on the polar axis, we consider the integral taken over the surface of a pole sphere with the exception of a small cap of angular radius 8 at each

To avoid

made

bounded at the poles, the integrals over the by taking S sufficiently small. small arbitrarily

If we perform

the integration of the second term on the right with respect

since the integrand

caps can be

*

on the

A sufficient condition is

that

left is

*

should be bounded when the variables assume all real values, Cf the corresponding two-dimensional investigation,

infinite values of the variables being inoluded.

Modern Analysis, t This

§ 12*41.

difficulty

was overlooked by Weber.

.

(


452 to

,

we

[CHAP. XIV

see that its integral vanishes because

The

valued function of position.

8/9<^> is supposed to be a oneterm on the right gives

first

-?/".[sin ^]r^' and

this can

be made arbitrarily small by taking 8 sufficiently small since d

continuous and therefore bounded.

is

made

can be

by taking

arbitrarily small

8 sufficiently small,

and therefore

it

is zero.

Consequently

£l£&a + *T.(r)-0,

(8)

,

,

.

,

so that

where of

A

and

values of

all

x

ts (r)

B

and

sin kr

+ B cos kr ,

since rst (r) and its derivate are continuous for must have the same constant values for all values

are constants

A

r,

A =—

B

;

r.

If

we make

r-*-0,

we

see that

B = 0,

A = 47r3> (x, y, z)/k.

Hence * u

and

=

t&w

this establishes

A

!o

exp (~

Weber's

3

sin

-

= exp ( - kH) * ( *

similar change to polar coordinates shews that, if

function of

f"

r,

and

if

r r ^(X

>

The reader

This integral

will

f(r)

is

O (x, y. z) is a solution

an arbitrary continuous

>

have no

is

difficulty in

-y

-i£i_'

y

r)

sm &r

exp ( -

r

-\ cos At

.

.

rdr.

enunciating sufficient conditions

make

the various changes in the

most easily evaluated by differentiating the well-known formula /

k.

*>' '

Y Z)fW{(X-xY+(Y-yy + (Z-zy}]dXdYdZ

concerning absoluteness of convergence to

with respect to

y

then

=

*

j

result.

of (1) of the type already considered,

(4)

kr rdr

dr = v/(irt) exp ( -

fc2 1 )

;

453

MULTIPLE INTEGRALS

14-3]

One such

integrations permissible.

bounded as the variables tend

set of conditions

to infinity,

3,

that 3> should be

and that

f(r) = 0(r-*), (rÔ); /(r)-0(f«).

where p <

is

(r

- oo

),

q > 1.

simpler formula established at about the same time by Weber* is that, a function of the polar coordinates (r, 0) which has continuous first and second origin is u differential coefficients at all points such that 0
A somewhat

if u(r, 6) is

,

then

k (r,

/

0) c£0

= 2n-tto ô (^")»

/ -JT

when O^râ. The proof of this

14*3.

is left

to the reader.

General discussion of Neumann's integral.

The formula F{R,&).Jo[u\/{B? + r*-2Rrcos(4>-)}lR(d
("info P°

(1)

J°

f" -'

J° j

=27rF(r,4>)

was given by Neumann in his treatise f published in 1862. In this formula, F(R, 4>) is an arbitrary functioD of the two variables (R, 4>), and the integration over the plane of the polar coordinates (R, <£)

is

a double integration.

In the special case in which the arbitrary function is independent of , we inreplace the double integral by a repeated integral, and then perform the to reduces formula tegration with respect to <£ the ;

f udu Jo(VCR) J, (uR) Jo («r) RdR = F(r),

(2)

Jo

closer resemblance to Fourier's integral J than (1).

a result which presents a

The extension

of (2) to functions of any order,

["udu

(3)

namely

FF{R) Jv (uR) Jv (ur) RdR - F(r),

was effected by Hankel§. In this result it is apparently necessary that though a modified form of the theorem (§§ 145— 1452) is valid for values of v

;

when v— ± \

The formulae and the proof of

(2)

and

(3) is

%

(3) is

v>-\, all real

actually a case of Fourier's formula.

(3) are, naturally,

much more

easy to prove than (1) (2), the

of precisely the same character as that of

• Math. Ann. i. (1869), pp. 8—11. Temperaturzustand eines homogenen t Allgemeine Ldsung des Problemes ilber den statiovaren wird (Halle, 1862), pp. 147— begrenzt KugelJUlchen nichtconcentrUchen KSrpers, weleher von zwei 409—410. 151. Cf. Gegenbauer, Wiener Sitzungsberichte, xcv. (2), (1887), pp.

% Cf. §

Modern Analysis, §9*7.

Math. Ann. vm. (1875), pp. 476—483.

'


454

[CHAP. XIV

arbitrariness of the order of the Bessel functions not introducing

any additional

complications.

Following Hankel, many writers* describe the integrals (2) and (3) as "Fourier integrals" or "Fourier-Bessel integrals."

On

account of

proving (1)

;

greater simplicity, we shall give a proof of (3) before this stage it is convenient to give a brief account of the

its

and at

researches of the various writers

As has already been general formula

He

(3).

stated,

v

-

investigated the formulae.

Hankel was the

first

writerf to give the

transformed the integral into

rRF(R)dR f Jv (uR)J

lim

who have

(ur).udu

rRF(R)[RJv+1 (\R)J (\r)-rJv+1 (\r)J (XR)-^—

lim

v

v

X-»oo Jo

sir

t

IT

and then applied the second mean-value theorem to the integrand just as Substantially the same proof was given by SheppardJ who laid stress on the important fact that the value of the integral depends only on that part of the JJ-range of integration which is in the immediate neighbourhood of r, so that the value of the integral is independent of the values which F(R) assumes when R is not nearly equal to r. in the evaluation of Dirichlet's integrals.

A different

mode

of proof, based on the theory of discontinuous integrals, who integrated the formula (§ 13*42)

has been given by Sonine§, t*+»

[V^ (ur) J„ (uR) du = Jo

after multiplication r" +1

[J

J"

&

r)\

\y,

by F(R)RdR, from

J,+1 (ur)

P

to oo

,

so as to get

Jv (uR)F(R) RdRdu =

J2»+»

J*

F(R) dR;

and then, by differentiating both sides with respect to r, formula (3) obtained but the whole of this procedure is difficult to justify.

is

at once

;

A proof of

a more directly physical character has been given by Basset

but, according to

Gray and Mathews,

it is

||,

open to various objections.

A proof depending on the theory of integral equations has been constructed by Weylf.

The extension

of Hankel's formula, which

is

effected

by replacing the

* See e.g. Orr's

paper cited later in this section. statement of a mode of deducing (3) from (1) when r is an integer was made by Weber Math. Ann. vi. (1873), p. 149, but this was probably later than Hankel's researches, since it is dated 1872, while Hankel's memoir is dated 1869. t

A

t Quarterly Journal, xxin. (1889), pp. 223—244. § Math. Ann. xvi. (1880), p. 47. Proe. Camb. Phil. Soc. v. (1886), pp. 425—433. See Gray and Mathews, A Treatise on Bessel Functions (London, 1895), pp. 80—82. ||

1 Math. Ann.

lxvi. (1909), p. 324.

MULTIPLE INTEGRALS

14*3]

455

Bessel functions by arbitrary cylinder functions, was obtained by Weber*, and

be discussed in

it will

§§ 14*5

—

14*52.

An attempt has been made by Orrf to replace the Bessel functions by any cylinder functions, the w-path of integration being a contour which avoids the origin but some of the integrals used by him appear to be divergent, so it is ;

difficult to

say to what extent his results are correct.

The same

applies to the discussion of Weber's problem in Nielsen's treatise.

shewn

(§ 14*5)

that

if,

criticism

be

It will

two cylinder functions under

as Nielsen assumes, the

the integral sign are not necessarily of the same type, the repeated integral

is

not, of necessity, convergent.

be stated that, if r be a point of discontinuity of F(R), the expressions on the right in (2) and (3) must be replaced by J It should

${F(r-0)+F(r + 0)\, just as in Fourier's theorem.

For the more recent researches by Neumann, the reader should consult

his treatise

Ueber die nach KreU-, Kugel- unci Cylinder-functionen fortschreitenden Entwicielungen (Leipzig, 1881).

Neumann's formula

(1)

was obtained by Mehler§ as a limiting case of a fact, it was apparently with this

formula involving Legendre functions; in

object in view that he obtained the formula of § 5"7l,

lim

but

it

Pn {cos (*/»)} = J (z), a

does not seem easy to construct a rigorous proof on these lines

(cf.

A

more direct method of proof is given in a difficult memoir by Du Bois Reymond|| on the general theory of integrals resembling Fourier's integral. The proof which we shall give subsequently (§§ 14*6 et seq.) is based on these researches.

§ 14*64).

Subsequently ErmakofflT pointed out that the formula is also derivable Du Bois Reymond which is the direct extension to

from a result obtained by

two variables of Fourier's theorem

=

»r. r. /-. f* {x

'

for

one variable, namely

Y) c°s

(a

(x

x)

+ ^ ( f_ y)]

•

(

dxdY> dadP' >

Ermakoff deduced the formula by changing to polar coordinates by means of the substitution a

and

= u cos to,

(3

= u sin at,

effecting the integration with respect to

to.

* Math.

Ann. n. (1873), pp. 146—161. t Proe. Boyat Irish Acad, xxvii. a, (1909), pp. 205—248. i The value of the integral at a point of discontinuity has been examined with some care by Cailler, Archives des Sci. {Soe. Helvttique), (4) xiv. (1902), pp.

Ann. v. (1872), pp. 135—137. Math. Ann. iv. (1871), pp. 362—390.

347—350.

§ Math. ||

f Math.

Ann.

v. (1872),

pp. 639—640.

:


456

[CHAP. XIV

and (R, ) be the polar coordinates corresponding to the Cartesian (x, y) and (X, Y) respectively, the formal result is fairly obvious when we replace V(X, Y) by F(R, <£>) but the investigation by this method is not without difficulties, since it seems to be by no means easy to prove that the repeated integral taken over an infinite rectangle in the (a, /3) plane may be replaced by a repeated integral taken over the area of an indefinitely great If

(r, <£)

coordinates

;

circle.

If the arbitrary function

F(R,

)

F (r,

not continuous, the factor

is

)

which occurs on the right in (1) must be replaced by the limit of the mean value of F(R, 4>) on a circle of radius & with centre at (r, when 8 -*-0. ) This was, in effect, proved by Neumann in his treatise of 1881, and the proof will

be given in

§§

146

— 14*63.

The reader might

what he knows of the theory of Fourier

A formula which

is

more recondite than

(3),

namely

has been examined by Bateman, Proc. London Math. Soc.

The

(2) iv. (1906), p.

484

;

cf.

-

§ 12 2.

Hankel's repeated integral.

14'4.

generalisation of

Hankel

anticipate this result from

series.

(cf. §

143)

Neumann's

integral formula which

in the case of functions of a single variable,

was

effected

by

may be formally

stated as follows

Let

F (R)

be

an arbitrary function of

the real variable

R

subject to the

condition that

\"F{R)s/R.dR exists

and

is

absolutely convergent;

be not* less than

— \.

and

let the

order v of the Bessel functions

Then

X

{"udu

(1)

\

F(R) Jv (uR) Jv (ur) RdR = \ {F(r + 0) + F(r - 0)1,

provided that the positive number r

lies inside

an

F(R)

interval in which

has


The proof which we shall now give is substantially Hankel's proof, and it same general character as the proof of Fourier's theorem it will be set out in the same manner as the proof of Fourier's theorem given in Modern Analysis, Chapter IX. It is first convenient to prove a number of lemmas. is

of the

* It

seems not unlikely that

;

it is sufficient

more extended range of values of

t>

for v to be greater than

would be more

difficult.

-

1

;

but the proof for the

:

]

MULTIPLE INTEGRALS

14*4, 14*41]

The analogue of the Riemann-Lebesgue lemma.

14*41.

A result, which resembles is

Letf

lemma of Riemann-Lebesgue*

the

of Fourier series, and which

theorem

457

in the theory

required in the proof of Hankel's integral

is

as follows

F(R)*/R.dR

|

and

exist,

{if

it is

an improper integral)

let it

be

a

absolutely convergent ;

and

v~^

let

— \.

X -»»

Then, as

oo

,

F(R)Jv (\R)RdR = o(l/
f J a

It is convenient to divide the proof into three parts

assumed that F(R)\/R the restriction that 6 that

F(R) *JR

(I)

is

bounded, and that

is

is finite is

bounded

is

removed

also

b),

in the first part

it is

second part

and in the third part the restriction

;

removed.

Let the upper bound of

j

F{R)\/R be K. |

Divide the range of in-

tegration (a, b) into n equal intervals by the points

xn =

;

b is finite; in the

xJt x2

,

...

{x

=a,

and choose n so large that

where e is an arbitrarily small positive number and and lower bounds of F(R) \/R in the rath interval.

the upper

F(R) *JR = F{R m_ ) V#m-i + «m (R), so that, when R lies in the mth w m (R) \Ûm — L m Now, when v ^ — \, both of the functions of x,

Write interval,

Um and Lm are

1

.

|

x*Jv {x),

i*tiJv (t)dt, Jo

are

Let

bounded when x > 0, even though the integral is not convergent as x •* oo A and B be the upper bounds of the moduli of these functions. It is then .

clear that *

F{R)Jv (\R)RdR

a

»

[*m

2 F{R m ^) V-R*,-, »=1

J

v

-

(\R) V-K

+ 2 »

^ 2

[

FjR^ ^R^W -^

2.Bwg

* * Cf.

R

\*

i

+

.

dR

Xm-l

j**r»

a>

|

w-l

.'

m {R)Jv {\R)^R.dR\

Xm-1

|

•

Jv {x)\Jx.dx + 2

AdR

/*« 0> I

m (i£)

Ae VX-'

Modern Analysis,

% 9-41.

t The upper limit of the integral may be infinite preserves the analogy with § 14-3 (3).

;

and aÔ. The apparently

irrelevant factor

"


458

By

X

taking

sufficiently large (n

last expression is o

made

can be

[CHAP. XIV

remaining fixed after e has been chosen) the than 2Ae/^X, and so the original integral

less

(1/^X). If the upper limit is infinite, choose c so that

(II)

r\F(R)\ s/R.dR
and use the inequality i

I

F(R) Jv (XR) Rd R

I

Ja

U

° f J a

F (R) J

(I),

we get

I

\

\

then, proceeding as in case

v

(XR)

RdR\ + ^- r\F(R)\
\

2BnK

J F(R)J (XR)RdR

2Ae

v

\

J a

The choice of n now depends- on e through the choice of c as well as by the mode of subdivision of the range of integration (a, c) but the choice of n is still independent of X, and so we can infer that the integral (with upper limit ;

infinite) is still o(l/V^).

If

(Ill)

unbounded

F(R) V-B

is

unbounded*, we may enclose the points at which

number p

in a

St \F(R)\^R.dR < 8

By

(II) to the parts of (a, b) outside these

is

I

now

that Ave can

14*42.

the upper bound of

The

|

F (R)

K and n now depend on

still

shall

V^

A,

J a

choices of both

We

and

(I)

we get

,

K

€.

J8

applying the arguments of

intervals,

where

e,

*JR outside the intervals h. The but are still independent of \, so \

infer that the integral is o (l/y/X).

inversion of Hankel's repeated integral.

next prove that, when v~^ —

\,

and

I

F(R)*JR .dR

exists

Jo is

absolutely convergent, then

Vudu rF(R)Jv (uR)Jv (ur)RdR Jo

it is

of intervals 8 such that

Jo K

=

lim

rF(R)

* Gf.

\

JV (uR) Jv (ur) udul RdR,

[JO

A-»ooJo

provided that the limit on the right

\

)

exists.

Modern Analysis,

§ 9-41.

and

MULTIPLE INTEGRALS

14-42, 14*43]

For any assigned value of hypothesi there exists a

A

If

we

it is

is

fi

e,

ex

such that

F(R)\JR.dR<

F where

and any arbitrary positive number

\,

number

459

the constant defined in § 1441.

F (R) J

write

v

=

(uR) J, (ur) uR

(R, u),

clear that*

I

["

f%(22,

j

=

I

u)duldR-

r|

r

&

u) dui

T

^J Since this result

dR~ r | f °v (r,

/

the integral on the

left

=

r

%>

^1 d U

?-\F(R)\
true for arbitrarily small values of

P° [ (R,u)dudR Jo Jo

to exist.

(R, u) dRi\ du

^ji? CR)|v #.<^<£R+J

j

is

Tj P

e,

we

infer that

r
Jo Jo

existing because the integral on the right

If the integral on the left has a limit as

X -*

oo

,

it is

is

assumed

evident from

the definition of an infinite integral that [" udu P° F(R) Jo Jo

J

¥

(uR) Jv (ur) RdR

=

lim f udu f° F(R) Jv (uR) J, (ur) RdR JO

K-m-aoJ

=

K

lim

I

*F(R) {

X-»oo.'o

and

this is the inversion formula

14*43.

f

J

v

(uR)

Jv (ur) udu\ RdR,

{Jo

)

which had to be proved.

The relevant part of the range of integration in Hankel's repeated

integral.

Next we

shall prove that, in

Hankel's integral, the only part of the .R-range

of integration which contributes anything to the value of the integral

part of the path in the immediate vicinity of r, provided merely that has an absolutely convergent integral. *

are X

The and

justification of the inversion of the order of integration for /3

presents no great theoretical difficulties.

is

the

F(R) \/R

& finite rectangle whose sides

—

.

[CHAP. XIV

THEOKY OF BESSEL FUNCTIONS

460

To

prove that,

effect this, it is sufficient to

interval*

r

(uR) Jv (ur) RdR

F (R) J

udul

t

JQ

We invert

if

not a point of the

is

then

(a, b),

v

= 0.

Ja

we

the order of the integrations, as in § 14*42, and

find that, if

the limits on the right exist, f

uduf F(R)Jv (uR)Jv (ur)RdR

J

J a X

b

=

lim

F(R)

i

X-ôo

J

J

(uR)

V

\ \

J

v

(ur)

udul

[JO

a

RdR

)

fb F(R) [RJy+1 (XR) Jv (Xr) - rJ^ (Xr) Jv (XR)]

=

*

lim \J. (Xr)

j£*£

J

-

•/,+.

-£—

TftlTf i

(XB) dR

XrJv+1

lim

\

(Xr)

J*

»£™^

(XB)
Since both the integrals

fEgV^figgn are

ea;

hypothesi absolutely convergent,

Lebesgue lemma

(§

("uduf J0

provided that r

is

F (R) J

¥

not such that

(uR) J, (ur)

a^r$

I

now be shewn

that, as

t f Ja J

J

v

\

J a

It will

follows from the generalised ;

and

Riemann-

so

RdR =

Ja

The boundedness of

14*44.

it

14*41) that the last two limits are zero

b.

(uR) Jv (ur) uRh dudR.

JO

X—

oo

,

the repeated integral

Jv (uR) Jv (ur) uR* dudR

remains bounded, provided that a and b have any (bounded) positive values. It is permissible for a and b to be functions ofXof which one (or both) may tend to

r as X -*

oo

Let us first consider the integral obtained by taking the dominant terms of the asymptotic expansions, namely 2

f

b

f

K

cos

I

7T\/r.J a

(

uR — \vtr —

\ir) cos

(ur — \vir — $ir) du dR

.'0

-— n

h ("sin

7T

X (R — r)

cos (X (R

R-r

sjrj a |_ CHb-r) si n f

1

—

VrL.'A(«-r)

#

x

dx

—

f

+ r) — vir\ — cos inr~ dR R+.r

A(6+r) cos

I

Jx(*+r)

-

(a* *

«

i>tt)

.

,

dx + cos

* It is permissible for 6 to be infinite.

vrr

log

—+—b

r*l

a + rj

.

MULTIPLE INTEGRALS

14*44]

The

first

integral

bounded because

is

second integral

is

bounded because

now under

&

x

consideration

is

dx

/

J

the integral

dx

I

J -00

461

bounded, and

is

convergent

is

;

and the

convergent ; and so

its limit,

as

\ -* oo

,

is

the limit of r/" A (*-»•) sin

1 :-

I

^r wr

a;

«

[J K(a-r)

.

b

.

dX + COS 1/7T

log

+ r~

°a + rj

,

provided that this limit exists.

But we may write f

J (uR)J„(ur)uR*dudR

I

¥

Ja J

= -A- f TTtJrJa

(w#)

f " [Ittw/,,

Jv {ur) J(Rr)

Jo

— cos (aR — \vir — \w) cos {ur — \vir — |7r)] dudR

— cos (uR — \vir — \tt) cos (ttr — |i>7r — \trS\ dudR H

I

\vir

— £71-) cos (ur — £i7r — J7r) c2u<£R. first is

function of r

any positive a continuous (and therefore bounded) positive and bounded.

R and r, and

when r

is

third integral has been

so it is

shewn

to

be bounded, and

it

converges to a

whenever

limit

rHb-r) sin# J A(a-r)

does

the integral with respect to

integral (with respect to u) which converges uniformly in

domain of values of

The

—

of the integrals on the right, the

Now,

R of an

cos (uR r / Tr^/rJaJo

,

X

so.

The second

—

—_

4„2 -

•inr

integral

J rb

\/r

}

a J A

where

^ (\),

<£ 2

\ -*

r

—j

d [_uK

sin

written in the form

(uR — hnr —

cos{uR-Ivtt-

-i

to zero as

roo

r-

may be

(X) and

oo

.

3

\ir) cos

lir) sin {ur

{ur — \vtr

— \ir)

- %vtt - \ir) + 0{l/u*) dudR

(X) are functions of

\ and

i2

which tend uniformly

[CHAP. XIV


462 Hence,

bounded positive values of

for all

the integral

a, b, r,

Jv (uR) Jv (ur) uR* dudR .'0

a is

bounded as \ -*•

oo

and

;

converges to a limit whenever

it

r\(b-r) s i n

does

^

®

.K(a-r) so.

Proof of HankeVs

14*45.

Now

that

the preliminary lemmas have been proved, the actual proof

all

of Hankel's theorem

F(R)

Since

integral theorem.

is

quite simple.

has limited fluctuation in an interval of which r

F(R)\/R; and

point, so also has

therefore

we may

an internal

is

write

F(R)s/R = Xi(R)-X*(R)> where X\ ( R ) an(^ X* ( R ) are nionotonic

(positive) increasing functions.

we choose a positive number After choosing a positive number fluctuation in the interval (r — 8, r -f 8) 8 so small that F(R) has limited total e arbitrarily,

and

also

+ S)- Xl (r + 0)
Xl (r

Xl (r

'

X2

2

-0)- Xl (r- 8) <

(r- 0)-

X2 (r- 8)

<

we apply the second mean-value theorem, we find that number £ intermediate in value between and 8 such that If

r+S f

X

f Xi

-'0

WJ

"

(w

ej '

ef

there exists a

R ) J" ( Mr ) Us/B" dudR fr+S r\

= Yj (r + 0)

J

¥

I

Jr

(uR)

J

v

(ur)

(Xi ( r

+ 8 ) ~ Xi

(

Afi

f

Since as

X -»• oo

Jo ,

r

+ °)1 J

sin

——

a;

x

,

\

it

.

-*- oo

J" ( uR ) J

»

(ur)u*JR. dudR.

,

— £7r, 1444

that the

(cf.

§ 1444).

if

lim

f

A-*.«

J r

f

J

v

(uR)J„(ur)uR i dudR = C / y/r, l

Jo

follows that

+ lim

P

(

*-*.*> J r

exists

and

first

is

equal to

X

JO X\ (r

term on the

while 8 remains fixed. And the second term in absolute value, where G is the upper bound

on the right does not exceed Ce of the modulus of the repeated integral Hence,

/

r+( J

8 remaining fixed, it follows from §

right tends to a limit as

JR dudR

r\

fr+S

+

u

•'

Xi( R ) J»( uR )

+ 0)/\/r.

J*( ur ) uRhdudR

—

MULTIPLE INTEGRALS

14-45]

We

treat

^ (R) in a similar manner,

the interval (r

—

r)

8,

lim A-*-

and we

;

I

also apply similar reasoning to

infer that, if

Jv (uR)Jv (ur)uR^dudR = C /^/r,

|

3

J r—8 J

oo

rr+8 fK

then

and

463

F(R) Jv (uR)Jv (ur)uRdudR

lim A-êo J r -S J

exists

and

is

equal to

C F(r+0) + C,F(r-0). 1

We

now have to we have

C

evaluate

C

and

t

2

By

.

the

theory of generalised

integrals*,

Jo Jr

S/f

=

f"

lim i>-»0

f

exp (—pû*)

Jo

r+ *

J

rr+S

(uR) Jv (ur) uR*

dRdu

roo

= lim

exp (- p*u 2 )

p-ôJr

Jv (uR) Jv (ur) uR* dudR

.0

p+

,.

1

v

J r

fi

1

R*

(

+

_

r*)

/flr\ Di ,_

by §1331(1).

Now, throughout the range

as

of integration,

^) -*- 0.

Hence

C^

—

=

exp x

lim s

p _fc0 2jt>V7rJ

= lim

1

r

-I—

.

ri«//>

-}—

exp (— #2 ) <&e

I

-

4p 2

|

=

p-*o "J"* Jo

I

dR

j

£,

and similarly C'2

=

1

lim —j—

r° I

exp (— xz) dx

= £.

p-^0 v"* J -}6/p

We

have therefore shewn that A

lim

/'

|

F(R)Jv (uR)J (ur)uRdudR y

A-»-» J r-S Jo

exists

and

is

equal to

t\F(r+0) + F(r-0)}. C,

* Hardy, Quarterly Journal, xxxv. and C2 se« §1452. ,

(1904), pp. 22

—

66.

For a

different

method

of calculating

[CHAP. XIV


464 But,

if this

by

limit exists, then,

§

1442,

f udu Jo(* F(R) J, (uR) J

v

(ur)

RdR

Jo

and

also exists

equal to

is

it

;

and so we have proved Hankel's theorem, as

stated in § 144.

The use of generalised integrals in the proof of the theorem seems to be due to Sommerfeld, in his Konigsberg Dissertation, 1891. For some applications of such methods combined with the general results of this chapter to the probleme des moments of Stieltjes, 81—88. see a recent paper by Hardy, Messenger, xlvii. (1918), pp. Note on HankeVs proof of his theorem.

14*46.

The proof given by Hankel of his formula seems inadequately. The first is in the discussion of lim f

|

two points somewhat

to discuss

F(R)Jv (uR)Jv (ur)uRdudR,

which he replaces by lim

f

^(/2)[^^ + i(X R)-/v(X/-)-rJ, + 1 (Xr)Jy (X/2)J^r-5 J

.

In order to approximate to this integral, he substitutes the first terms of the asymptotic expansions of the Bessel functions without considering whether the integrals arising from objection to the proof), the second and following terms are negligible (which seems a fatal and without considering the consequences of \R vanishing at the lower limit of the path of integration.

The second

which

point,

is

lim *-».«>

after proving

and

is

$

if

J

Jv (uR)Jv (ur)uRdudR

it

for granted that it

this is zero if £ tends to a positive limit

must be bounded

if

£-*-0 as X-ôo

;

and

seem prima facie obvious.

Extensions of Hankel's theorem

14*5.

We

J

J r+t J o

by the method just explained that

{-=0, he takes

this does not

of a similar character, is in the discussion of

shall

now

to

any cylinder functions.

discuss integrals of the type

l"

uduT F(R) <@

v

(uR)

K (ur) RdR,

in which the order v of the unrestricted cylinder function &,(z) is any real* number. The lower limits of the integrals will be specified subsequently,

since

it is

convenient to give them values which depend on the value of

For definiteness we

where *

suppose that

shall

<&, (z)

=

v.

{cos a

.

Jv (z) + sin a Yv (*)}, .

and a are constants.

The subsequent

discussion

ie

simplified

and no

generality is lost by assuming that

p^Q.

465

MULTIPLE INTEGRALS

14-46-14-51]

The analogue

of the

Riemann-Lebesgue lemma

f F(R) e (

(XR)

v

(§ 14*41),

namely

that

RdR = o (l/y/\),

Ja rb

provided that

F(R)*JR.dR

\ .

<>.

and is absolutely convergent, may obviously be proved by methods of § 14*41, provided that a £ b ^ oc and

exists

precisely the

,

faÔ [a

O^v^l,

if

>0

>\.

if v

The modified

The theorem theorem

is

of § 14*44 has to be modified slightly in form. that the repeated integral

P <£v (uR)
(

u r) u s/R

V (

.

du dR

J a J t is

bounded as

functions of

though

A,-h»oo

while t remains fixed; as in §14*44, a and b may be The number t is positive, finite limits as X -*- oc

X which have

.

be zero when 0î>^£.

it is permissible for it to

Also the repeated integral and the integral r

A <6-

r)

8ina?

dx

/:

both converge or both oscillate as X -* [Note. type,

i.e. if

If the two cylinder functions we considered the integral

oo

.

in the repeated integral were not of the

same

A

/:/: it

<&9 (« R) <(c

v

(«r)

u sjR du dR, .

would be found that the convergence of this integral necessitates the convergence of the

integral

/A(6-r) 1-cos.r

and

so, if

X (6-r)-*-oo as X-*-
14-51.

,

,

dx;

the repeated integral

is

divergent*.]

The extension of Hankel's theorem when

Retaining the notation of

*5

v

%

|.

14*4—14-5, we shall now prove the following

§§

theorem. Let

\

F(R)*JR.dR

exist

and

be

an

absolutely convergent integral,

and

Jo

letO^v^^. Then (1) -,

„,

(* udu

n/

^

* This point

r F(R) „/

<&.

«s

(uR) <@v (ur) 2

>

2
RdR

o sin (a

+ vir)

f

00

R —r iv

2

"

viv\ fjp

was overlooked by Nielsen, Handbuch der Theorie der Cylinderfunktionen

(Leipzig, 1904), p. 365, in his exposition of Hankel's theorem.


466

provided that the positive number r

an

lies inside

[CHAP. XIV

interval in which

F(R)

has


As

in § 14*42,

we may shew that

r udu Jor F(R) 9?

v

RdR

(uR) <@v (ur)

Jo

=

F (R) f

lim l" A.

-». oo

<@v (uR) <@v (ur)

uRdu dR,

J

.'


exists.

But now we observe that k(

(

&,(itRy&y (ur)udu

Jo 1

R

uR <@v+1 (uR) c@v (ur) -

-i*

2

X

R -r 2

2

r t% v+1 (ur)

^„ (uR)

[R <@, +1 (\R) <%„ (\r) - r <@y+1 (\r) <&, (\R)]

2
a sin (o

+

R

vnr)

R

TTsini/TT

Hence we

ur

v

rv

(R -r )' 1

2

infer that, if r is not a point of the interval (a, b), then

K

f

F (R) \

<@,

(uR) <@v (ur) 2
n\ as

X. -»-

oo

Now

and so the

;

last

uRdudR 2

sina sin (a +

yTr)

/-

6

R*»-r*>

v/

repeated integral has a limit when \

number

choose an arbitrary positive

e,

-*-

^ p^ p oo

.

and then choose 8 so small 8, r + 8) and so that

that F(R) has limited total fluctuation in the interval (r —

(\F(R)-F(r + 0)\
Now

("

take

F(R)

Jo

*

I

if

r
if

r- 8 ^R
8,

#„ (uR) <@v (ur) ududR,

Jo

and divide the 22-path of integration into four (0,r-8), (r-8,r), (r,r

parts,

+ 8),

Apply the second mean-value theorem as

namely

+ 8,).

(r

in § 14*45,

and we find that

K<

(* Jo

F(R)\ @¥ (uR) <@

v

(ur)

uRdudR

Jo

+ *-7r)jf'-«

2a*sinasin(a

-|l

7T Sin VTT

/* R» - r* ^ /pw -rUl &-**>{» -ft*™** |

Cr+S r\ fr+S

+ F(r + 0)
+ F(r - 0)

+ V,

V

r .

<@y

(uR)<@y (ur)uRldudR

<&,

(uR) <@v (ur) uRHudR

Jo

( ( J r-tJ

X

^>

MULTIPLE INTEGRALS

14-51]

where

\r)\

has an upper bound which

arbitrarily small

The making

when

independent of \ and which

is

\

after

is

e is arbitrarily small.

integrals on the right converge to limits e -*•

467

-* oo

we

,

X

r udu Jof

when \ -*- oo and ,

so,

by

infer that

F(R) <$ v (uR) <@, (ur) RdR

Jo is

convergent and equal to 2
+

a sin (a

IT

R - r"' 2v

=°

vir)

f

Sin VJT rco

+ F(r + 0) Vr

•

fr+6

lim «-*0.'0

+ F(r-0)
r

@

uR*dRdu

<@v

(uR)

r tfv

(uR) <@v (ur) uR* dRdu,

v

(ur)

Jr

I™ I*

provided that the limits on the right exist.

To prove that the limits exist and to evaluate them simultaneously, take F(R) = R ¥ when r
We

thus find that <@v

r'+i lira

(uR) <$ v (ur) uRSdRdu

t-»0J0 Jr

rr+S

r

=

lim / / S-*oJo Jr

provided that this repeated limit exists

r'+Mim f" «-»0.'0

f

<@ v

;

^{uRY^ûûR'^dRdu,

and similarly

(uR) r$ v (ur)uR*dRdu

Jr-S

-

r

lira I" ( «-»-0.'o Jr-S

For brevity we write b in place of r I" f &* (uR) ^r (ur) uR' "

-

+

f

>

<@„(uR)<@¥ (ui')uR '+ 1 dRdu. ,

+ 8. We

then have

dRdu

{6" +l

^h-i (ub)

- r»+ ^, +1 (iir)} #r (ur) du l

Jo

=

lim P -*.+oJo

{b'»<&„ l (ub)-i^ Vrt(ur)}<& p (ur)--, u l

since the second of these three integrals lutely convergent

Now

when

is

convergent, and the third

is

abso-



-

;


468

[CHAP. XIV

and these are all absolutely convergent. They may be evaluated as cases of Weber's discontinuous integral of § 13 "4, and hence we find that °°

{6-+ 1 |

K+

_
2

i

(ah)

- f+» ^„ +1 (ur)\

r* sin (a

err

The

"

.

+

—

+

H

vir)

.

vir) sin vir

when p -* 0,

limit of this expression, /,

r*\

\

^

'log 1-t-2

=

7T Sin

+ pir 4- vir) T (1 — p) ,r(l —v)T(v + p + 1)

sin a sin (a

sin (p7r

a*/* sin a sin (a

%

+ pir) sin (a + vrr ) .T(v + l—p) vir T (v + rJT (p + 1)

2* sin pir sin

2%>

<&„ (wr)

J/7T

L

/

is

6

.

+2 log r ft

reducible to

„ /, , i*^(l, -»; l-v;

62"

"^

V

+ \ir cot a — |7r cot (a + vir) — some algebra

after

and the

;

limit of the last expression,

yfr

(1)

when

r*\ t-J

W

+ ty (— v)

b

-*»

r

+ 0,

\,

is

simply ^^r".

In like manner

it

may be shewn

that

r

lim (" 6++0J0

and [ °°

so

we have proved °°

udu j

<@v (uR) <&p (ur) uR*+' dRdu

( J

that

F (R) <&. (uR) <@

v

RdR

(ur)

2 ^ Sma Z^{F{r + 0) + F(r-0)}n x

provided that

§%v%\, F(R) is

this is the general

14*52.

^ (a+ ^r- f ~^ , F(R)dR, Br-îlfr-r*)

Trsini/TT

<@¥ (z)

and

= \
r-S

=

er

v

Jo

'

subject to the conditions stated in § 144, and {cos

aJ¥ (z) + sin aF„ (z)}

theorem stated at the beginning of the

section.

Weber's integral 'theorem.

It is evident

from

§ 14-51 that, if

|

F(R) *JRdR

exists

and

is

absolutely

Ja

convergent, where a (1)

lim

[°°

> 0,

then

F{R) [R<@v+1 (\R) V. (Xr) -

=

+

r<@v+1 (Xr) •*„ (Xi2)]

^^

0) + F(r-0)}, provided that r lies inside an interval in which F(R) has limited total fluctuation and F(R) is defined to be zero when ^R < a, if the order of the

$
cylinder functions lies between

—£

and

\.

Y

,

MULTIPLE INTEGRALS

14*52]

We shall

now

469

establish the truth of this formula for cylinder functions of

unrestricted order.

[R<&p+l (XR) <®w (Xr)

Let It is

so,

r

;

X)

- <£,,_> (R,

r

X)

;

= -—- OVi (\R) <$

v

(Xr)

(R, r

;

X).

+ %Vi (Xr) £?„ (Xft)]

by the analogue of the Riemann-Lebesgue lemma lim I"

(2)

A-»-oo 7

[3>„

(R, r ; X)

-

4^ (R,r;

lim

(3)

where n

is

T

r ; X)

[4>, (R,

any positive

r

r

-
X) 22F(22)

;

values of

is

r

X)] 22F(22) dft

;

= 0,

practically

v,

<*22

between ±

lies

- |
we deduce from

2

[F (r

$,

and then from

(1)

+ 0) + F (r - 0)},

(3) that

X)RF(R)dR = $
lim P°„(22,r*

This result

dR = 0.

integer.

„± n (22,

so, for all real

(4)

JRF (22)

we have

result,

Choose n so that one of the integers v ± n lim

X)]

(§ 14*41),

a

Hence, by adding up repetitions of this

and

<*>„

an easy deduction from the recurrence formulae that

and

- r%% +1 (Xr) r$v (XR)] ^-—^ =

due to Weber*, and

it

was obtained by the method

indicated in § 14*46.

To obtain the

Weber's form,

result in

Y (,) = Y

{<$v (z)

<°>

\tf „

Then u r^, (l

=

let

v

(r)Jv (z)-Jv (r)Y,(z),

v

(R) J, (z)

- Jv (22) Yv (z).

_

^)^^-» <«r)^^ r

and the expression on the

left is also

equal to

[22iU (uR) <@¥ (ur) - r<@v+1 (ur) f „ (uft )] = w22 [ Yv (22) J, +1 (u22) - Jv (22) 7, +1 («22)] [ Yv (r) Jv (ur) - Jv (r) Yv (ur)] -ur\Yv (r) Jv+1 (ur) - Jv (r) Yv+l (ur)] [ Yv (R) Jv (uR) - Jv (R) Y, (u ft)] = wFv (22) Yv (r) [RJ, +1 (uR) Jv (ur) - rJ v+1 (ur) Jv (uR)] + $u{Jv (22) Yv (r) - Jv (r) Yv (22)} [22J, +1 (uR) Yv (ur) - R v+l (uR) Jv (ur) - rYv+1 (ur)J„(uR) + rJ¥+l (ur) Yy (uR)] [RD v+l (uR)D v (ur) - rD v+l (ur)D v (uR) (R)\ (r) Y -\u{Jv (22) Yv (r) + Jv v - RD v+l (uR) D v (ur) + rD v+1 (ur) D v (uR)] - uJv (22) Jv (r) [RYV+1 (uR) Yv (ur) - rYv+1 (ur) Yv (uR)], \Dv (z) = Jv (z)+Yv (z), where \D v (z) = Jv (z)-Y,(z).

u

*

Math. Ann.

vi. (1873),

pp. 146—161.

:


470

Now

[CHAP. XIV

suppose that

r/(R)RdR J

exists

and

is

a


and consider

;

(* <&,(ur)W.(uR)uRdudR. lim \* f{B)

Carry out the integration with respect to u, and replace the integrated part by the sum of the four terms written above, divided by R* — r2 .

J,(B)F.W-J.(r)r.(B)

Since is

bounded near

containing

r, it

of terms tend

to

r,

and has limited

total fluctuation in

any bounded

interval

follows that the integrals corresponding to the second group zero as \->- by the generalised Riemann-Lebesgue lemma. ,

Corresponding to the third group of terms we get a pair of integrals which

happen to

cancel.

When we

use (1),

we

are therefore left with the result that

A

p / (R)

lim

f

<&¥ (ur) 9$ v

= that

is to

${J„*(r)

.

dudR

+ Yv>(r)}.{f(r + 0)+f(r-0)},

say

"

uduT f(R)<@

v

(6)

(uR) uR

(ur)9$ v (uR)RdR

f

-!{/,« (r) + in in

F„» (r)}

which the cylinder functions are defined by which f(R) has limited total fluctuation.

Apart from

(5),

.

{/(r

and r

+ 0) +f(r - 0)},

lies inside

details of notation, this is the result obtained

an interval

by Weber

in the

case of functions of integral order. 14'6.

We

Formal statement of Neumann s

shall

now

state precisely the

integral theorem.

theorem which

will

be the subject of

discussion in the sections immediately following. It is convenient to enunciate the theorem with Du Bois Reymond's* generalisation, obtained by replacing

the Bessel function by any function which satisfies certain general conditions.

The

generalised theorem

(I)

is

Let "^(Z, Y) be

variables (X, Y), which

is

exists *

and (t),

pair of real

V(X, Y).(X* + Y*)-l.(dXdY)

—oo

is absolutely convergent.

—

390. Neumann's formula (cf. § 14-3) is obtained by writing iv. (1871), pp. 383 and the conditions (I)— (III) are substantially those given in Neumann's treatise

Math. Ann.

g{t)sJ

J

the

such that the double integral

("

I -00

as follows

a bounded arbitrary function of

published in 1881.

,

MULTIPLE INTEGRALS

14*6, 14-61]

When *&(X, Y) F(R, ), and

(II)

denoted by

of $> between ±

F(R, and

7r),

the interval (0, oo

) ;

is let

471

expressed in terms of polar coordinates, let it be F(R, 4>) have the property that (for all values

qua function of R, has limited fluctuation and also F (+

4>),

let this

total fluctuation in 0, 4>) be

integrable

functions o/. (III)

(±

0,

R),

If Q(R,

)

fl(R,

3>)

let

denote the total fluctuation of F(R, ) in the interval tend to zero uniformly with respect to 4> as R-*-0,

throughout the ivhole of the interval (— in

a number of

sectors the

with the exception* of values of may be assumed arbitrarily

it, ir),

sum of whose

angles

small.

Since

F (R;

F( R,

-*

)

(IV) that

\

)

F (+ 0,

Let

- F(+

!

0, )

),

this condition

is

be a continuous function of the positive variable R, such bounded both when R-*-0 and when R —~ oo .

rR

/•»

g(t)tdt

= O(R), and

let

j

dt

G(t)—

be convergent.

r u du Jl™-acJ["-x V (X, Y).g{u
Then

necessitates that

4>) uniformly except in the exceptional sectors.

g(R)

g (R) \/R

Let

^ Ci(R,

.

(dX d Y)

JO is convergent,

and

is

equal to dt

r°°

2*-.[i*W(+o,)] Jo

where ffiKF (+

0, 3>)

ffW?. t

meansf

Lll n F(+0 '* )d
'

2tt,

Before proving the main theorem,

we

shall prove a

number

of

Lemmas,

just as in the case of Hankel's integral.

14'61.

The analogue of the Riemann-Lebesgue lemma.

we have the theorem that if T is an unbounded domain^, surrmmding the origin, of which the origin is not an interior point or a boundary point, then, as A. -*- oo Corresponding to the result of

JIT *

The

§ 14'41,

F(R,4>)G(XR) ( *5*?> = o(l).

object of the exception is to ensure that the reasoning is applicable to the case (which

which & (X, Y) is zero outside a region bounded by one more analytic curves and is, say, a positive constant inside the region, the origin being on the boundary of the region. f The discovery that the repeated integral is equal to an expression involving the mean value of F[R, <£) when the origin is a point of discontinuity of F(R, $) was made by Neumann, Ueber die naeh Kreis-, Kugtl- und Cylinder-functionenforUchreitenden Enticickelungen (Leipzig, 1881), pp. 130—131. J For instance T might be the whole of the plane outside a circle of radius 5 with centre at the is

of considerable physical importance) in

or

origin.

.

!


472

[CHAP. XIV

a theorem of a much weaker character than the view of hypothesis (II) of § 14*6. The reason of this is the fact that (,/A) for certain values of R, and this seems to make arguments of the

It will be observed that this is

theorem of §

14*41, in

G (\R) may be* type used in

of

-

14 41 inapplicable.

§

To prove the lemma, suppose F(R, <£>) may be expressed

,

first

tonic functions Xi (-^> ®)> %2 (-^» ®)> F(R, 3>) in the interval (0, R).

If

R

and

R

R

and

bounded. Then, for any value

two (increasing) mono-

the total fluctuation of

is

any particular value of , it some value of R% between

for

that, for

lf

rKR

*„,.,dt

J kR„

x

dt

.

,„

Ax

+ X» (*i.*) 0(*)y l

f J

kR „,.,dt <

G(t)

kRt

t

.

G (t)—

Since

R

mean- value theorem

= Xl (£„,<*>) C

T is

wnos e sum

are the extreme values of

1

follows from the second

R

that

as the difference f of

is

convergent,

an arbitrary positive number, we can

if e is

t

choose

X

so large that

dt

„ for all values of

i

£ not

less

Xx (R,
|

^

{

ff((

<€,

>

than the smallest value of Xl (R, 4>)

R

.

Also

- IF(+ 0, *)}+*! F{+ 0, <*>)

.*)-i-P(+o.*) + *|^(+o

J

*)!,

and similarly

IttCJR,*)!** whence

it

Jf

X

(< >,)

+ $F(+0,) + $\F(+0,®)\,

follows that

F(R,<&)G(\R)

(dRd®)

R

<2eT

{*<«>, *)

+ *(ao,
{n(oo,)

+

J -ir

= 2e r F(+ 0,

bounded, this can be made arbitrarily small by taking it is independent of the outer boundary of T. Hence proceed to the limit when the outer boundary tends to infinity.

and, since

3>) is

e sufficiently small,

we may *

some

This

\F(+0,)}d;

is

the case

and

when g (R)sJ {R) then G (R)=RJt (R). imposed on J*' (/?, *f>) are superfluous,

of the conditions

t Cf. Modern Analysis, § 3-64.

;

It is

by no meaus impossible that

MULTIPLE INTEGRALS

14-62]

We infer that, if T has no outer boundary,

473

the modulus of

can be made arbitrarily small by taking \ sufficiently large

;

and

this is the

theorem to be proved.

The inversion of Neumann's repeated

14*62.

integral.

We shall next prove that the existence and absolute convergence of the integral

r r are

x

t)

(dXdT)

sufficient conditions that

f— « ¥ (X,

fttdttf JO

J

— CD

+ F )} (dXdY)

Y) g {u V(X 2

2

.

.

J

-

K

T

Y)\ g{u>J{X*+Y*j\udu{dXdY),

¥(X, f" A-».oo J — oo J — oo lira


./

exists.

For any given value of \ and any arbitrary positive value of a number

such that

/9

,

J where

J. is

We

there exists

e,

JV(B *)|2»((Md*)<^, >

the upper bound of

(w) j

Vw*

then have k

f r [ F(R, )g (uR) udu R (dRd
j

=

r-vJr JOr F R (

i

I

J

>

fi

^ ^r o^)

-f

*

< ji

f*

["

J -irJ

fi

/"*!

R (dRd®)udu

J -it J/3

!

-F(#, <&) w*dw !

(dRd
itf

Jo

+

aT i'-n r J

Since this

I

(<^*&)

f" ^(ii, <&)g(uR)

f

Jo

•

I

J

J

!

F(R,

4>)

R* (dRd^)

u*

d«

I

fi

<

€.

is

true for arbitrarily small values of

e,

we

infer that

f !'-wor F(R, <5>)g(uR) R (dRd
Jo

J

=

lim A-*>

the integral on the

left

f

oo ./

["

-WO

P F(R,®)g(uR)udu.R(dRd), Jo

existing because the limit on the right

is

assumed

exist.

w.

b. f.

16

to


474

Hence

it

follows that, if the limit

on the right

exists,

[CHAP. XIV then

\~uduf' [" F(R Q>)g(uR)R(dRdQ>) )

-lim

f-v

K-*.aoJ

-ft

14*63.

The proof of Neumann's integral theorem.

We

now

are

in a position to prove without difficulty the theorem

due

to

We

Neumann e

f>(ie,4>)(?(X^)^^. J

stated in § 14*6. first take an arbitrarily small positive number then choose the sectors in which the convergence of (R, 4>) to zero is

O

and

way that

uniform, in such a

sum of their angles exceeds 2ir — e. We then 4>) < e in these sectors whenever R ^ 8 and we

the

choose 8 so small that II (R,

;

take the upper bounds of

n (R,
F(R,

|

B and C. We then apply the second

)

|

and

I

f°G(u)

^

to be

mean-value theorem. "We have

f* dR r* fJJ? -Xk(+0,*)/ O(XB)^+to(8,*)- Xl (+0,*)}J^(xfi)^

f

o

where

^ f ^ 8. f*

Now

dR

/"** I

G(\R)^\ =

R

Jt

\

t

I

dii

I

«

I

G(u)-\<2G.

\

J a*

Hence f Jo

where and is

tj |

|

less

Hence I

f

"

F(R,)G(\R)t§ = F(+0, Jt u Jo

is less

than

than 2eC inside the sectors in which convergence

2BG in

it follows

I

l'F(R, ») Q

(MJ)^*> _ 2,J«J(+ 0,*) f
I

^

27r.2etf-l-e.250

= 2eC {2tt + £}.

for large values of X,

/" /* *<* ») ./

-WO

(Mi)

- ftriW( + 0, «)

-tt

f© (.) JO

<2eC(2ir + that Urn"

is

I

uniform,

that

< Hence,

is

the exceptional sectors.

2?)

*« U

+ o(l),

to say '

f JF(#,

f K-^colJ-wJo

4>)

^?> -

(\R) (

Jti

2,r

JWF<+ 0, 0»)

f G («) -

Jo

**

< 2eC (2tt + 5).

MULTIPLE INTEGBALS

14*63, 14-64]

Now the small,

expression on the

we

left is

independent of c

That

infer that the limit is zero.

is

is

e is arbitrarily

f f *(**) 0(XS)<-^

Km and

and so since

to say,

A-»xJ-irJO exists

;

475

-"-

equal to

2-rJ»lF(+O Applying the result of been proved.

§

f

*)J"0(iO ^' » Jo

1462, we see that Neumann's theorem has now

In the special case in which g (u)

= J (w), we

have

ru

tg(t)dt

I

= uJ

x

(w),

Jo

G (u) = uJ

so that

x

(u),

r g wu *» -H- /.-(.)} «. -c

and

Jo

Jo

Hence we have (1)

puduj

9 I"

V(X,Y).J {u*J(X*+Y*)}.(dXdY) = 2?r jffcW (+ cos + ,

.

If (2)

we change the

origin,

).

fuduf" f° V(X,Y).J,lu4[{X-*Y + (Y-tf\].(dXdY)

finally,

(3)

sin

we deduce that

= 2ir0Vfr (oc +

and

.

cos

,

y+

sin 4>),

changing to polar coordinates,

r

ruduT

[u
F(R,
J-wJ-aa

Jo

where 0lF(r,

)

)]

RdRd®

„ = 27rJ*LF'(r,0),

now means the mean

of the values of

F(R,

)

when

traverses the circumference of an indefinitely small circle with centre

14*64.

(R, 3>)

(r, ^).

Mehler'8 investigation of Neumann s integral.

Neumann's

integral has

been deduced by Mehler* from the formula

5 ?!L±2 f ' f" /(S, ) P„ (cos 7 ) sin @d4>de f(0 £) = »=0 **?r Jo J -n t

by a limiting process cos

The formula

is

in this formula

;

7

= cos

cos

© + sin 6 sin @ cos

(

— $).

obtained f by constructing a solution of Laplace's equation, which has an assigned value f(0, ) on the

valid inside a sphere of radius k,

surface of the sphere. * Math. Ann. v. (1872), pp.

135—137 ;

t Of. Modern Analysis, § 18*4.

cf.

Lamb, Proc. London Math.

Soc. (2) n. (1905), p. 384.


476

The § 5'71

by Mehler

limiting process used

the radius of the sphere

;

made

is

is

[CHAP. XIV

that suggested by the result of

indefinitely large,

and new variables

R, r are defined by the equations

R = k%,

r

= k0,

so that R, r are substantially cylindrical coordinates of the points with polar

coordinates

(«,

®,

noted by F(R,

3>), (k, 6,

)

the function of position /(©, ) is then debecomes approximately equal to J (nvr/ic),

;

Pn (cos7)

and

),

where ra-

We are

= Km 1 «-*.«>

now we

=

R?

+•

r2

- 2Rr cos (4> -

write

±lf *Tr

»=0

w//c ^= u,

Kn

2n

JO

r J

But

Neumann's

many

Thus, although we 00

S

n=0 \*/

F(R,
be made the basis of a rigorous proof, be-

steps which require justification.

know

/o\ n 2« 4-1

(-)

)^d^. K*

result.

this procedure can hardly

cause there are so

(n^/ K

we get

F{r,$)=±-ûdu^ j" is

F(R,^)J

-n

and replace the summation by an integration (taking

l/« as the differential element),

which

).

thus led to the equation

*>,<£) If

2

that /"" /""

-IT-Î *T Jo

a potential function (when

f(®,®)Pn(cosy)sm@d®d® .'

-a-

r< k\

which assumes the value f(0, ) on the that we may put p = k in the series necessitates a discussion of the convergence of the series on the surface of the sphere and the transition from the surface of a sphere to a plane, by making k -* oo , with the corresponding transition from a series to an integral, is one is

surface of the sphere, the theorem

;

of considerable theoretical difficulty. It is possible that the method which has just been described is the method by which Neumann discovered his integral formula in 1862. Concerning his method he stated that "Die Methode, durch welche ich diese Formel so eben

abgeleitet habe, ist nicht. vollstandig strenge."

CHAPTER XV THE ZEROS OF BESSEL FUNCTIONS 15*1.

Problems connected with the zeros of Bessel functions.

There are various

classes of problems, connected with the zeros of Bessel

functions, which will be investigated in this chapter.

We shall begin by proving

quite general theorems mainly concerned with the fact that Bessel functions

have an infinity of zeros, and with the relative situations of the zeros of different functions. Next, we shall examine the reality of the zeros of Bessel functions (and cylinder functions) whose order is real, and discuss the intervals in which the real zeros lie, either by elementary methods or by the use of PoissonSchafheitlin integrals. necessarily real,

product.

We

Next, we shall consider the zeros of Jv (z) when v is not this function as a Weierstrassian

and proceed to represent

then proceed to the numerical calculation of zeros of functions

of assigned order, and finally consider the rates of growth of the zeros with the increase of the order, and the situation of the zeros of cylinder functions of

A

unrestrictedly large order. full discussion of the applications of the results contained in this chapter to problems of Mathematical Physics is beyond the scope of this book, though references to such applications. will be made in the

course of the chapter.

—

Except in §§ 15*4 15*54, under consideration, is real.

The

zeros of functions

it is

supposed that the order

whose order

is

v,

of the functions

half an odd integer obviously lend

themselves to discussion more readily than the zeros of other functions. particular the zeros of —^

,7**

'*

In

have been investigated by Schwerd

and by Rayleigh*; and more recently Hermitef has examined the zeros of Jn+l (a?). The zeros of this function have also been the subject of papers by RudskiJ who used the methods of Sturm; but it has been pointed out by Porter and by Schafheitlin§ that some of Rudski's results are not correct, and, in particular, his theorem that the smallest positive zero of Jn ^(x) lies between J (n + 1) tr and \ (n + 2) tt is untrue. Such a theorem is incompatible with the inequality given in §153 (5) and the formulae of §§ 15*81, 15*83. >

*

Sohwerd, Die Beugungsertcheinungen (Mannheim, 1835); cf. Yerdet, Lecons d'Optique i. (Paris, 1869), p. 266; Bayleigh, Proe. London Math. Soc. iv. (1873), pp. 95—103. t Arehiv der Math, und Phyt. (8) i. (1901), pp. 20—21. % M6m. de la Soc. R. det Sci. de Liege, (2) xvm. (1895), no. 3. See also Prace MatematycznoFizyczne, in. (1892), pp. 69—81. [Jahrbueh liber die Fortschritte der Math. 1892, pp. 107—108.] 8 Porter, American Journal of Math. xx. (1898), p. 198; Schafheitlin, Journal fur Math. cxxn. Phytique,

(1900), p. 804.


478

The Be88el-Lommel theorem on

15*2.

the zeros

of J,

[CHAP.

XV

(z).

It was stated by Daniel Bernoulli* and Fourierf that

J

(z)

has an infinity

and a formal proof of this result by an analysis of Parseval's due to BesselJ. It was subsequently observed by Lommel§ that Bessel's arguments are immediately applicable to Poisson's integral for J, (z), provided that — \ < v ^ \. A straightforward application of Rolle's theorem to x ±v Jy (x) is then adequate to prove Lommel's theorem that J,(z) has an of real zeros

integral

infinity

;

is

of real

zeros,

for any given real value of v. investigation consists in proving that

The Bessel-Lommel

when - \ <

v

< \,

between mir and (m + 1) ir, then J¥ (x) is positive for even values of m, (0, 2, 4, ...), and is negative for odd values of m, (1, 3, 5, ...). Since Jv (x) is a continuous function of x when x > 0, it is obvious that Jv (x) has an odd number of zeros in each of the intervals (\ir, it), (§7r, 2ir),

and x

lies

(fir, Sir),....

Some more

precise results of a similar character will

To prove Lommel's theorem,

let

x = (wi +

obvious transformations of Poisson's integral, r

t/ "

sgn

so

Now

\6)

ir

<

where

—1536.

^ 1;

J

v

{x)

the last integral

= sgn

^

then,

by

we have

p+« cosfrm *(*«* M _~r(i/ + |)r(|).(2m au w {(2m + ey-u*}*-> + ^J rim+0

and

be given in §§ 15-32

'

cog i 7rM

{(2w

+ g), _ „,),->

•

may be written in the form m 2 (-ft!, + (-)"»», r=l 8r

/

\

(-)' vr

where

=

^

f .'

\m

J

If

COSîm

,

^—^^^--^ du,

8 - ***}*-' 2r _ 2 {(2m + 0) r2m+0 cog

|

^m

{(2m+ey-u*}i-'

du.

write u = 2r — 1 + U, and then put 0y-(2r-l + UYy-i-{(2rn+OY-(2r-l-Uy}'-*=fr (U), +

now we

{(2m it is clear

that tV

and, 8ince||

v^

t

fr(U)

is

= Cf^^sinîrU.dU, Jo

a positive increasing! function of r.

Sci. Imp. Petrop. vi. (1732—3) [1738], p. 116. Analytique de la Chaleur (Paris, 1822), §308.

*

Comm. Acad.

+

La TMorie

% Berliner Abh., 1824, p. 39.

BeueVsehen Functionen (Leipzig, 1868), pp. 65—67. This is the point at which the condition r ^J is required ; the condition v > - & ensures the convergence of the integral. f The reader will prove this without any difficulty by regarding r as a continuous variable and then differentiating /, (U) with respect to r. § Studien iiber die ||

'

ZEROS OF BESSBL FUNCTIONS

15-2-15-22]

479

It follows that

and

so

sgn

Jv (nnr + 10tt) = sgn [(-) m

{•*•„'

-J-

(«„,

- ^-0 + («m_, - w_ ) + ...}] i*

8

= sgn(-l)"-, since vm'

is

obviously not negative.

That

is

to say,

when — J < sgn

and from The

v «/„

this result Lomrael's

<

\,

(mir

+ \6v) = j_

theorem follows in

zeros of Ji(x), as well as those of

—

J

-0,2,4, ...) (m = 1,3,5,...) the manner already stated. (to

have been investigated by

(x),

Baeh'r, Archives

Nlerlandaises^vu. (1872), pp. 351 358, with the help of a method which resembles the Bessel-Lommel method. Baehr's result for Ji (x) is that the function is positive when x lies in

the intervals

(0,

n), (fw, 3»r), (§«-, 5tt), ...,

the intervals

(§7r, 2a-), ($«-, 4jr),

gated in this

way by

The

(Qir,

C. N. Moore,

6n-), ....

and that

it is

negative

The function Ji(x) has

Annals of Math.

(2) ix. (1908), pp.

when x

lies in

also been investi-

156—162.

results just stated are of a less exact nature than the results obtained with the aid

of slightly

more

refined analysis

by Schafheitlin

(§§ 15*33

—

15-35).

was noted by Whewell, Trans. Camb. Phil. Soc. ix. (1856), p. 156, that zero between 2 and 2 ^2, and that the function Hq (*) has some real zeros. It

J

(x)

has a

The non-repetition of zeros of cylinder functions.

15*21.

It is easy to prove that 9%, (z) has no repeated zeros, with the possible exception of the origin *. For, if $v (z) and ^y(z) vanished simultaneously, it would

V„
by repeated

differentiations of the differential equation (z)

The interlacing of zeros of Bessel functions.

15*22.

now be shewn

that if j¥ lt j ¥>2 ... are the positive zeros of arranged in ascending order of magnitude, then, if v > — 1, It will

,

,

This result

is

<>,3 <

•••

Jy (x),

•

sometimes expressed by saying that the positive zeros of J, (x)

are interlaced with those of J,+ x

(x).

To prove the result we use the recurrence formulae j£ the

first

J¥ (a,)

[*-Jp (x)} = - x-"J,+1 (x),

of these shews that

^

{tf"

+*

J^ (x)} - ar» J

v

(x)

;

between each consecutive pair of zeros of

at least one zero of x~" J, +1 (x), and the second shews that between each consecutive pair of zeros of x*+1 J, +1 (x) there is at least one zero a?-'

there

is

of x*+x J¥ (x)\ and the result

is

now

obvious.

* This is a special case of a theorem proved

by Sturm, Journal de Math.

i.

(1836), p. 109.


480 If

i>^— 1, the zeros are obviously

The

interlaced but the smallest zero of

still

nearer the origin than the smallest zero of

[CHAP.

XV

Jv +i(x)

is

Jv (x).

result concerning interlacing of positive zeros is obviously true for

cylinder function* <$¥ (x) and the contiguous function

^v+1

any

real

(x).

This fundamental and simple property of Bessel functions appears never to have been proved until about a quarter of a century agof,

when

four

mathematicians published proofs almost simultaneously; the proof which has just been given is due to GegenbauerJ and Porter §; the other proofs, which are of a slightly more elaborate character, were given by Hobson|| and

van VlecklF. It has

been pointed out by Porter that, since r = 2(i/ + Jr/\ v (x) + Jv+2 (x) /

\

l) r

.

.

«/„ +1 (X),

any positive zero of Jv (x) the functions Jv+1 (x) and Jv+2 (x) have the same sign but at successive zeros of Jv (x) the function J„ +1 (x) alternates in sign, and so there are an odd number of zeros of Jv+i (x) between each consecutive pair of positive zeros ofJv (x) interchanging the functions /„+2 (x) and Jv (x) throughout this argument, we obtain Porter's theorem that the positive

at

;

\

zeros of

Jv+Q (x)

are interlaced with those of

J

v

(x).

15*23. Dixon's theorem on the interlacing of zeros.

A result of a slightly more general character than the theorem of § 15'22 due to A. C. Dixon**, namely that, when v> — 1, and A, B, 0, D are constants such that AD j* BC, then the positive zeros of A Jv (x) + BxJv'{x) are interlaced with those of CJ, (x) + DxJJ (x), and that no function of this type can have a repeated zero other than x = 0.

is

The latter part of the theorem deducible from § 5'11 (11),

is

an immediate consequence of the formula, Jy

f*

Jo

when x

is

positive

would vanish at a repeated zero of A J, (x)

in

A

real cylinder function is

(X) f

d \xJv (x)} dx

dx

for the integral is positive

*

Xjy

(X),

J* (t) tdt - - \a d{J¥ (x)}

and the expression on the right

+ BxJv

f

(x).

an expression of the form aJy (x)+pYy{x)

a, /3 and v are real, and x is positive. Gray and Mathews, A Treatise on Bessel Functions (London, J Monatshefte fUr Math, und Phys. vm. (1897), pp. 383—384. § Bulletin American Math. Soc.vr. (1898), pp. 274—275. Proc. London Math. Soc. xxviii. (1897), pp. 872—373.

which t

II

Cf.

—

1895), p. 50.

American Journal of Math. xix. (1897), pp. 75 85. ** Messenger, xxxn. (1903), p. 7 ; see also Bryan, Proc. Camb. Phil. Soc. vi. (1889), pp. 248—264. IT

ZEROS OF BESSEL FUNCTIONS

15-23, 15-24]

To prove the former

part of the theorem,

we

481

observe that,

if

_ CJv (x) + DxJ '(x) (x) = AJv (x) + BxJv'(x)' v

*'<•)--

then

2

£2l/;*
x{AJv (x)+BxJ,'(x)y

(x) is monotonic. The positive zeros of (x) are therefore interlaced with the positive poles, and from this result the former part of the theorem

and so is

obvious.

Jv (x) is replaced by a real cylinder function aJ

If the function

v

(x)

+/3Yv (x),

we have af@:(x)

#„(*),

2vft (a sin inr

%<

-\ @?i$)tdt = \x

dx

provided that

d

d<@v (x)

Jo

—1 < v<

1

;

/9

+ /3 cos vv)

sin vrr

and and C<@v (x) + Dx<@J (x)

so the theorems concerning non-repetition

interlacing of zeros are true for

provided that

ir

dx

'

and

{rf@v ' (x)}

A<@v (x) + Bx r@¥

'

(x)

(o sin vir+ /S cos vnr) is positive.

Again, since <@¥ {x),

x<@v\x)

d{x<@¥'(x)} dx

\* d<@¥ (x) dx the theorem

and

is

= - \ {(a? -

true for zeros exceeding

+

i^)

<@* (x)

+ &<@:* (*)},

VV. whether v

lies

between

—1

1 or not.

The

result of § 15'22 is the special case of Dixon's

theorem in which

4 = 1, 5 = 0, C = v, D = -l. The interlacing of zeros of cylinder functions of order

15*24.

Let ^?r (x) and

we

^P„ (x)

v.

be any distinct cylinder functions of the same order

shall prove that their positive zeros are interlaced*.

W

If

9

(*)

= aJw (w) + 0Y,(*),

^

then

¥

W„(x)

= yJ¥ (x) + 8Y¥ (x),

(x)^ ¥'(x)-^ ¥ (x)^¥'(x)^^^^. irx

Now it is known that, at consecutive positive zeros of #r (x), <$,' (a?) has opposite signs,

and

therefore,

from the

last equation, <&„ (x)

has opposite signs; that

is

to say & 9 (x) has an odd number of zeros between each consecutive pair of positive zeros of 9^¥ (x); similarly <@¥ {x) hasan odd number of zeros between each consecutive pair of positive zeros of 9% 9 (x) and so the zeros must be <

;

interlaced.

If we take one of the cylinder functions to be a function of the first kind, we deduce that all real cylinder functions have an infinity of positive zeros. * Olbricht,

cussion of

Nova Acta Caes.-Leop.-Acad. (Halle), 1888, this result with some instructive diagrams.

pp.

43—48, baa given an elaborate

dis-


482

Lommel's theorem on

15*25.

An

real,

J

— 1,

if the order v exceeds are not real.

To prove Lommel's theorem, suppose, not

is

v

{z)

Jv (z)

for

has no

has no zeros which

that a

if possible,

from the series

It follows

real.

J

then the function

(z)

The extended theorem

has been effected by Lommel*f*.

is that,

which

of the zeros of Jv (z).

extension of a theorem due to Fourier*, that the function

which are not

zeros

the reality

XV

[CHAP.

is

J

a zero of

that a

is

v

{z)

not a pure

imaginary, because then

,„

would be a Let

or

1)

be the complex number conjugate to

J

>—

Since v

v

a real function of

(z) is

follows from §

1, it

tM«t)JAa since a2

±«

51 1

= ^-^[

t)dt

j so,

+ ?« +

series of positive terms.

J„ (z), because

and

=0 «i!r(i>

so that a

a,

is

also a zero of

z.

(8) that

Max)—_^_ ^(c^-g—j

,

2 tf

,

tJv (at)Jv (a t)dt

= 0.

Joo

The integrand on the left is Hence the number a cannot

exist,

may be used A Jv (z) + BzJv (z) has all its when (A/B) + v<0.

Similar arguments J function zeros

'

These results follow from the

and so we have obtained a contradiction. and the theorem is proved.

positive,

to

shew

A

that, if

series for

-y- {z

A
(z)}

B

and

zeros real, except that

it

are real and v

> — 1,

the

has two purely imaginary

combined with the formula

l

i

which

is satisfied if

15*26.

£ and

/So

tJv {Pt)Jv (fot)dt=0,

are any zeros of

AJv (z) + BzJr

'

(?)

such that

/3

2

^/8

2 .

The analogue of Lommel's theorem for functions of the second kind.

It is not possible to prove by the methods of § 15*25 that§ Yv (z) has no complex zeros in the region|| in which arg2 < ir. But it has been proved by SchafheitlinU that Y (z) has no zeros with a positive real part, other than |

|

the real zeros. * La TMorle Analytique de la Chaleur (Paris, 1822), § 308 ; see also Steam, Quarterly Journal, xvu. (1880), p. 93. t Studien iiber die BesseVschen Functionen (Leipzig, 1868), p. 69. % See A. C. Dixon, Messenger, xxxn. (1903), p. 7. § Or,

more

generally,

<$v

(z).

Y,(z) = e T

'"ri

Yy {-z)±2i cosi'T«7„(-«), and hence, by §3-63 (1), Yv (z) cannot vanish unless v is half of an odd integer. This type of reasoning is due to Macdonald, Proc. London Math. Soc. xxx. (1899), pp. 165—179. In this paper Schafheitlin also sub**I Archiv der Math, und Phys. (3) i. (1901), pp. 133—137. I!

When arg*=±ir,

jects the

complex zeros of Yi

(z)

to a similar treatment.


15*25-15'27]

For

let /9

so that ft

is

483

Y (z), and let j3 be the conjugate complex, Then, by § 511 (8) and 351 (1),

be a complex zero of

Y (z).

also a zero of

l't7.(/3t)Y,(frt)dt

= and

/s^ft2 L

Fo(/3ar)

= pe*", we

so, if /9

I,(A * )

-is

"£r-J-^09--^ u «g'

have

and the expression on the left is positive while the expression on the right negative when to is an acute angle. The theorems of Hurwitz on

15*27.

is

Jv (z).

the zeros of

The proof which was given by Fourier that the zeros of J (z) are all real was made more rigorous and extensive by Hurwitz*, who proved (i) that when v

>—

the zeros of

1,

«/„

(z) are all real, (ii) that, if s is

a positive integer or

— (2s + 1) and —(2*+ 2), Jv (z)

has 4s + 2 complex which 2 are purely imaginary, (iii) that, if s is a positive integer and between — 2s and — (2s + 1), Jy (z) has 4s complex zeros, of which none

zero and v lies between zeros, of

v lies

are purely imaginary.

To

establish, these results,

we use the

notation of § 97.

m

We

take the function g2m,»(0 which has, in the respective cases (i) (ii) in — 2s — 1 positive zeros, 1 negative zero and 2s complex

positive zeros, zeros, (iii)

m — 2s positive zeros and

2s complex zeros.

We

now prove

that, if

——

\»

/

qo

2 f¥ (f) = «=o n 1n (y + !

has at least as

many complex

where m (£>

£,

rj

V)—

1'

shewn

to

ii

zeros as g
are real and f = f +

v) are easily

yn

i-q,

§"

-r

yr then the function ,

i)

= — it}. The

write

'

—t

(:

we

After Hurwitz,

terms of highest degree in

be

£m (m + 1) (v + m) (v + m + 1) {(/ + m) (2m + 1) + in - 1} (£* + rff^ and since g2Mt¥ g-2m,*(0> so a^ so

is *

s

a real function, K

' \

it

it is

not

w+ i (fc

difficult to

•*)

follows that if £

is

1 ;

a complex zero of

an(i therefore the complex zeros satisfy the equation

f (£)

*?)

=

<>.

deduce from the recurrence formulae

- (* + 2m + 2)^m+1§r (f)^,,, (£' ) + (f +

(§ 9*7)

that

a *?

)

<*>„,

(ft

»/)•

246—266; cf. also Segar, Messenger, xxa. (1893), pp. 171—181, a discussion of the Bessel coefficients. The analysis of this section differs in some respects from that of Hurwitz ; see Watson, Proc. London Math. Soc. (2) xix. (1921), pp. 266—272. * Matft. Ann. xxxin. (1889), pp.

for


484 Hence,

for sufficiently large values of

the curve negative

m when

(£, 17)

=

m+1 (f

rj) is

zero so the curve $„+,. (f,

rj)

=

zeros of

g^, (f)

lie in

m (ff, rj)

is

wholly inside

lies

one or other of the closed branches which compose the curve

Hence asm-ôo, the complex

XV

m (i.e. those for which v + 2m is positive),

the finite part of the plane, and

lies in ,

[CHAP.

m (f

,

rj)

— 0.

bounded regions of

the f-plane, and consequently have limit-points.

Now,

since,

by §§965, 97,

^

Jv

}

r(!/+2m +

l)

can be made arbitrarily small in any bounded domain of the f-plane, by taking m sufficiently large, it follows from Lagrange's expansion * that the number of

number of zeros ofg2mv (£) and so /„ (£) has 2s complex zeros. None of these zeros is real, for if one of them were real it would be a limit point of two conjugate complex zeros of gtmiV (£), and so it would count as a double zero of/„(£); and/„(£) has no double zeros. zeros off, (£) in in that area

any small area

when

is

at least equal to the

m is sufficiently large

;

Again, from the series for/„ (f ) it is seen that, when v lies between - (2s + 1) and — (2s + 2),/„ (£) must have one negative zero, and it cannot have more than one negative zero, for then gm> v (f) could be made to change sign more than once as X varied from to — 00 [since g*m,,»{C) can be made to differ from /„(£) by an arbitrarily small number], and this is impossible.

For similar reasons /„(£") cannot have more than 2s complex

zeros.

If we replace f by \z*, so that negative values of £ correspond to purely imaginary values of z, we obtain the results stated in the case of Jv (z).

For a discussion of zeros of Besscl functions in association with zeros of polynomials based on rather different ideas, the reader should consult Lindner, Sitz. der Berliner Math. 5. It may be mentioned that Hurwitz has extended his results to Ges. xi. (1911), pp. 3

—

generalised Bessel functions in pp.

a brief paper, Hamburger Mittheihingen, n. (1890),

25—31. 15*28.

Bourget's hypothesis.

It has

been conjectured by Bourgetf

(zero included), the functions

than the origin, for It

all

that,

Jv (z), Jv+m (z)

when

v is

a

positive integer

have no common

positive integral values of

zeros,

other

wi.

seems that this theorem was not proved before 1929, except in some asm = l and 2 (cf. § 15'22).

m—

simple special cases, such

The formula Jv+m (z) * Cf.

— Jv (Z) Rm, v (Z) — Jv -\ (z) Bm-i, H-l (z)

Modern Analysis,

§ 7 '32.

t Ann. Sci. de VEcole norm. sup. in. (1866), pp. 55

—95.


15*28, 15*3]

shews

that, since

of J„(z) and

Jv (z)

«/„+,» (z)

and Jv _x (z) have no common must satisfy the equation

485

zeros, the

common

zeros

Rm-i,y +1 {z) = 0, they must be algebraic numbers.

i.e.

been proved by Siegel* that J,(z) is not an algebraic number and z is an algebraic number other than zero; hence follow theorems which include Bourget's hypothesis as a special case. It has, however,

number when

v is a rational

Jp (z) and Jv+m (z) have no common zerosf; for such zeros are algebraic numbers and it is known that no algebraic number % can satisfy the equation When

v is half of

an odd integer,

cot (z

easy to shew that

it is

- \vir - ±tt) -

since the right-hand side is algebraic in z

pfo

when

.

y)

v is half of

The proof % given by Lambert and Legendre that applied to § 5*6 (6) to prove that

Jv (z)

ir

%

an odd

integer.

is irrational

has no zero whose square

is

may be rational

inspection of the polynomial R m -Jt v +i(z) now immediately an elementary proof of Bourget's hypothesis in the cases tn — 3 and m = 4.

when v is rational an ;

yields

15*3.

Elementary properties of the zeros of Jv (x).

It is possible to acquire a considerable

amount of

interesting information

Jy (x) and related functions, when v is positive, differential equation satisfied by Jv (x) together with

concerning the smallest zeros of

by a

discussion of the

the recurrence formulae;

we

theorems concerning such

zeros.

shall

now

establish the truth of a selection of

The reader will find a more systematic investigation § of these theorems in various papers by Schafheitlin, notably Journal fur Math. cxxn. (1900), pp. 299 321 Archiv der Math, una" Phyt. (3) I. (1901), pp. 133—137; Berliner Sitzungsberichte, in. (1904),

—

pp.

;

83—85.

J¥ (#), JJ (a), J" (x),

For brevity, the smallest positive zeros of "

called j ¥ , j,', j¥ The smallest positive zeros of " will similarly be called yv , yv\ y v

We first prove (1) It is obvious *

Y¥ (x),

F„' (x),

will

be

Y " (x),

...

. . .

v

that

j*>v,

jj>v.

from the power series for

Abhqndlungen Akad. Berlin, 1929, pp.

1

—

70.

J¥ (x) and JJ (x) that these functions This abstruse and important memoir contains

numerous applications of SiegePs fundamental theorem. t This was noticed by Porter, American Journal of Math.

zx. (1893), p. 203.

51—53. I Gf. Hobson, Squaring the Circle (Cambridge, 1913), pp. 44, § Some related results are due to Watson, Proc. London Math. Soc. (2) xvi. (1917), pp. 171.

165—


486

are positive for sufficiently small positive values of

x

[CHAP.

XV

and, from the differential

;

equation

<

evident that, so long as x

it is

increasing,

and so

J

v

v

(x) increases

J

and

(x)

v

with

positive,

is

J,,

'

(x) is positive

and

x.

< x < v, both Jv (x) and xJJ and jj cannot* be less than v.

Therefore, so long as

functions so that

xJv

(x) are positive increasing

Again, from the differential equation

vJv"{v) = and

J "(x)

so

-J '(v)<0, v

has become negative before x has increased to the value v from

v

Hence, when! v>\,

zero.

j,"
(2)

Next, since

JT

.,

,

„ +2

(x)

=

2(i/ v

+ l)L >

x

|1

-

v(v

+ 2)\

-^—

J

,.

L ~

-

T Jv (*)

2(,/

+1)0/ +2)1

^

|1

}

-A

(«),

the expression on the right is positive so long as x < v + 2. Now, if ^V were less than V{" (" + 2 )} the expression on the right would be negative when x is equal to jj (which, from a graph, is obviously less than jy ), and this is not the case. (

Therefore

jJ>
(3)

Now, from

+

2)).

§ 15*22 it follows that

Jv< jv+\ < jv+2> and, as has just been stated,

JJ so that «/„(>)

and

Jv+*{j*)

,

are both positive.

If

now we put

x=j

v

'

in the

formula

it is

obvious that

jv'
(4)

Similarly, («/

by putting x =jv

+ 3) Jv (x) + 2 {v +

we deduce

2) |l

-

+

l)}.

in the formula

2 {V

+

^

+ 3) |

+ (» + 1) J* («) = 0,

that

jr

and therefore

*/{v(v+2)}
(5) * Cf.

t

Riemann,

Partielle Differentialgleichungen (Brunswick, 1876), p. 269.

When 0
is

negative for sufficiently small values of x.

f


15-3]

In

487

manner, we can deduce from the formulae

like

^> _(l -!*£*>} and

J'„+1 (x)

=-

j. (*)

+ •/»

- ^=i j Jv (x) - (v +

v jl

1)

J/ (*)

that

J\v( v

(6)

Some

-l)}
rather better inequalities than these are obtainable by taking more

complicated formulae

thus, from the equation

;

Jv+% (x) = Jv+1 (x) R

5i

- Jv (x) R

(x)

„ +1

4 „ +3

(x),

Schafheitlin* deduced that

R 3jv*

i.e.

s,

- 16 (v + 2) {v + 4) j„a +

v+ i(j„)>0,

16 (v

+

+

1) (v

2) (y

+ 4) (v +

5)

> 0.

Since j„ is certainly less than f (v + 2) (v + 4), by results already proved, j v must be less than the smaller positive root of the equation 2

3a; 1

-

16 (v

+

2) (p

+ 4)^ + 16 (v +

1) (p

+ 2)(i/ + 4) (p + 5) = 0,

and hence, a fortiori,

j,
(7)

+

!)("

5)}.

Similarly, from the equation 4J'h-4 («)

- «/r (*) {R*. v (#) + R

- ^s, 0) - ^1, M-2 (*)}

,+a (#)

3,

,

- 2,/; (a?) {^ r+1 (a?) - i2

4 , „ +1

(x)}

,

Schafheitlin deduced that

j;>V^(^ + 2)),

(8)

and,

when

i>

>

4,

j/WK' +

(9)

S)},

these inequalities being derived from the consideration that jv the positive roots of the equation x*

The

- 3 (p +

discussion of

2 2) x*

+ 2i>

y v requires

(i>

+

1)

slightly

(i/

+

3) (v

'

lies

between

+ 4) = 0.

We

more abstruse reasoning.

use

the result that

J*(x)+Y*(x) is

a decreasing function of

that

x

;

this is obvious

Y

from

§

13 73.

Hence

it

follows

and so y v exceeds j,'; again, follows from §363(1) that Yv (j ) is

'\ v*(x) decreases through the interval (0,j v

in this interval positive, since

Y

y

(x) is negative,

JJ (jy )

is

and

it

ll

obviously negative.

Hence (10)

jJ
v

This inequality (withj/ replaced by v

+ \) was established by Schafheitlin

with the aid of rather elaborate analysis. * Berliner Sitzungsberichte, in. (1904), p. 83.

t Journal filr Math. cxxn. (1900), pp. 317—321.


488

[CHAP.

XV

Stationary valves of cylinder functions*.

15*31.

It has already

been seen that the cylinder function

J

v

(x) cos

a-Y, (x) sin a,

and so there are an infinite or 9%, (x), has an infinite number stationary. Such values of x number of positive values of x for which it is /i which exceed the order v (supposed positive) will be called lf 2 fi3 ., where of positive zeros,

//.

/x 1

We

shall

The

first

now study some

<

,

,

. .

....

of the simpler properties of the sequence

theorem which we shall establish

is

that

I^WI >|^rWI>l^r(A*,)l >.... To prove

A (x) defined as

observe that the function

this,

has the negative derivate

-2x*<@; 2 (x)/(x*-v*Y,

and

A (fa) > A (/t

so

Since

A (/*„) = <^„

A

2

(/*„),

more interesting

> A (fa) >

2)

the truth of the theorem result

is

....

is

now

evident.

suggested by Hankel's asymptotic formula

(§7-21)

^r(*)«(^ycoB(* + a-W-iir)+o(^). This indicates the possibility of proving inequalities consistent with i^,(/*«)|

when

shewn that

It can in fact be

The values assumed by (aj2 — 9 )* c<@v (x) when x takes the values fa, ...form an increasing sequence whose members are less than *J(2/tt).

(I) fa,

= 0(l/vV«)

/in is large.

1/

\

|

(II) A*r+2» •••

The values assumed by x^\9^,(x)\ when x takes the values fa, fa+i, decreasing sequence whose members are greater than \f(2jir)

form a

provided that (i)

v>± V3,

(ii) fa'

> * {4i>2 + 4 + ^(48^ +

13)}/(4i*

- 3).

Consider the function

A (x) <@S (x) + 2B (x) <@, (x) <&,' (x) + G (x) <$;* (x) = @ (*), where

A (x), B(x),

©' (x)

=

G(x) are to be suitably chosen.

[A' (x)

-2(x*-v*)B (a-)/* ^V 8

8

}

We

have

(*)

+ 2 {R (x) + A(x)-B (x)/x - (a3 - v*) G («)/*»} + {C (x) + 2B (x) - 2G(x)/x] <&„'* (x)

#„ (x) #„' (x)

= D(xY$J*(x), where

fa,

D (x) = G' (x) + 2B(x)-2C (x)/x, * Cf. Proc.

London Math.

80c. (2) xvi. (1917), pp.

170—171.


15*31, 15*32]

provided that

A (x)

is

489

B (x)

chosen arbitrarily and that

and C(x) are then


2B (x) = a? A' («?)/(** - v%

=

C(x)

«•»

+A(x)-B (as)/*} /(a? -

[B' (x)

UA(x) = (x -v )\ then ID {x) (a? — v )* = i

(I)

2

and so

@ (a?)

is

a?

3

(3a*

1, 2, ...,

+ 14a? a + 4*»4) > 0, z/

an increasing function of x which

<$ (jin ) =* (A* n s

— v9)* *^

2

we

(/*«)

then the numbers (fi^

— v )* 2

\

therefore less than

is

= 2/7r.

lim @(a?) *-»*

Since

v*).

l

see that 9%v (/*„)

|

when n assumes the

values

form an increasing sequence

than V(2/?r).

less

(II)

If

A (x) = x,

then

2D (a?) (a? - 1^) = - x* {(4i> - 3) x* - 8v* (v* +l)x* + v* (4i/ - 1)} 2

2

4

<0, provided that 4n?

> 3 and x

exceeds the greatest root of the equation

(4v -3)ar*-8i/2 a

(^ + l)a* +

i/*(4»/

8

- 1) = 0.

In this case ® (a?) is a decreasing function and we can apply arguments, similar to those used in theorem (I), to deduce the truth of theorem (II). Schafheitliris investigation of the zeros

15*32.

o/J

(x).

By means of the integrals which have been given in § 6*12, it has been shewn by Schafheitlin* that the only positive zeros of J (x) lie in the intervals (m7T + f tr, mir +îr) and the only positive zeros of Y (x) lie in the intervals (mir + \ir, mir + §7r), where m = 0, 1, 2, ....

We

shall first give Schafheitlin's investigation for

modifications,

and then we

shall

J

(x),

with slight

prove similar results for cylinder functions

of the type

J

v

(x) cos

ol—Yv (x) sin a

—\

and $), by the methods used by Schafheitlin. and %ir of a. Schafheitlin's investigations were confined to the values (where v

lies

From an

between

inspection of the formula of §

Jo( it is

obvious that,

6

and

'

'"ttJo sinflVcostf

when mir
so

612 (7),

{sin

(a»

+ \ir,

+ \G)) = sgn (- l)m

sgn Jo («)

= sgn (- l)m

,

.

Consequently Jo (x) has no zeros in the intervals (mir, mir * Journal fUr Math. cxiv. (1894), pp.

31—44.

+ £ it).


490

To prove

that

J (x)

has no zeros in the intervals {mir

=

/.(.)

last

integrand

is

+ \ir, mir + ir), write

^±^ P"**'*'y ' *> r-~'M. ir

The

XV

— (m 4- 1) ir — 0,

x

and then

[CHAP.

J o

sin

cos

negative or positive according as

<

< 20

or

20 <

Since <îr, the second of these intervals

<

\tt.

the longer; and the function

is

»—23 cote

ycos

sin is

when x >îr and

an increasing function* of

Hence

is

an acute angle.

between and 20 there corresponds a value which sin {\0 — 0) has the same numerical value, but has the positive sign, and the cofactor of sin {\0 — ) is greater for the second set of values of than for the first set. The integral under consideration is consequently positive, and so J (x) cannot have a zero in any of the intervals (mir + fair, mir + ir). Therefore the only positive zeros of Jo (x) are in the intervals (mir + £ ir, mir + %ir). to each value of

between 20 and \ir

15*33.

We

for

Theorems of Sckafheitliris

shall

now extend

ir

and

— § < v ** \.

Schafheitlin's results to functions of the type

=

^P„ (x)

where

when

type,

Jv (x) cos a— Y

v

(x) sin a,

— \ < v ^ \.

We shall first prove

the crude result that the only positive zeros of

^f„ (x)

in the intervals

lie

(mir

where m — 0, 1, 2, which shew that

—

+ f 7r + \vir — a,

mir

+ ir — a)

This result follows at once from the formulae of § 612,

*A) ~~r{v

sin-^ + i)r{i)i — mir a < x < mir + £ ir 4- \vir — a, sgn [sin (x + a — v0 + \0)~] = sgn (— l)w

when

for,

we have and

r

$

e

aff>

,

Consequently the only zeros of 9£y (x) lie in the specified intervals, and there are an odd number of zeros in each interval, with the possible exception of the first if a > f ir+ \vir. so, for

such values of

x,

Next we obtain the more lie

v

(x) is

not zero.

precise result that the only positive zeros of <$„ (x)

in the intervals

(mir+ f 7r + \vir—

a,

mir

+ \ir + \vir —

* Its logarithmic derivate is

(2x

- sin

cos 0) cosec 2

+ J tan 0.

«)


15*33, 15*34]

where

m = 0,

1, 2, ...,

except that,

if

a

is sufficiently

491

near to

ir,

there

may be

zeros* in the interval (£""

We

+ ivw — a,ir — a).

prove this result by proving that

shall

<$, (x)

has a fixed sign throughout

each of the intervals^ (mir + |7r + \vir—a, mir +

is

With

and

an angle between

£tt.

this value of x,

(-)«*2"*«» f* cos-*flsin

To each and

— a).

x = (m + l)ir-a-(l-2v)4>,

Write

where

ir

\ir for

and

between

value of

- 2„) (\0- $)}

^^.^

there corresponds a value between 2<£

2

sin{(l — 2v)(\0-4>)}

which

{(1

has the same numerical value, but

has the positive sign. e-2* cot *cos '-*0/sina, +1

Again is

provided that


> max

2a?

and

(2i/

this condition is satisfied

Hence,

if

this proves the

15*34

+ (v - 1) ^-^

since v

,

^ J.

+ \vir — a
= sgn (- 1 ) m+1

We next consider the

type,

when

\
function

*$w (x) ir,

,

more precise theorem.

Theorems of Schafheitlin's

â<

cos

when x > %

sgn <$v (x)

we have

where

+ 1) sin

x > § and mir + \ir

and

'

,

= J¥ (x) cos a—Yv (x) sin a,

as before, in which

it is

now supposed

that

\
W e shall first prove the crude result that the only positive 7

lie in

(mir —

where % m = 0, 1, 2, which shew that

. . . .

o,

mir — \ir

when *

9!> v

(x)

+ \vir — a)

This result follows at once from the formulae of § 612,

*+'*' (r\ ^(*)r

for

zeros of

the intervals

(,,

co8"-i0sin(x+a-v0+hd) ^ sin-»0 + *)ra)Jo

mir — \ir + \vir —

By taking as an alternative function J\ v

there cannot be

more than one such

\

e

tfl

^.

a< x < (m + \) ir — a, (x)

and applying the theorem of

zero.

t If x(if- \)t, the interval for which m=0 is, of course, to be omitted.

§ 15-24,

we

see that


492

we have

sgn

[sin (x

whence the theorem stated

lie

+ a — v0 + $ 0)] = sgn (— l) m

,

precise result that the only positive zeros of 9BV (x)

in the intervals

\vrr-a., mir — \tt

— £tt +

(mir

m = 0,

where 1, 2, ..., except a zero in the interval (0, %vir — îr (tt

XV

obvious.

is

Next we obtain the more

[CHAP.

— a,

We

that, if

^7r 4- \vir

a

— a),

that,

is-

near to

and there may

use the same notation and reasoning as in 2"+i

§

there

may be

be one in the interval

15 33

< ty < 20,

;

only now,

if

6 =/($),

not necessarily an increasing function of 0; but

when

ir,

— a). e -2*cote C08 ^-j 0/ s i n

f(0)

+ %vir — a),

is sufficiently

it is sufficient to

prove

then

f(2-ir)+ylr).

To

obtain this result, observe that

4l

log

T^r^rl - **

{

cose°2 ( 2 <*> + *) + cose°2 ( 2 * ~ *)}

v-\ r ^ Lcos (20 + +) cos (20 - +) [cosec 2 (20

But is

an increasing function of [cosec2 (20

is

when ^=0, and

is

I J )

and

ijr,

therefore,

a fortiori,

+ ip) + cosec2 (20 - ^)] cos (20 + ^) cos (20 - ^)

an acute angle

i.e.

+i

+ +) sin (20 - ^

+ f) + cosec2 (20 - VO] sin (20 + +) sin (20 - $)

an increasing function, since

because 40

2i>

sin (20

;

tbis function exceeds tbe former

and so

%_T(

j-r log' >^ 9

is

by an increasing function

alwayt positive

if it is

positive

if

this is the case

when

Hence, when \ <

v

4* > {(v - i) tan2 20 + (2v + 1)} sin 40, x> £v+ \.

,

the only zeros of 9By (a?), which exceed f v

+ &, lie

in

the intervals (»i7r

The method seems

— \tr + \inr — a,

mar

— \tr + \inr — a).

inapplicable for larger values of v on account of the

oscillatory character of sin (a;

+ a — v0 + \0)

method which

these larger values will

15'35.

is effective for

as

increases from

now be

to \ir\ a

explained.

Schafheitlin's investigations of the zeros of cylinder functions of

unrestrictedly large order.

We shall now

prove that,

if v

> J,

those zeros

of the cylinder function

Jy (x) cos a — Yv (x) sin a which exceed

where

(2i/

+ S)/ir lie in the intervals (mir — a + %vir + \tt, mir — a + \vtt + f tt)

+1)

(2v

m assumes integer values.


15-35]

The method used

to obtain this result

considered the case of functions of the his reasoning is

first

493

due to Schafheitlin*; but he

is

kind and of integral order only, and

made lengthy and obscure by the use of arguments equivalent mean-value theorem when the explicit use of that

to the use of the second

theorem

As

is

obviously desirable.

in the preceding analysis, write

= Jv (x) cos a —

9BV (x)

F„

(x) sin a,

so that '

(, "r(.+j)r(i)l (

r (v

Now

cot2v+1

.

f )T

It will

with

cot

33

(|) J o

e- 2*^49 increases as

increases from

as

2

to

e

sin^tf

£77-,

be observed that

2 is

cos*+i

1

increases from

where

= arc tan-

2

dtf

to

2

dtf -

e

)

and then decreases

x .

nearly equal to \tr

.

when x

is

large

compared

v.

Now

suppose that # rrnr

and then choose

between

lies

— a + \ir (v — |) and

mir — a + \ir

(v

— |) 4- f 7r,

^ so that X + a—

{V

+

§)

0!

=

W7T.

It is easy to verify that

2y-l

7T

2i/+3'2

.

< ^1<

2i/

so that 0! is a positive angle less than v

arc tan

We

2y

2,

+h x

+ 2 ir + 3'2'

provided that

v +6

<

4i>

suppose now that

x>(2v + l)(2v + 3)/ir, so that

X

is

certainly less than

2.

Then, by the second mean- value theorem, there exists a number and X) such that {

+%)

!o

sin2

"^

e

J»,d0\ co

* Journal

^+i

fUr Math. cxxn.

between

atf

cos'+itf

*-"«*•,! (cosJww + ^) foot** 1 ox .b -(cot .j| ^

O,

-

cos(x

(1900), pp.

J

+ a-v0o -l0 cogi>+}

299—321.

^

o)

)

|.


494

Now

qua function of

[CHAP.

XV

0,

+ a - v0 - £0)/cos K+ * when sin (x + a — v0 + \&) = 0, and for cos {x

such values of

the

cannot exceed numerically the greatest value of l/cos*"""**? in the interval

(0, 0^),

is

stationary

fraction is equal to

±

-

1

/cos* * 0.

co S (x

Hence nenCe

-W

+ a-v0

o

cob*+»0,

and therefore (

SDT1 Sgn

)

cos ( m7r

+ft)

_ cos (g + «-

cos-**

Therefore, since the sign of sin (x lies

between

X

and

we

^7r,

"fl>

_ gM/_ ! vm }- sS£m 1>

-*fl>) |

cos"+*0o

+ a—

+ \0)

v0

the sign of (— l)m

is

see that, for the values of

sgn 9B9 (x) Hence, when x exceeds (2v+ 1) (2v+

l) m

= sgn (-

S)/ir, 9$v {x)

•

a;

when

under consideration,

.

has no zeros in intervals of

the type (rrnr

— a + £ vir — \ir,

—a+

mir

and so the only zeros of <$* (x) which exceed

\ vir + \ir), + (2j/ 1) (2i/ + 3)/tt lie in intervals

of the type

(mir

and

— a + \vir + \tr,

— a + \vtt + J ir),

mir

this reduces to Schafheitlin's result*

when a =

and v

an

is

integer.

The reader will observe that this theorem gives no information concerning ^v {x) when v is large it will be apparent in § 15*8 that there are a large number of zeros less than (2i» + l)(2i/ + S)/tt, and that interesting information can be obtained concerning them by using Debye's the smaller zeros of

;

integrals.

15*36.

Bdcher's theorem

A result of

f

on the zeros

of^

(x).

a slightly different character from those just established was

discovered by Bocher from a consideration of the integral formula § 11*41 (16). The theorem in question is that 9^ (x) has an infinite number of positive zeros,

and the distance between consecutive zeros does not exceed smallest positive zero of Jo

To

2;

where j

is

the

(x).

establish this result, write v

= 0, z=j

f ô (») d - tt^

(Z)

in § 11-41(16),

J

(>)

and then

= 0.

.'o

Hence %\{^t) cannot be one-signed as increases from Z — to Z+j and so ^(nr) must vanish

to-7r, i.e. as •w

increases from

value J of

number

m

;

in the interval

(greater than j

),

,

).

Bocher's theorem

* Schaf heitlin gives (2v + 3) (2r lie

(Z—j Z+j

+ 5)jr

Since is

now

t Bulletin American Math. Soc. Cf.

Modern Analysis,

§ 3-63.

is

one an arbitrary positive

evident.

as the lower limit of the values of

in the specified intervals.

X

Z

v. (1899),

pp. 385—388.

for at least

x

for

which the zeros


15-36, 15*4]

By

[Note.

a form of Green's theorem,

u

I

J

where

— as = cv

two solutions of ^—?

u, v are

oar

coefficients inside the closed

By

taking

must vanish

v=J

curve

«,

I

,

CV ,

and

with continuous second differential

d/dv indicates differentiation along the normal.

{J(a?+y )} and the curve to be x2 +tf2 =*jo2 Weber* deduced that u on any circle of radius^. ,

at least twice

subtending an angle

Hence the

apart less than 2/ -ll

_

v

2

Bocher inferred from this result that since if a circle of radius j

exceed j

du \

J

+ ^-s+«=0 cy2

except at the origin,

circle.

495

less

than

ir/n at

positive zeros of ,

^n

(r) cos n6 satisfies the requisite conditions drawn with centre on the axis of x and

is

wn {r)

$n (r)

must vanish somewhere on the

are such that consecutive zeros are at a distance

and the distance from the

+ cosec 5-

r

the origin,

origin of the smallest of

them does not

J-

These results are of interest on account of the extreme simplicity of the methods used to prove them.]

On

15*4.

We shall

the

number of zeros of Jv (z) in an assigned

next give the expression for

Jv (z)

strip of the z-plane.

as a Weierstrassian product,

and then develop expressions involving quotients of Bessel functions in the form of partial fractions but as a preliminary it is convenient to prove the following theorem, which gives some indication as to the situation of those zeros of Jv (z) which are of large modulus. In this investigation it is not supposed that v is restricted to be a real number, though it is convenient to suppose ;

that v

is


When

v

is

real the results of § 15*2 to

extent take the place of the theorem which will

Let

G be

now be

some

proved.

the rectangular contour whose vertices are

± iB

+

\iril (y), ± iB

+ mir + %vir + \ir,

B is a (large) positive number. We shall shew that when in is a sufficiently large integer the number of zeros of z~" J {z) inside G is precisely equal f to m. Since z~* J (z) is an integral function of z, the number of its zeros inside where

v

v

Cis

1_[ d\og{w-"Jr ('w)} dw _

dw

2TriJ c * Math. Ann.

1.

1 [ liri) c

Jr±lM dw

.

J»(w)

(1869), p. 10.

a real negative number (and for certain complex values of v) there may be pairs of zeros on the imaginary axis in such circumstances the contour C has to be indented, and each pair of zeros is to be reckoned as a single zero.

t

When

v is

;


496

We now on

C

consider the four sides of

[CHAP.

It is first to

in turn.

XV

be observed that

the sides of C,

all

= (—) *«-+- **> {1 +fhA W )h

#„(« (w)

H® (W) = l-Â «r«*-*"-W where

ij

„

lt

Now,

(w) and

r)

2< „

(l/w)

(w) are

since the integrand is

27rtJ tjB+ |jrt/ ( „)

when

|

{1

w

||

+1?2 ,„(W)], is large.

B -* oo

an odd function*, we have, as

,

ZTriJiB-hniiM J*{w)

%/,(«>)

fiB+lmHo) /""

\ 1

idw=\il{v). iB-lniI(v)

Next take the

•

.

-

C

upper horizontal side of

integral along the

this is equal to

;

tB+»l»+Jnr+Jir

1

fiB+mTT+Ji/ir+iir

1

r

Q

i

_

/^Y)

2v

+ l.

s [«» + *»*(>) + ^+-5.7>v

i

\

.

i

.

log

iB + mir + ^vTr +

^ + ^-

~|

n/ 1/m +Q(i/J)J

lir

/(l/)

-iw + iflfcO + i,

as

B — oo.

Similarly the integral along the lower side tends to the so the limit of the integral along the three sides

Lastly this

we

we have

first

same

now considered

is

value,

and

m + \v + £.

to consider the integral along the fourth side,

and to do

investigate the difference

%^>- tan («/-**7r -**), which,

when w |

|

is

large, is equal to

riB+mv+iw+ir

Now

tan(w— \vir — ^7r)dw = 0,

I

J

— tB+«Hr+J>w+J»

and so 1_ riB+mw+ivw+i* 2iri J

-u iB+mv+iv*+\w

jr+i ( W ) T ^kW

dw

~-i(2i/ + l)+0(l/m). Hence the

limit of the integral

* Allowance is

round the whole rectangle

made for the indentations, just specified

,

is

m+

(l/m).

in the first step of the following analysis.


15-41] If

we take

(1/w)

is

is

must be an

m

sufficiently large,

numerically integer, it is

497

ensure that the expression which

we can

than 1 and since the integral round the rectangle equal to m.

less

;

That is to say, the number of zeros of z~*Jv {z) between the imaginary axis and the line on which

R (z) = W7T + {$ R (v) + 1} ir is

exactly

ra.

Note. The approximate formulae quoted for the functions of the third kind shew that the large zeros cannot have a large imaginary part and so all the zeros of «/„ («) lie inside a strip whose sides are parallel to the real axis and at distances from it which are ;

bounded when

v

\

is

bounded.

The expression of Jv (z) as an

15*41. It

|

possible

is

to

express

Jv {z)

infinite product.

as a product

of 'simple

of

factors'

Weierstrassian type, each factor vanishing at one of the zeros of Jv (z). In order to express J,(z) in this form, it is convenient first to express the logarithmic derivate of z~ v J¥ {z) as a series of rational fractions by MittagLeffler's

theorem*.

The zeros of z~y Jv (z) R(h,n) >0 and R(j r,i) < |

|

jy<3

,

being

...

all

are |

taken to be ±jv
^(>

>2)

unequal (§15*21).

I

,

±jv,s>

•••

wheref

^ & O'm) £ — the values of j We draw a (large) rectangle D, •

!

I

Wtl

,

jv<2

,

whose

where A and B are positive, and we suppose that ±j¥>m are the zeros of highest rank which are inside the rectangle. vertices are

We now

±

A

±

iB,

consider z 2irtJ*J)W(W 2

«/,+,

(w)

— Z)' J¥ (w)

dw,

where z is any point inside the rectangle, other than a zero of J¥ (w), and v not a negative integer.

The only poles of the integrand

inside the rectangle are

z,

±>,i, ±>.a»

±.Jv,m'

The

residue at z

is

J¥+1 (z)/Jv (z) [2

since

and the residues at ± j„„ are

+ Jv,n

JJ (z) = — J¥+1 (z) when z •« ± >,n,

Jy,n)

by

§ 3*2.

It follows that

=

—

2-rri

* Acta Soc. Seient. Fennicae, xi. (1880), pp.

t

If

R

(

±jy>n )=0

for

any value

of n,

*

]j)W(w — z)

J*+1 (W) dl». J ( w) v

273—293. Cf Modern Analysis, § 74. to have its imaginary part positive.

we choose jvn

.

is

•••

>


498

We

next shew that, by giving

increase without limit,

Since this function

A

XV

and B suitable sequences of values which can be taken to be bounded on D.

Jv+1 (w)/J,(w)

an odd function of w,

is

[CHAP.

it is sufficient

to consider the

right-hand half of D.

We

A = Mir + M (\v \) where M a positive integer and then M to be at least so large that M = m, which possible by § 154, and

take

we take

4-

is

ir,

;

is

also to be so large that

be

§ 15'4, to

Then

we can take the

functions

iy r

,i(w),

*7„ f2

(w), defined in

J in absolute value.

less than, say,

Jy+1 {w)jJv (iv) is

bounded whenever g»»

(W—\mr~J»)

L

and when the expression does not lie within bounded and w is not arbitrarily near a zero of J, (w) so that, from the asymptotic expansion of § 7*21, J, +1 (w)/Jr (w) is bounded on

is* less than \ or greater . han 2

these limits,

I (w)

;

is

;

the part of the rectangle witl;in this strip.

That

is

to say

the rectangle

Jv+l (w)jJ

v

D as B and

(w)

is

bounded on the whole of the perimeter of

M tend to

infinity.

Hence

«_ JjtiW,,,,^

if

2itijdw(w — z) J p {w) and therefore f

1 K

Jy(z)

'

When we

l*->.n

n=l

integrate,

we

jv,n)

«=1 \*

find that

^t-i:»*}-fi{( i -ja-»(rJiM i+ a-'(-aand hence

This

is

the expression of J, (z) in the specified form.

The formula may

also

be written in the modified form

This formula was assumed by Euler, Acta Acad. Petrop. v. pars 1, when v=0, aud subsequently by various writers for other values of v

(1781) [1784], p. 170, ;

cf.

§§

15*5, 15'51.

due in substance to Graf and Gubler, Eirdeitung in die 130, and it was given T/ueorie der BetseVschen Funktionen, I. (Bern, 1898), pp. 123 explicitly by Kapteyn, Monatshefte fUr Math, und Phyt. xiv. (1903), pp. 281—282.

The

*

analysis of this section

Because

t If we

is

—

TTa

**

5•

take the rectangle to have its vertices at

on the right converge separately.

A±iB, -A'±iB, we

see that the two series

—

,

:

'


15*42]

The expansion on the

499 power

right of (1) is evidently expansible in a

series;

the

such a series have tieen expressed as determinants by Kapteyn, Proc. Section of Sci., K. Acad, van Wet. te Amsterdam, vm. (1905), pp. 547 549, 640 642 Archives Neerlandaises, (2) xi. (1906), pp. 149—168. Some associated formulae have just been published by Forsyth, Messenger, L. (1921), pp. 129 149.

coefficients in

—

—

;

—

The Kneser-Sommerfeld expansion.

15*42.

An expansion which, in some respects, resembles the partial fraction formula obtained in § 15*41 is as follows

x and

in which

X are positive numbers such that ^x^X^

while z

and v are

to take

R (z) > 0.

1

unrestricted (complex) numbers, except that

it is

convenient

The expansion was discovered in the case i>=0, as a special form of an expansion occurring in the theory of integral equations, by Kneser, Math. Ann. lxiii. (1907), pp. 511 517. Proofs of this and of related expansions for integral values of v were published later

by Somnierfeld, Jahresbericht der Deutschen Math. Vereinigung, xxi. (1913), pp. 309 method of proof has been criticised adversely by Carslaw, Proc.

353, but Sommerfeld'8

London Math.

Soc. (2)

xm.

(1914), p. 239.

be noticed that the expansion has some connexion with the expansions which will be discussed in Chapter xvni.

may

It

To obtain a proof of the f ffr « (Xw) 1 2irij

in which the path of integration

The

#,<'>

(w)

zt-w*

supposed that the

Fourier-Bessel

expansion, consider the integral

H® Q) - H® (Xw)

it is

•

left side

is

Jv (xw) J,(w)

dw W

'

a rectangle with vertices ± Bi, A ± Bi, and is indented at the origin.

of the rectangle

integral round the indentation tends to zero with the radius of the

indentation, whether v be an integer or not

;

and the integrals along the two

parts of the imaginary axis cancel. Also,

when x and \R.® (Xw)

X

satisfy the specified inequalities, the function

H® (w) - HJ» (Xw) H® (w)} J (xw)/J (w) y

v

remains bounded on the other three sides of the rectangle when

when

A

-*-oo

Hence the limit of the

limit of the integral

sum

and

round the rectangle

is zero,

and so the

of the residues of the integrand at the poles on the right of

the imaginary axis

Now

B -*- oo

through the values specified in §1541.

is zero.

the residue at z

is

FTE7 {J

"

{Xz) Yy {z) ~ Jv

{z)

F" (Xz)]

'


500

while the residue atjVin

is

- 2* J, (j„nX) Yv OV, n ) Jv (jv n x)l{Jv _ -2lJv (j n X) Jv (j w x)

'

( j„,

,

4t Jv (j*,

,

v>

Vi

°v \Jv,n/\ z

=

XV

[CHAP.

~~

nX)J

v

J

-j\ n )}

M ) (*»

y

\_J

\J'(A

(X

\Y'(i

V

v,n)

(j v< n x)

and on summing the residues we at once obtain the stated expansion.

Jv (xw)/Jv (w) by the contour integral, see Carslaw, Proc. London Math. Soc. (2) xvi. (1917), pp. 84 93; Carslaw has also constructed some similar series which contain Legendre functions as well as Bessel functions, and these series represent the Green's functions appropriate to certain physical problems. See also Beltrami, Lombardo Rendiconti, For a generalisation of this expansion, obtained by replacing

96 V {xw)l<$v {w) in

—

(2) xin. (1880), p.

15*5.

An

336

;

and Lorenz, Oeuvres

Scientifiques, n. (1899), p. 606.

Euler's investigation of the zeros of

J

(2

*Jz).

ingenious method of calculating the smallest zeros of a function

was

devised by Euler*, and applied by him to determine the three smallest zeros of

J

(2Jz).

If the zeros arranged in ascending orderf of

then by

magnitude be

aôt,,

a„

...,

§ 15*41,

j (2v*)= n(i-£). w

As has already been is

stated (§ 15'41), this formula was assumed

by Euler;

if it

differentiated logarithmically, then

-£log/ dz

(2\/*)=i

*

oo

oo

w=1 an

- 2 n=l

provided that

|

z

\

t ;

and the

last series is

—z

X

mÔ^n then absolutely convergent.

00

2

Put

l/a»m+1

= orOT+1 and change

-£-J Replace

J

(2>s/z)

(2
the order of the summations

=J

(2Jz)

2 a^z™.

2s

"~

1*

.

2s 3 s

+

'

—

§ 15*25 it follows that the zeros are positive

"

.

* Acta Acad. Petrop. v. pars 1, (1781) [1784], pp. 170 et teq. Math, xxxiii. (1846), pp. 363 365 should also be consulted.

From

then

on each side by

p+p

t

;

A

paper by Stern, Journal /fir

and unequal.


15-5]

501

multiply out the product on the right, and equate coefficients of the various

powers of z in the identity; we thus obtain the system* of equations

=
2

lt

x

,

-1^

•

<*4

2

1

,

<>

whence o-i

<

Since

0^

= 1,
<

02

<

a3

<

. . .

8

,

it is

<*!

<
extrapolating from the following Table

m

^ml^m+l

1

1000 000

2-000 000

2

1-414 213 1-442 250 1-445 314

1

3

4 5 6

Euler inferred that ax

By

= ^$r,

,

By

o-6

,

evident that

m <
and so

y

1 1

-500

000

1-454 545 1

-447 368

1-446 089

-445 724 -445 785

— 1*445795, whence l/a, = 0-691661, 2Vfli = 2-404824.

adopting this value

writing

for e^,

£ l/On^d-™', »-2

and then using the inequalities l/a2m

< am

,

m+1

<

m /a.z

,

Euler deduced that a 2 = 7*6658, and hence that o3 =1863, by carrying the process a stage further. These results should be compared with the values ax

= 1-445796,

a2

= 76178,

a3 = 18-72,

derived from the Tables of Willson and Peirce, Bulletin American Jiath.,Soc. in. (1898), pp. 153—155.

The value of a, is given by Poissont as 1 446796491 (misprinted as 1 46796491); according to Freeman J this result was calculated by Largeteau for Poisson by solving the quartic obtained by equating to zero the first five terms of the series for J (2 Jz) the magnitude of the sixth term is quite sufficient to account for the error. ;

* This

t

M4m.

system

is

an obvious extension of Newton's system

for

an algebraic equation,

de I'Acad. R. des Sci. xn. (1833), p. 330.

—

t Proe. Camb. Phil. Soc. ra. (1880), pp. 375 377. Thiorie Analytique de la Chaleur, p. 310, footnote.

Of.

Freeman's translation of Fourier's

La


502 15*51.

[CHAP.

Rayleigh' s extension of Eider's formula.

The method just described was used independently by Rayleigh * late the smallest positive zero of

Taking the formula

(§

v (z).

x

i

2 -^ »=i J"

a-

1 .

2a

,

(i'

v,n

Rayleigh, that

find, after

'"=

to calcu-

J

15*41)

and writing

we

XV

a<

2>

1

=

+ l)*(i' + 2)' 5v +11 w = <7„ 2 8 (k+ 1)4 (i/ + 2) (i- +

+

l)'

24

"

2

28

'

(i*

3)(v

(l/

+ l)8 (I/ + 2)(»;^-3)

,

+ 4)'

7i/+19 29

The

(i/

+ iy(v +

smallest positive zeros of

J

2)<(v

and

(z)

+

+

S)(v

«/i

(z) are

4)(j/

+

5)'

deduced to be 2*404826

and 3831706. Immediately afterwards Cayley f noticed that o-„( r can be calculated rapidly when r is a power of 2 by a process which he attributed to Encke J, but which is more usually known as Graeffe's§ method of solving an equation. >

The method consists in calculating
Cayley thus found

ov< 8 > to

be

429» 5 + 7640i> 4 + 53752v 3 + 185430» 2 + 31 1387» + 202738 2»« (v

It

+ l)8

(i-

+ 2)« (i>+3) 2 (v + 4)2

(i/

+ 5) (v+6) (v + 7) (v + S)'

was observed by Graf and Gubler|| that the value of ov

easily be checked

by the

formula o-

where

Br is

the rth Bernoullian

W=22*-

J

number

;

1

i?r/(2r)!,

this formula is

an evident consequence of the

equation

J *w-(s)* 8ini

-

Extensions of some of these results to the zeros of zJv (z)+hJv (z), where h have been made by Lamb, Proc. London Math. Soc. xv. (1884), p. 273. '

is

a constant,

and 1, has recently been The smallest zero of Jv {z\ for various values of v between tabulated by Airey, Phil. Mag. (6) xli. (1921), pp. 200—205, with the aid of the RayleighCayley formulae. * Proc.

London Math. Soc. v. (1874), pp. 119—124. [Scientific Papers, i. (1899), pp. 190—195.] t Proc. London Math. Soc. v. (1874), pp. 123—124. [Collected Papers, ix. (1896), pp. 19—20.] t Journal fur Math. xxn. (1841), pp. 193—248. § Die Aufibsung der hoheren numerischen Oleichungen (Zurich, 1837). Einleitung in die Theorie der Bessel'schen Funktionen, i. (Bern, 1898), pp. 130 131. I!

—


15-51, 15-52]

The procedure

[Note.

of calculating the

sum

503

of the rth powers of the roots of an

equation in order to obtain the numerical value of

its largest root seems to be due to Waring, Meditationea Analyticae (Cambridge, 1776), p. 311 other writers who were acquainted with such a method before Graeffe are Euler (cf. § 15*5); Dandelin* Mem. de VAcad. R. des Sci. de BruxeUes, in. (1826), p. 48; Lobatschevsky* Algebra, or Calculus ;

of Finite* (Kazan, 1834),

§ 257.]

The large zeros of J*{x).

15*52.

The most

method

effective

functions (when the order v

of calculating the

not too large)

is

though subsequent writers have,

to

some

is,

large zeros of cylinder

in substance,

due to Stokes f,

extent, improved on his analysis.

Stokes' method will be sufficiently illustrated by his own example J whose zeros are the roots of the equation

with the notation of expansions of

P (x,

0)

zw P( *'

Q

For

1.9

,

nt *' Q(

0) are

(x,

m 0) ~ X ax 0)

-27(8^ +

function cot (x

— \ir)

and so

it is

P (x, 0)

in each of the intervals

(n-n-

Q (x,

is positive,

is

\tt, wit

+ \ir),

iV,

J

if u r , v r

negative

denote the (r

+

that there

(x) has precisely

one zero

and that the distance of the zero

from the left-hand end of the interval tends to zero as n -* Again,

is

— \v)

obvious from a graph of cot (a?

—

0)

a negative increasing § function of x. a decreasing function which vanishes when is

N such that when n >

exists a positive integer

-•

^!(8^ 1.9.25

1

sufficiently large values of x,

x = mr — \ir,

1.9.2.5.49

~-TT^ + 3W~----

and the quotient Q(x, 0)jP(x, 0)

The

(&),

be remembered that the asymptotic

It will

§ 7*3.

and

J

l)th terms of

P (x,

0)

oo

and

.

Q (x,

0)

we may

write

P (x, 0) = where 6 and * I

1

m-l

to-1

2

ur

+ 0um

,

are certain functions of

owe these two

Q (x,

0)

= 2

vr

m which

# and

-I-

lie

l

vm ,

between

and

1.

references to Professor Whittaker.

t Camb. Phil. Trans, ix. (1866), pp. 182—184.

[Math, and Phys. Papers, n. (1883), pp.

350—

353.]

| Stokes also considered Airy's integral

(§ 6-4)

and

J

x

(x), for

the purpose of investigating the

position of the dark bands seen in artificial rainbows. §

The reader may

verify,

by

§ 3-63, that its derivate is

{1-P2-Q2}/P2, where P, positive.

Q

stand for

P (x, 0), Q (x, 0)

;

and, by the asymptotic expansions, this

is

ultimately


504

Now

[CHAP.

XV

consider the equation Wl-l

2

cot(a?-^7r)

=

^2

V,

+

ur

+

x

Vm ,

0um

r=0

in

which

it is

any numbers which

values, are

and 1} instead of having their actual between and 1.

temporarily supposed that

The equation now under

lie

consideration involves no functions more compli-

If a? were supposed complex, there would be a number of contours in the a?-plane each of which enclosed one of the points nir — \ir and on which cot (a? — \ir) exceeded the modulus of the quotient on the right.

cated than trigonometrical functions.

\

|

Biirmann's theorem * the modified equation would have one root inside

By

mr — ±tt, and

the part of the contour which surrounds

panded

We

in descending

is

'

and

independent of

- Sir)™

f

lt

;

so that,

when

f

r

Now

r=0 (nir

and X r (0, 0i) are independent of n but depend on readily perceived that the first in of the coefficients are actually

in which the coefficients it

can be ex-

thus obtain an expansion for the root of the equation in the form *

and

this root

mr — \tt.

powers of

sum

{0

r x

t

<

m, we

may

write

)=f

r.

is a bounded function of and and 1 vary between and 1 and it is clear that the upper bound of the modulus of the function in question is 0(n~2m_1 ) as n~*- oo Hence, when and X are given their actual values which they have at the zero under con-

the

of the terms after the with

as

X

;

.

sideration, the

That to nir

is

sum

of the terms after the with

to say, it has

— ^7r),

and

x

its

been proved that there exists one zero (nearly equal may be written

value

= (mr-%Tr)+ 2

J ,

Xf^nir

— \ir

•¥

w+1

\

Hence the asymptotic expansion of the

zero

where

remains to calculate the

yjr

*

as x

-*•

» ,

,

tan

first

+

is

— îryr+1

few of the coefficients fr

then ,

1

y oo 8a? * Cf.

(n-»»-*)-

^~

2 ,=o (nir

It

(w~ am_1 ).

is still

3417

33 512a?

'

16384a?6

Modern Analysis,

% 7'31.

.

If

.

ZEBOS OF BESSEL FUNCTIONS

15*53]

25

1

505

1073

,

^~8^-384^ + 5l20^"-* ,

sothat

therefore the equation to be solved assumes the form

and

1073

25

1

T «-(nir-i7r)~Q--KQTZ,+ 384a» 8a? The

5120a?

result of reverting the series is

This series

3779

31 1 t^ + 8(w7r _ i7r) - 384 (n7r _ i7r) + i5360(n7r-i7r)» _ -'

-( x-{mr

i

.

is

adequate for calculating

all

the zeros of

places of decimals, except the smallest zero, for which

J

n—

{pe),

to at l eas*

nve

1.

The large zeros of cylinder functions.

15*63.

It is easy to see that the large zeros of

any cylinder

function,

Jv (z) cos a — Yv (z) sin o, where v and a are not necessarily real, may be calculated by Stokes' method from a consideration of the equation cot(s

- \vrr - i^ + «) = p^

v

y

It seems unnecessary to prove the existence of such zeros (with large positive real parts) or the fact that they may be calculated as though the (z, v) and Q (z, v) were convergent, because the proof differs from series for

P

the investigation of the preceding section only in tedious details.

The expression for the large zeros of a cylinder function of any given order was calculated after the manner of Stokes by McMahon*; but the subsequent memoirs of Kalahnef and Marshall J have made the investigation more simple and have carried the approximation a stage further with no greater expenditure of work in the calculation. Following Marshall we define § two functions of equations

M cos

ylr

M sin

= P(z,v),

yfr

z,

called

M and

ty,

by the

= -Q(z,v),

m the understanding that M -* + 1 and «f -*-0 as s -* +

oo

It is then clear that

Jv (z) cos a - Y¥ (z) sin a = ( — *

AnndU

of Math.

ix. (1895),

pp.

23—25

;

if cos (z

— \vtt — \nr + a - ^r).

J

see also Airey, Proe. Phys. Soc. 1911, pp.

219—224,

225—232. t ZeitschriftfUr Math, und Phy*. lit. (19Q7), pp. 55—86. t Annals of Math. (2) xi. (1910), pp. 153—160. § Cf. Nioholson, Phil. Mag. (6) xix. (1910), pp. 228—249. W.

B. F.

17


506

[CHAP.

XV

Again

{j$} -z-hw-l'*-*,

aTC tan

and,

when we

differentiate this equation,

cty

and use

363 (3), we

§

find that

2/(irz)

_

dz ~JS(z) + Y/(z)' so that,

When ljz 9,

by

§ 7*51,

the expression on the right

we

is

expanded

as far as the

term involving

find that

dz

_ fi-1 _

~

3

2 *

-1) (ft -25) _ (/*-!) Q*a -1 14/^ + 1073)

(p

2V

2

(n

2 10 ^

- 1) (5/r>

-s-

1535/* 2

+ 54703/a - 375733)

'"'*

2 15 zs in this equation

/x

has been written in place of 4v 2 for brevity.

It follows,

on

integration, that

_rr ~ /*-!

,

(/i-DQ*-25)

Qi- 1X^-114^ + 1073)

3.2 7 23

5.2 10 * 5

23 s

(/i

+ and

If

/3

Zrvft

-M7T - \VTT

= (n, + |v -

^-1 2

^)

( /i

s

7.2

be solved

so the equation to

*

- 1) (5/x3 - 1535/*2 +

+

7r

^TT

— a,

is

3 2

54703/* *

-

375733)

7

+

'"'

*

+ a ~ - -^

•"

g-^V

the result of reversion

-l)(7 /A -31) 7

15

is

Qt-l)(83/t a - 982/*

+ 3779)

15 2 10/38

3

£ 3 (a* 1) (6949/i - 153855/* + 1585743/t - 6277237) lOS^/S7

ft

.

.

2

Therefore the large zeros of J, (z) cos a

— Yv (z) sin o

by the

are given

asymptotic expansion (n

*

+ \v — \)ir — This equation

1797, but

no clue

lvii. (1902), p. 19.]

a

—

-l 8 {(n + \v - \) ir - a}

(in the case v

is

4i/

= \)

2

(4p»-1)(28i/ 2

384

{(n

-31)

+ %v- \)

ir

-

aj

3

was given by Gauss in his notebook with the date Oct. 16, it. [Cf. Math. Ann:

given concerning the method by which he obtained

——

;


15-54, 15*6]

The fact that J* (z) + F„2 (z) has a simple asymptotic expansion shortens the manner which was not noticed by Marshall he used the equations

[Note.

analysis in a

;

l+

and he solved the

latter

dz M*' |_
Zeros offunctions related

15*54.

The method

of Stokes

those just investigated.

and of

9$v ' (z)

507

—

is,

z*

J^

to cylinder functions.

when the

»

M.]

of course, applicable to functions other than

Thus McMahon* has

-

J/3'

series for

calculated the large zeros of

cylinder function

is

a Bessel function of

CLZ

the

or second kind.

first

The general formula

for the large zeros of 9$J (z) is fi

^ where

fi 1

zeros ot

*

?

The

fi

+7

7/a2

8&

+ 154/a + 95 s

384ft

Jv (z) Yv (kz) - Yv (z) J„ (kz),

zeros of is

+ 82fi - 9

384& 8ft a, while the corresponding formula for the large

—*-* is

Awhere k

8

7/x,

8

= (n + ^v + \)7r —

——

+3

constant, and of

Jv'(z)Yv'(kz)-Yv'(z)Jy'(kz) have been treated in a similar manner by McMahon. Kalahnet has constructed tables of the zeros of the former function when k has the values 1*2, 1*5, 2*0 and v is 0, £, 1, $, 2, # while it has been proved by Carslaw, Conduction of Heat (London, 1922), p. 128, that these zeros are all real when v and k are real. The zeros of Jv (z) Yv (kz) - YJ (z) Jv (kz) have been examined by Sasaki, Tdhoku Math. Journal, v. (1914), pp. 45 47. '

—

15*6.

order

The mode of variation of

the zeros of a cylinder function

when

its

is varied.

The equation

in z

has an infinite number of roots, the values of which depend on v since /„ (z) is an analytic function of both z and v, so long as z ± 0, it follows that each ;

root of the equation is (within certain limits)

similar statement holds good

when the

by any cylinder function of the type

an analytic function of

function of the

J

v

(z) cos

a—

first

kind

y„(,s)sina,

is

v.

A

replaced

where a

is

any

constant.

is

If j denotes any particular zero of Jv (z), the rate of change of j, as v varies, given by the ordinary formula of partial differentiation

«

*'o->| +

P^

= 0.

* Annals of Math. ix. (1895), pp. 25—29. t Zeitschrift fUr Math, und Phys. liv. (1907), pp. 55

—

86.

:


508

J

Since zero,

y

(j)

= Q,

it

follows that J^'(j)

and hence, from §511

(15),

= — Jr +i(j)j*0,

when R(v)>

[CHAP.

XV

so long as j is not

0,

|=^|V,'(«)* wr dp jj\ jJ\+i(j)Jo (j)j

(2)

"

+1

This formula shews that when v

is positive, the positive zeros

of

Jv (x)

in-

crease as v is increased.

Equation

was stated without proof by Schlafli, Math. Ann. x. (1876), p. 137 and the was established in a different manner by Gegenbauer*, M4m. de la Soo. de Liege, (3) n. (1900), no. 3, in the case of the smallest zero of Jv (x). (2)

deduction from R. des Set.

We

;

it

proceed to extend the results already obtained to the positive zeros of 9$v (z)

where p

is

= Jv (z) cos a —

an unrestricted real variable, and a

The extended theorem

is

Any positive zero, c, of $v (z) of the real variable

To prove

this

constant

is

(i.e.

independent of p).

as follows

r

is definable

as a continuous increasing function

p.

theorem we observe that arc tan

is

Yv (z) sin a,

<

a function of p such that

c is l

r ;

.

{Jv (c)

constant, so that

dc[d

A*

\Tv (z)Y]

.

(Y,(2

,

= 0,

W'JJ and therefore vx

ircdp

Hence, by

~ = 2c dp

Since the integrand function of less

v

dp

'

dp

] z=0

§ 1373 (2), we have

(3)

A

'

[_

" f

K (2c sinh

t)

r**dt.

Jo

is positive,

this formula

shews that

an increasing

c is

p.

general theorem, namely that,

the order p (supposed positive),

then

c is

if c is

a zero which

is greater


p,

than

has been

proved by Schafheitlinf with the aid of very elaborate analysis. It will be observed

when

p tends to

from the definition of F„ (z) that

any negative value which sin (o

satisfies

c

tends to zero only

the equation

— pit) = 0.

The reader should note that the analysis

in the latter part of Gegenbauer's memoir is by his use of Rudski's erroneous results (§ 15*1). t Berliner Sitzungsberichte, v. (1906), pp. 82 93 ; Jahretberieht der Deutachen Math. Vereini*

vitiated

—

gung, xvi. (1907),

pp.272—279.


15*6]

It should be noticed that (3) shews that,

when

509

v is taken to be a

complex

a (complex) number, with a positive real part, then c is an r analytic function of v and so, as v varies, the zeros of @v (z) vary continuously, and they can only come into existence or disappear when c fails to be an

number and

c is

;

analytic function of

v, i.e.

when

c

= 0.

^

W

(z) It follows that the positive zeros of p (z) are derived from those of except that one positive zero varies, as variation v continuous a process of by disappears whenever v passes through one of the specified negative values.

we now choose a

If

so that*

variation of

0$a<7r, we

v,

of Stokes' type (§ 15*53)

(a/rr)

is available,

.

the formula

.

— a — ^— ±tt mr + *vtt * 2

8(rwr

gives the nth positive zero,

when the

4^-1 — —^ + $ Pir-iir — a) ;

c

...

positive zeros are regarded as arranged

magnitude.

in order of

however, v has varied so that it finally lies between (a/7r) — k and 1, where & is a positive integer, k zeros have disappeared, and so

If,

(a/Tr)

see that, as v varies from \

— l,

no zeros disappear during the process ofzeros which are so large that the formula of case the and so in

to any value exceeding

— k—

the formula just quoted gives the (n

— &)th

positive zero.

is due to Macdonald, Proc. London Math. Soc. xxix. (1898), was applied by him to the discussion of the zeros of Bessel functions of the

This type of argument pp. 575 first

— 584

;

it

kind of order exceeding

— 1.

If we draw the curve ^* (y) = 0, it evidently consists of a number of branches starting from points on the negative half of the #-axis and moving upwards towards the right, both x and y increasing without limit on each

branch. If

we take any

that the curve

point with positive coordinates

and a

line to the right

line

(«<>, y<>)

downwards terminated by the

and draw from

it

a

#-axis, it is evident

^x (y) — ^

meets each of the lines in the same number of number of zeros of $*„ (y ), qua function of v, which the number of positive zeros of ^, (y) qua function of y

It follows that the

points.

exceed v

is

which are

less

who took

v

Pig.

33

equal to

=

This is a generalisation of a theorem due to Macdonald -f> and the cylinder function to be a function of the first kind.

than y

.

illustrates the general

shape of the curves

of the sides of the squares being 2 units.

A much

Jx (y) = 0,

larger

the length

and more elaborate

diagram of the same character has been constructed by Gasser j, who has also constructed the corresponding diagram for F„(y) = 0. The diagram for <$x (y)

=

is

of the

same general character

as that for

Jx (y) — 0,

* This does not lead to any real loss of generality. t See a letter from Macdonald to Carslaw, Proc. London Math. Soc.

t Bern Mittheilungen, 1904, p. 135.

(2)

except that

xni. (1914),

p.

239.


510

[CHAP.

the portions of the curves below the axis of x consist merely of a isolated points

on the

on which 2x

lines

XV

number

of

an odd integer.

is

y

r

o

I

I

L Fig. 33.

The reader will

[Note. of 9BJ

(2),

find it interesting to

deduce from § 13*73

(3) that, if

d is a

zero

then

dd

2d ^ = ~-^

(4)

and hence,

if

C* (c'»

cosh 2t -

r 2)

K

(2d sinh t)e~ **

dt,

J

the variables are real and

d> v |

\

> 0, then d increases with

v.

The sign of dd/dv has also been discussed (by more elementary methods) by Schafheitlin,

—279;

Jahresbericht der Deutschen Math. Vereinigung, xvi. (1907), pp. 272

used by Schafheitlin

15*61.

is

but the analysis

extremely complicated.]

The problem of the vibrating membrane.

The mode of increase of the zeros of Jv (x) when v is increased has been examined by Rayleigh* with the aid of arguments depending on properties of transverse vibrations of a membrane in the form of a circular sector. If the membrane is bounded by the lines 6 » and 6 « tr\v (where v > ^), and by * Phil.

Mag.

(6) xxxii. (1916),

(6) xxi. (1911),

pp.

544—54B

pp.

53—58

[Scientific

Papen, yi.

[Scientific Papers, vi. (1920), pp.

(1920), pp. 1—5],

444—446].

Cf. Phil.

Mag.


15*61, 15*7]

the circle r

= a,

placement in a

511

and if the straight edges of the membrane are normal vibration is proportional to

fixed,

the dis-

Jv (rp/c) sin vd cos {pt + e), If the circular boundary zeros of Jv (x), while if are the of apjc values the of the membrane is fixed, zeros of Jv (x). are the they transversely the boundary is free to move

where

c is the velocity of propagation of vibrations.

'

The effect of introducing constraints in the form of clamps which gradually diminish the effective angle of the sector is to increase v and to shorten the periods of vibration, so that p is an increasing function of v, and therefore and

(since a

c are unaltered) ap/c is

say, the zeros of

J

v

and

{x)

«/„'


(x) increase with

That

v.

is

to

v.

using arguments of this character, Rayleigh has given proofs of a number of theorems which are proved elsewhere in this chapter by analytical

By

methods.

The zeros of

15-7.

K

v

(z).

K

number (zero v (z), where v is a given positive and z lies in the domain in which axgz < § ir, have been studied qualitatively by Macdonald*. The

zeros of the function

included),

From

K that K that

v

|

|

the generalisation of Bessel's integral, given in

(z)

has no positive zeros; and

it

§ 6*22, it is

obvious

has been shewn further by Macdonald

&rgz \^\tt. This may be proved at once v (z) has no zeros for which from a consideration of the integral given in § 1371 ; for, if z = reia were such a zero (r > 0, — \ir < a < £tt), then z = re~ia would be another zero but the |

;

integral shews that

K rr

which

is

If a

v

equal to +

K

v

(z)

„ j*„ -)

/r*\dv

K

v

(re^) = £tt J[J* (r) + F„ a (r)}, |

has no purely imaginary zeros.

either between \tt It

—

^cos2a)

we have

\tr,

Next we study the

is

f

^ Jo >0,


is

|

and so

^ = If" exp {- v^

K

isTjr / / {re-) v (re-~)

and

may be shewn

zeros for which ir

or

that the total

the even integer f nearest to v

the number

In the

is

R (z) is negative, the phase

of z lying

between — \ir and — ir.

number

— \,

of zeros in this pair of quadrants

unless v

—\

is

an integer, in which case

v — \.

first place,

there are no zeros on the lines arg z

* Proc.

t This

— ± ir,

London Math. Soe. xxx. (1899), pp. 165—179. not the number given by Macdonald.

is

unless v

—\

K THEORY OF BESSEL FUNCTIONS

512 is

an integer

K

for

;

= j*"* Kv (r) +

(re ±1ri )

v

Iv (r),

iri

.

K

v

(r)

= 0,

sin vir

.

K

v

(r)

XV

and, if both the real and

the imaginary parts of this expression are to vanish, cos vir

[CHAP.

we must have

+ irlv (r) = 0.

Since the Wronskian of the pair of functions on the left of the equations (rr/r) cos vir, they cannot vanish simultaneously unless cos vir = 0.

Now

consider the change in phase of zv

K

v

(z) as z describes

is

a contour

by the lines arg z = ± tt, terminated by the circular arcs.

consisting of arcs of large and small circles terminated

together with the parts of these lines (Cf. Fig. 15 of §7-4.)

T and 7, their equations being z = R and z = 8, number of zeros of v (z) in the pair of quadrants under equal to the number of zeros of zv Kv (z) inside the contour,

If the circles be called

j

consideration

and

is

this is equal to l/(2rr) times the

|

J

K

evident that the

it is

change in phase of z v

K

|

(z) as z traverses

v

the contour.

Now

the change in phase axg{z"

(z)^ -

v

is

|ârg {** JT r <*)}]

+\ug[rK

arg {*"#„(*)} JjRexpirt

9

(s)}

L

J«exp(-ir»)

two terms* tend to 2tt(v — |) and respectively, because when \z\is large or small on the contour,

As

and 8-*-0, the

i2-*-oo

z»

K

v

(z)

first

~ s—*e-V(!*-),

zv K> {*)

~ 2--

V (v)

1

respectively f.

The

last

two terms become r lim

4.

/«-o\

K

Now

v

{r) is

o 2 Tarc tan L

„

K

v

1R

TrcoBV7r.Iv (r) v .

.

(r)

'

:

,

.

+ 7ramv7r.Iv (r)j s

.

a positive decreasing function of r while I*(r)

is

a positive

increasing function, and so the last denominator has one zero if sini^r negative, and no zero if sin vir is positive. If therefore

when

r-*-30

is

we take

R

total

number of zeros of < it is

negative and arg z |

*

This

arg z

I

<

is

K

v

(z) in

the pair of quadrants J in which

\

v

I

r-*-0, its limit

vir),

vir.

Hence the (z) is

less

when

the value assigned to the inverse function than two right angles and having the same sign as the

arc tan (cot

being numerically sign of cos

the inverse function to vanish

is

— \ + — arc tan (cot V7r),

evident from the consideration that the asymptotic expansion of § 7*23

t.

t The second of these approximate formulae requires modification when r=0. X The two zeros of 2 (z) are not very far from the points - 1-29 ±0-44/.

K

is

valid

when

-


15*8]

and the reader will which is nearest to

When v — $

is

find

it

easy to verify that this

513

number

is

the even integer

— \.

v

an

integer,

K

v

a polynomial in z multiplied by a

(z) is

function with no zeros in the finite part of the plane, and so the zeros for which

R (z) <

is

exactly v

Next consider the portion If

we

write z

K

— v

ê***,

{z)

number

of

— \.

of the plane for which

ir

< argz ^ 2w.

we have

= - ^-»-[F„(D + *(l + 2e 8™) /„(£)],

K {z)

has a sequence of zeros lying near the negative part of the imaginary axis. The zeros of large modulus which belong to this sequence are given approximately by the roots of the equation

and so

v

tan (f it

\vir

- {ir) = -i(l + 2e 2"* ) i

;

may be verified that they are ultimately on the right or left of the imaginary

a axis in the s-plane according as cos

i>7r

is less

than or greater than £;

according as v differs from the nearest integer by more or less than

sequence does not exist when e 2vni =

— 1,

i.e.

when

\.

i.e.

The

v is half of an odd integer.

a corresponding sequence of zeros near the

line arg z

There

is

15*8.

Zeros of Bessel functions of unrestrictedly large order.

= — § ir.

The previous investigations, based mainly on integrals of Poisson's type, have resulted in the determination of properties of zeros of Bessel functions, when the order v is not unduly large. This is, of course, consistent with the fact that Hankel's asymptotic expansions, discussed in Chapter vn, are significant only when v* is fairly small in comparison with the argument of the Bessel function.

The

fact that

Debye's integrals of § 8'31 afford representations of functions

of large order suggests that these integrals may form an effective means of discussing the zeros of Bessel functions of large order; and this, in fact, proves to be the case *.

Moreover, the majority of the results which will be obtained

are valid for functions of any positive order, though they gain in importance

with the increase of the order.

We shall adopt

the notation of § 831, so thatf

H ® (v sec £) =

gKi(tan/3-0) rao+wi-ifi

v

where

e-"dw,

-.

— t = sinh w — w + i tan (3 (cosh w — 1), and the w is chosen so that t is positive

the complex variable

contour in the plane of

on

it.

• Watson, Proc. Royal Soc. xciv. a, (1918), pp. 190—206.

t

We shall

use the symbols

x and

v sec

j8

indifferently

when x > v.


514 If

w = u + iv,

where u and

u and

v are real,

[CHAP.

XV

v both increase steadily as

w

describes the contour, so that e~ VT dv

e-^du, J —oo

—3

J

are both positive.

Hence,

if

we regard

/3

as variable, raa

and

define

+ ni-ip

arg

e

- "7

dw

J _oo_j(5

# = 0, and to vary continuously with /3, it remain a positive acute angle for all values of /3 between and ^v: and, moreover, by § 8*32, it cannot exceed J ir, since dv/du < \/3to

be a positive acute angle when

will

This positive acute angle will be called %, and then

^ will be defined by

the equation

¥ = i/(tan/8-£) + x-£ir. It is then evident that

H <»(vBec/3) = ffl#*, w

where Jtt

is

and

positive (not zero);

M

Jv («) =

cos

9,

Yv (x) sin a,

^„ (a;) = Jv (x) cos a -

If

= Jtt sin ¥.

F„ (a?)

it is clear that the only zeros of 9i> v (x), greater than v, are derived from the values of "9 which make "9 + a equal to an odd number of right angles.

It is easy to

shew that

SP increases with x,

we have

when

v remains constant.

^

2 - ^, /(?ng) ^ > - arc tan ^ >00. *-arctan^ Y ^ + ^ ^ as x increases, W increases steadily, and so, to each of ,

(fl;)

Hence, of

(

the values

W for which y\T

= (m +

ir

£)

- a,

corresponds one and only one positive zero of 9£v

Next we

shall prove that

theorem of a much to prove

it

;

%

is

deeper character, since the result of §

dx

^ and

so

li

m

[x

2

2/(wvp) 8

J, (x)

From Hankel's asymptotic expansion

=

This

x.

1374

is

is

+

it is

*J(x 2

F„

-v*)

#

(a?)

>

clear that

— ^ptt — Itt — \J(ot? — v )+v arc cos (v/x) + \tr + 2

arc .tan

which the expression on the

[

<

—Jy

»( V )

left is

:

\ .

>

»

a positive acute angle.

a

required

we thence have

dx~ dx

lim x

(x).


also

dx^dV_ V d (tan ft- ft) _

in

For

(l/x)]


15*8]

To form an estimate

x when

of the value of

v is large,

Jv (v)=-Y„(v)ta.ny

v

hence, from § 8*42,

when

we

write

;

we have yv =

lim and,

515

£tt,

v is large,

r<|)

V3( so that,

when

v

is large,

y„

is

210

rm.^)* r Q).

)


v.

following Table gives the sexagesimal measure (to the nearest half minute) of the angle whose circular measure is yv it exhibits the closeness of 7„ to its limit, even when v is quite small:

The

\

V

1

t

6

4

3

2

f

12

10

8

00

30° 28° 39' 29° 23£ 29° 38' 29° 45' 29° 51' 29° 54' 29° 56£' 29° 57$' 29" 58' 29°58£'

yy

The

table suggests that

values of

v\

yv

is

to prove that this

is

an increasing function of v for all positive we need the theorem that

the case

J* {v) —dv this inequality has already

and

We

are

now

m

is

Yv (v)

~^r > °

been established in

§

in a position to infer that <@v {v sec

v (tan

where

—

1374. has a zero for which

fi)

£ - /3) + % « mntr - o,

any positive integer (unity possibly excepted)

;

and therefore at

the positive zeros of <@y (v sec £), with the possible exception of the v (tan & — £)

first,

has a value between

(w —

i)

— a and

it

mtr

— yv —'a,

and the difference of these expressions (when v is not small) slightly exceeds -fair.

From

the result of % 15-3(10),

increases from

and

so, if

it

follows that the phase of

ia

v

a-^Trto^ + oasZ

< a < tt,

the number

to any value exceeding v, increases from of zeros of
greatest integer contained in ("¥

+

a)lir

+ £.

A simple theorem which may be noted here is that, when that, if 8 is

H ^(X)e

any positive number,

P

+

*^| rf*

vs

> £, it follows from

decreases as 8

is

increased.

§

1374

This result

the shews that the interval between consecutive zeros of <$y (x) decreases when the rank of zeros is increased.

A different proof of this theorem is due to

Porter;

cf.

§ 15-82.


516

Jv (x) and Y„ (x). It has been seen (§ 15'3) that Jv (x) and Yv {x) have no

XV

The smallest zeros of

15*81.

(0, v),

[CHAP.

when

v is positive,

obtained in

§

842

and it

is fairly

zeros in the interval

obvious from the asymptotic formulae

that they have no zeros of the form v

The asymptotic formulae which were quoted

+ o (v*) when v is large. shew

in § 8*43

that, with the

notation of § 15*8,

where the inverse tangent denotes a negative acute angle. Hence, at the smallest zero of J, (x), tan {„ (tan

As 1

-

(l/*Jv),

(

and f 7r, so that

we

=-

$$£%$

which v (tan

for

- 0) = £y tan

v (tan if

.)|

while the expression on the right decreases from 0*2679 to 0; the

smallest root occurs for a value of

Hence,

W

to îr, the expression on the left increases from

increases from

+

» - J, +

8

+

— 0)

lies

between f tr

(v ~ *).

solve the equation

tan(f-f7r)=-Q(£,i)/P(£,£), the value of £ so obtained

which

is

(i>~ J ).

is

the value of £i>tans

The value

smallest positive zero of

of £

is

#

at the zero with an error

approximately 2*383447, and hence the

Jv (x) is v + v*x 1855757 + 0(1).

In like manner, by solving the equation

= -Q(£,£)/P(M), approximately £ = 0*847719, we

tan(£-i7r) of which the smallest root is

smallest zero of

v

The formula xxxiv. (1917), of § 8*42 for

find that the

Yv (x) is + v^x 0-931577 + 0(1). Jv (x) has been given by Airey, Phil. Mag. (6) was derived by using Debye's asymptotic expansion

for the smallest zero of

p. 193.

Jv (x)

Airey's formula

when x has a value such that

x-v = 0{v*) + o(yi). For such values of the variables, it has not been proved that Debye's expansion is valid, and although Airey's method gives the two dominant terms of the smallest zero of «/„ (x) correctly, the numerical result which Airey gives for the smallest zero of J¥ (x) is not the same as that of § 15-83. The reason why Airey's method gives correct results is that Jv (v + f) is expansible in powers of £ so long as { is o (v), and in this expansion it is permissible to '

substitute Debye's formulae for

A

Jv (v), Jv

formula for the smallest zero of

where between

'

iy\

Jv" (v),

J- V (x) was

and the smallest zero of

Jv (x),

....

given by Airey.

This zero

according to the value of

v.

may

lie

any-


15-81, 15*82]

517

seem to be possible to make further progress by the methods used in this now make a digression to explain the methods of Sturm (which have been applied to BessePs equation by various mathematicians), and we shall then give an investigation which leads to the fascinating result that the two expressions which, in this It does not

section.

section,

We

shall

-1

were proved to be 0(1) are in reality

for the smallest zeros of

become negligible when

Jv (x)

and

Tv (x)

in

(i»

),

so that approximations are obtained

which the errors are 0(v~*),

i.e.

the errors

v is large.

[Note. An elementary result concerning the smallest zero of Jv (x) has been obtained from the formula of § 5-43 by Gegenbauer, Wiener Sitmngsberiehte, cxi. (2a), (1902), p. 571; if /* = v + e where e < 1, then the smallest zero of Jiv + (x) is less than twice the smallest zero of Jv (x), because, for the latter value of x the integrand cannot be one-signed.]

<

15*82.

Applications of Sturm's methods.

Various writers have discussed properties of Bessel functions by means of the general methods invented by

Sturm*

differential equation of the second order.

for

the investigation of any linear

The

results hitherto obtained in

manner are of some interest, though they are not of a particularly deep character, and most of them have already been proved in this chapter by

this

other methods.

The theorem which

is

at the base of the investigations in question

given a differential equation of the second order in

its

is that,

normal form

in which the invariant / is positive, then the greater the value of /, the

rapidly do the solutions of the equation oscillate as

As an example

x

more

increases.

an application of this result, we may take a theorem due to Sturm Ann. sci. de Pltcole norm. sup. in. (1866), p. 72, that, if v2 - i be positive and c be any zero of
(ibid. pp.

which

is

of

— 175) and Bourget,

next greater than c does not exceed

c+

This result follows at once from the consideration of the facts that the function 3&9&V (#)

is

annihilated

by the operator dx*

(§ 4-3)

+ \ —A

1,2

and

pp.

that,

when x > c,

1

-$=

x* /»

>

c2_„2_i.l ^

-

.

A slightly more abstruse result is due to Porter, American Journal of Math. XX. (1898), 196—198, to the effect that, if v 2 > \ and if the zeros of <@„ (x), greater than ^(v 2 - £),

magnitude are clf c2 c3 ... then cn+1 — cn decreases as n increases. This has already been proved in § 15*8 by another method. in ascending order of

,

,

Other theorems of like nature are due to Bdcher, Bulletin American Math. Soe. in. 205 213; vn. (1901), pp. 333 340; and to Oasser, Bern Mittheilungen, 1904, pp. 92—135. (1897), pp.

—

—

—

* Journal de Math. i. (1836), pp. 106 186; an account of recent researches on differential equations by Sturm's methods is given in a lecture by Bdcher, Proe. Int. Congress of Math. i. (Cambridge, 1912), pp. 163—195..


518 15*83.

Applications of Sturm's methods

to functions

[CHAP.

XV

of large order.

We

proceed to establish a number of results concerning cylinder functions of large order which are based on the following theorem of Sturm's type :

Let U} (x) and u2 (x) be solutions of the equations

d2 u

d2 u2

_

T

zt + i^-o, such

that,

x

when x = a, Wj (a)

and

let

r

^? + /.«.-o

I and I2

= u2 (a),

=m

(a)

Mi'

(a),

a^x^.b, and

be continuous in the interval

I

'

2

u

also let

x

(x)

and

u 2 (x) be continuous in the same interval. Then, if J 1 ^ I2 throughout the interval*, u2 (x) exceeds u (x) so long as t between a and the first zero of u x (x) in the interval, so that the first zero j

x

j

of u

(x) in the interval is

x

on the

left

of the first zero of u 2

Further, if w/ (a) has the same sign as u x

u (x) in the interval moreover i

j

j

lies

x

is

|

on the

max

I

(a), the first

maximum

left of the first

Uj (x)

To prove the theorem f, observe

|

< max

|

(x).

u 2 (x)

maximum

point of

point of'j u 2 (x) \,and,

j

that, so long as

u

x

(x)

and u 2 (x) are both

positive, Ul

and

so,

when we

d2 u2 ~ d2 u^ U2 ~dxv dxJ

=(T ~

T

,

1

.

UlU*

*'

^

°'

integrate,

[du Since the expression

du

2

~\

x

.

1

now under

consideration vanishes at the lower limit,

we have du

du 2

n \ (1)

x

^dx-'^dx-^

-

Hence we have djujuj)

dx and therefore

W%0, LMiJa that

is

to say,

M2 (a:) u 2 (a) > u (x) " Ui (a)

(2)

=

1

x

*

To

of u, (x)

simplify the presentation of the proof of the theorem,

and u 2

(x), if

necessary, so that u x (x)

signs indicating moduli

t

The theorem

is

may

is

it is

convenient to change the signs on the right of x = a; the

positive immediately

then be omitted throughout the enunciation.

practically due to Sturm, Journal

<}e

Math.

i.

(1836), pp. 125

—127, 145 — 147.

ZEBOS OF BESSEL FUNCTIONS

15*83]

It follpws that just before

and

positive,

value

vanishes for the

1*1 (a?)

has remained positive

it

first

part of the theorem

Again, if w/(a)

is positive,

and at

before it vanishes,

so that « 2'(/*i) is positive

Therefore the Finally

first

still

as well as v^ (a), then v^{x)

this point,

fi lf

we have from

and v^(x) must be

maximum

point

/*2

When

is

/*,).

must be on the right of /t^.

2

(ft*)

= max u^ (x),

completely proved.

two functions,

and that w2 (x)

Uy,

(x)

and

it,

(x),

is less oscillatory

manner postulated more oscillatory* than

are related in the

than

i*j

(x) is

v^ (x).

now apply the theorem just proved and F„ (x) when v is large and x — is

shall

Jv (x)

(1)

positive in the interval (0,

of u^ (x)

in this theorem, it is convenient to say that

We

must have a maximum

we have

and the theorem

ing

is

therefore proved.

is

max %! (x) = w, (jj^) < w2 (/Hi) $ u

(x)

time Ma(#)

while x has increased from the

a.

The

i/ 2

first

519

1/

to obtain results f concern(?*).

Our procedure

will

be to construct pairs of functions which are respectively slightly less and slightly

more

oscillatory

than the functions in question.

first place we reduce we then have

In the

x=

ve°;

[^ +

(3)

A

function which

is

small positive values of

"*

(**

" 1 >] *'

W

normal form by writing

= °-

obviously slightly less oscillatory than is

^(ve9)

for

obtainable by solving the equation

[^+2^]u-0,

(4)

since e* - 1

Bessel's equation to its

> 20 when 6 > 0:

The general

solution of (4) is

.-**,(?£*); and the constants implied in this cylinder function have to be adjusted so that u and its differential coefficient are equal to <@¥ (ye°) and its differential coefficient at *

6 = 0.

The reason for the use of these terms is obvious from a consideration of the special case in which 2} and I2 are positive constants. f These results supersede the inequalities obtained by Watson, Proe, London Math. Soc. (2) xvi. (1917), pp. 166—169.


520

which

It follows that a function

when x ^ v,

[CHAP.

is (slightly) less oscillatory

XV

than 90¥ {x),

is

We

now endeavour to construct a function which is (slightly) more oscilthan latory ^?r (x), in order that we may have 9$ v (x) trapped between two functions which are more easily investigated than <$„ (x). The formula

for

the

where from

but

yfr

(0) is

combined with the result

less oscillatory function,

we should

stated in § 8*43, suggests that

construct a function of the type*

to be determined.

a function of

It

might be anticipated

that the suitable form for ty (0) would be £ tan3 fi, where sec /3 = e e ; appears that this function leads to a differential equation whose solution

§ 8'43

it

its degree of oscillation depends on the relative values of v and and we are not able to obtain any information thereby. is

such that

The

0,

by

invariant of the equation determined

v{t(WW}îW(«)l is

known

to

be

(§

4*31)

and it is requisite that

this should slightly exceed

natural to test the value of

i/r

(0)

which

is

value of

_ 3 (^(0))

2^'(0)

4.Vf'(0)J

1

When we

for this

+"'(0)

replace e* by sec $, •f'(0)

we

(5)

*(0)-0, yfr

(0),

5_ | +'(0)1

is

2

36J^(0)j

= tan/3,

T/r(0)

> "

"

= tan£-£,

*"-<*>=-

J;*^

-

to test the truth of the inequality

3(tan/3-£)^

sin s

cos

2

^3(tan£-/3)

negative .when tan 2 /3

#

/3V(l+itan2 /9) sin3 /3

»'•

Now

It is consequently

find that

+-«>-«•*«•« a and hence we have

2

— 1).

2 (e *

given by the equations

*'(*)- <•(*•-!* by determining whether^

2 i/

cos

3

< V24 — 3, and

£V(l + £tan2 £)_ it

is

positive for greater values

of tan2 /9. *

The multiple of the cylinder function cf. § 4*31 (17). its normal form

equation in

;

is

taken so that the product

satisfies

a differential


15*83]

Hence, since (5) is true when #«= 0, it is true when a certain angle between arc tan V{V24 — 3} and \ir

is

measure of

O

is

V{3 (1-/9 cot 0)} [r (|) (£*)* .

slightly

We

more

-*

£

**

/9

»

where

/S

The sexagesimal

59° 39' 24"-27.

Proceeding as in the former case,

is

521

oscillatory

we

find that the function

9

W (?) J_j [v (tan fi - /3)} + r (t) (**)* »/ (») Jj {* (tan£ - 13)}]

J¥ (v sec /8),

than

so long as*

•£

# •$ #>•

can now obtain an extremely important result concerning the smallest

zero of 9BV (x) which

is

greater than v

;

for let £\,*

be the smallest value of f

which maês

r (|) a*)* *, GO /-» (I) + r a) (H8 »/ oo /j (?) Then

vanish.

6o£& of the equations

20 « \,a/i/», give

a;

Since,

v (tan

= + Â,

2

i/

!/ 1

/3

+

we

is expressible

^?r (x) is equal to

+ $\M +

«/"„

0(v~i).

(x) it is easy to verify

\,

= T926529 + 1-855757

+ is

v

i/

maximum

of

+ v*x

(ir*),

is large, is

manner, the smallest zero of F„ (x)

first

from a Table of Bessel

± £ that

and so the smallest zero of Jv (x), when v

The

between two expressions of

in the form

functions of orders

like

lies

see that the value of the zero of *@v (x) which is next greater than v v

When

V

(v~ J ).

by Sturm's theorem, the zero of 9£v (x)

this form,

In

- £) = J

+

«/„ (a;)

i/*

+

x 0-931577

may be

0(jt*).

0(ir*).

obtained in a similar manner, by differ-

entiatingf the two expressions constructed as approximations.

The

r (|) vanish J, then the i.e.

which makes

result is that if $fi v* is the smallest value of f

(W

first

J. (v) J% (f)

maximum

- r (i) (\vf j: (v) jl, (f

of t/"„(#)

is

at the point

at the point i/

+

i/J

x 0-808618

+

(i/~i).

maximum of the function Yv (x) cannot be treated in this manner because its The first maximum is on the right of its first zero this follows at once from § 15*3, because F„ (x) increases from - oo to O as x increases from to the first zero. For an investigation of the maximum value of Jv (x) qua function of v the reader first

;

should consult a paper by Meissel, Astr. Nach. cxxvni. (1891), * This restriction is trivial because i.e.

small values of

we have

cols.

to consider values of

j3

435

for

—

438.

which

*/3

3 is

bounded;

/3.

t The permissibility of this follows from the second part of Sturm's theorem just given. X This value of £ is approximately 0*685548.

—

CHAPTER XVI NEUMANN The

16*1.

The

SERIES

AND LOMMEL'S FUNCTIONS OF TWO VARIABLES

definition

of Neumann


series.

and of Chapter xvn

is

the investigation of

various types of expansions of analytic functions of complex variables in series whose general terms contain one or more Bessel functions or related functions.

These expansions are to some extent analogous to the well known expansions by the theorems of Taylor and Laurent. The expansions analogous to Fourier's expansion of a function of a real variable are of a much more recondite character, and they will be discussed in Chapters xvni and XIX. of an analytic function

Any

series of the type ac

2

a n Jv+n (z)

n=0 is

called a

although in fact Neumann considered* only the which v is an integer the investigation of the more due to Gegenbauerf.

Neumann

series,

special type of series for

general series

To

mann

is

;

distinguish these series from the types discussed in § 16*14, the description Neukind has been suggested by Nielsen, Math. Ann. lv. (1902), p. 493. '

series of the first

'

The reader will remember that various expansions of functions as Neumann have already been discussed in Chapter v. It will be sufficient to quote here the following formulae

series

:

ni

»=o

J.(z

+ t) = £

m= -oo

«•

where

«r2

We

J^m (t)Jm (z), "

TO =o

= Z + z* - 2Zz cos

z

1

.

expanding an arbitrary function then we shall investigate the singularities of the analytic function defined by a Neumann series with given coefficients and finally we shall discuss the expansions of various particular functions. into a

shall first discuss the possibility of

Neumann

series;

;

For a very general discussion of generalisations of all kinds of series of Bessel functions, may consult memoirs by Nielsen, Journal fiir Math, cxxxn. (1907), pp. 138

the reader

146; Leipziger Berichte, lxi. (1909), pp. 33—61. * Theorie der Bessel' schen Functionen (Leipzig, 1867), pp. 33

+ Wiener Sitzungsberichte, lxxiv.

(2),

(1877), pp.

125—127.

—35.

NEUMANN

16-1, 16*11]

SEBIES

523

Various expansions of types which resemble Neumann's (other than those given in this Webb, Phil. Trans, of the! Royal Soc. cciv. (1905), p. 487 and

chapter) are due to H. A.

490—497.

Nielsen, Atti delta R. Accad. dei Lincei, (5) xv. (1906), pp.

16*11.

Neumanns expansion* of an arbitrary function in a series of Bessel

coefficients.

Let f(z) be a function of z which is analytic inside and on a circle of R with centre at the origin. If C denotes the contour formed by this circle and if z is any point inside it, it follows from Cauchy's theorem that radius

Now, by

§9-1,

-i

U0.(*)./.W;

—z

t

M=0

this expansion converges uniformly

and

on the contour. It follows at once

that

/(#)- I an Jn (z),

(1)

»=o

where an =

(2)

and

this is

^fc f(t)O

n (t)dt;

Neumann's expansion.

If the Maclaurin expansion of f(z) is

/(*)- 5

we

b n z»,

see that

-^,

i

bj

t™on {t)dt

m — 1)! m

(n>l)

=M t*2»-(î-^ 6_.

(«>1)

« _^_ n.(n — '*-**•-

^k**

w=0

and

\

so fOo

L

<3>

=b

, !

This result shews that the

Neumann

series corresponding to a given function

assumes a simple form whenever a simple expression can be found

«=0

The given

is

construction of the

consequently

Neumann

338—340.

sum

W*

series

now merely a matter

when the Maclaurin expansion

is

of analytical ingenuity.

* Theorie der Bessel' $chen Functionen (Leipzig, 1867), PP- 33

. (1872), pp.

for the

—35.

See also Konig, Math. Ann.


524

[CHAP. XVI

Neumann's* analogue of Laurent's theorem. Let f(z) be a function of z which is analytic and one- valued in the ringshaped region defined by the inequalities 16*12.

r^\z\^R. Let

C and

c

be the contours formed by the \z\

= R,

circles

= r,

\z\

both contours being taken counter-clockwise; then, if z be a point of the region

between the

f(£)= JK '

we have

circles,

i

/•/(*)&

2-TriJc

1

,

t-z

/(«)*

f

2ttiJ c

z-t

^^ ^M^f/(t)On(t)dt+XÔn (z)j

f(t)Jn (t)dt.

c

o

Consequently f(z)

is

expansible in the form

/(#)- i an Jn (z)+ % an'On (z),

(1)

where

an =

(2)

^j

f(t)O n (t)dt, c

an'=^ff(t)Jn (t)dt.

If the Laurent expansion o£f(z) in the annulus

/<,)-

we

is

i £. ik*+ n=l *

*=0

have, as in §1611,

w

I a„

ttn

(**)

=n 2 —

—

2 n-*w-

*-

(n

ft*-™,

^ 1).

~v,\t&'»+»»+i ^.On+sm^t/^ = 2 n+2m m!(w + «i)! .

1 6*13.

By

Oegenbauer's generalisation of Neumann's expansion.

using the polynomial

A n>v (t)

defined in §9*2,

Gegenbauerf has

generalised the formula given in §16*11. If f(z) is analytic inside and on the circle \z\ contour formed by this circle, we have J

x

'

1m Jo

t

= R,

and

if

G denotes

the

—z

and so

*"/(*)=

(1)

5

an J,+n (z),

n=0

—

* Theoric der BeueViehen Functionen (Leipzig, 1867), pp. 36 39. t Wiener Sitzungsberickte, lxxiv. (2), (1877), pp. 124—130. See Wiener Denk$chriften, (1884), pp. 298 816 for some special cases of the expansion.

—

xlviii.

NEUMANN

16*12-16'14]

SERIES

525

where

a.-^.Jo /(*)-A«.r(0*

(2)

provided only that v If,

is


as in §16"11, the Maclaurin expansion of f(z)

/(*) =

S Kz\ »=o

then On = (v

(3)

+ n)t2'+»-™

r{v + n - m) W»!

m=0

Neumann's expansion

may be

of §16*12

is

K

generalised in a similar manner.

16*14 The Neumann-Oegenbauer expansion of a function as a

series

of

squares or products. of § 9*5, namely

From the expansion

—— -= 2 B

n .^ ¥ (t)JIIL+in (z)Jp+ ^n (z)

-

Z

J>

which

when

when z <

is valid |

z

|

< r,

n =0

|

\

\

t

we can

, |

z*+ >f(z)=

5 »=o

when z < |

\

r,

and the an =

(2)

C being

at once infer that, if f(z)

analytic

an Jli+in (z)J¥+in(z)

;

an expansion


coefficients are

^jc f(t)B

ni

the contour formed by the circle

Gegenbauer*

is

then the expansion

(1) is valid

)

|

^{t)dt, z

j

closely connected

= r. with

This expansion this,

is

due to

namely that

f(z)= I an'Jn*(z),

(3)

n=0

where

a^ =

(4)

^

tf{f)Cln {t)dt,

c

and Q„ it) is Neumann's second polynomial (§ 9 "4), is valid provided that f(z) this expansion was obtained by Neumann f. is an even analytic function ;

Gegenbauer's formula has been investigated more recently by Nielsen, Nouv. Aim. de Math. (4) II. (1902), pp. 407—410.

A

type of series slightly different from those previously considered

derived from the formula of § 5*22 (7) in the form z-

{ = 2>r(l v + l)2, -¥± r-Jiv+n (z),

»=0

ni

* Wiener Sitzungsberichte, lxxv. (2), (1877), pp. f Math. Ann. in. (1871), p. 599.

218—222.

is


526

[CHAP. XVI

which shews that

-2

b n z»+n

i

(5)

»=0

n=0

a n Qz)i <"+»> JHy+n) (z),

where t — ^-m + l) On-2m>

_+"-wt r(|i/ + an — 2,

(6)

Expansions of this type have been the topic of a detailed investigation by Nielsen*.

5 an Jy+niz)

Let

and

Pincherle's theorem

16*2.

be any

generalisations.

its

Neumann series, and

let

the function defined by

n=0 this series

and

its analytic

continuations be called f{z).

Let also

The

by f(z)N and

function defined

its analytic continuations will

be called

the associated power series off(z).

The Neumann

series converges

throughout the domain in which

Urn %\{an J^n(z)}\
and

this

domain

is identical

with the domain in which

lim

*/ =rp-

nôcV ire + n+l)

>

by Horn's asymptotic formula (§81). It follows that a

Neumann

series has

a

circle of convergence, just like

power series, and the circles of convergence of a associated power series are identical.

Neumann

The theorem that the convergence of a Neumann a power series and, in fact, it

is

Nyt

we

i.

write

r> + ,» + t)r(tr * w

Tidsskrift, ix. (b), (1898), pp.

493—496.

further,

.

this theorem,

t Bologna Memorie, pp.

much

can be proved that f(z) has no singularities which are not also

2 *

a

and of the

series resembles that of

to Pincherlef; but it is possible to go

due

singularities oif{z)N

To prove

series

77—79. 151—180;

(4) in. (1881), pp.

-

see also Nielsen,

Math. Ann.

lv. (1902),

NEUMANN

16*2, 16*3]

and then, inside the

circle of

SERIES

527

convergence*,

cos

{W(l

-«•)}•

^

From the theory of analytic

continuation

0(^)^3^. follows that, if

it

(z) is analytic

any value of z, so also is f(z), provided that the path of integration is suitably chosen and so all the singularities off(z) must be singularities of $ (z). for

;

Now

the series denning

cp

(z)

2£ S and a theorem due

F

1

(z)

may be

= 2

bn zn

,

all

F,{z)= 2

t

on zn

F (z)= 3

,

»=0

the singularities of

some singularity of F

(z)

F

s

is

some

b n cn zn ,

I w=0

(z) are expressible in

and y

n

states that, if

n=0

then

T(v + n+l)

an

Hadamardf

to

written in the form

the form 8y, where

singularity of

/3 is

F

2 (z).

Since the only finite singularity of the hypergeometric function

g

Tfr+n + l) + n + %)

n=0 r(v is

z=l,it follows that all the singularities of (z) are singularities and therefore all the singularities of f(z) are singularities cff(z)N is the theorem which was to be proved.

at the point

of f(z)x

and

;

this

(f>

;

The reader should have no difficulty in enunciating and proving similar theorems \ connected with the other types of expansions which are dealt with in this chapter.

Various special

16'3.

Neumann

series.

The number of Neumann series, in which the coefficients are of simple forms, whose sums represent functions with important analytical properties is not large we shall now give investigations of some such series which are of ;

special interest.

By

using the expansion

{V-n* cos 20 + 1)"*= 1 * It is

assumed that R(r + £) is positive; by the omission of the terms

to be truncated

argument

is

if

for

r »- Pn (cos20), 2

2

not, the several series under discussion have

which

R (v + n + %)

is

negative, but the general

unaffected.

t Acta Math. xxn. (1899), pp. 55—64 Hadamard, La Serie de Taylor (Paris, 1901), p. 69. t For such theorems concerning the expansion of § 16-14, see Nielsen, Math. Ann. m. (1899), ;

p.

230

et seq.

:


528

[CHAP. XVI

Pincherle* has observed that

J Î Jm» where the contour

^p{^(<-l/Q}^ m =- JL ^lj ^ _ + (0+)

(z)P f cos 20) (*) /*» (COS

lies

f

wholly outside the

2ti

circle

1

|

C08 2 ^

,

1}

= 1.

1

write £ (t — ljt) = w, so that the contour in the w-plane (large) closed curve surrounding the origin, we find that If

now we

5 JU.(«)P.(co.M)-^,j* and

we

so

V{(*'+l)(w'+sin«0)}'

e-faMtt du = -^-

n=0

where the modulus of the

The

a

obtain the formula

X Jm+1 (z)Pn (cos 20) = -f*""

(1)

is

^""•"•du,

*7r J

J

elliptic function is sin 0.

interesting expansion

W

^vW-*o

{2"+»I>

2.4...(2n)

who proved

has been given by Jolliffef,

I)} + n + l)

2

that the series on the right satisfied

This expansion is easily derived as but the following direct proof is not without

the same differential equation as J,* ($z).

a special case of § 11*6

(1),

interest

By Neumann's

formula

J" and,

if

we expand J» (z VO

r / u\ J* <* VO -

2t "

Ti

* 2

find

we have J

^-7ri

V{<(l-0}'

into the series

(2i/ o

(

+ 2w + l)r(2i/ + w + l) m!r(2* + l)

x

we

5 '43)

(§

& (- m,

2v

+m+

1

;

2v

+1

;

<) «/»,+»„+! (s),

on integration that _

*

.

9/1 Jv\h z)™ ^

2(2i/

»»=0 * Bologna Memorie, (4)

vm.

(1887), pp.

+ 2m+l) 1

*

m

°"

, v r 9H-sm+i\*/i

125—143. Pincherle used

elliptic

functions of modulus

cosec 6 in his result.

t Messenger, xlv. (1916), p. 16.

by Nielsen, J If

The corresponding expansion of z* J„- j (J*) Jv ($z) was obtained

Jfyt Tidsskrift, rx. b, (1898), p. 80.

B + i) < 0, (j»

we use loop

integrals instead of definite integrals.

NEUMANN

16-31]

SERIES

529

where

r

a>» =

7T

.

- *>~* F W9 f '^ (* \ZVt*i\ + 1) Jo >

If ml

i

.

(-rn,2v

+ m + l;

2,

+

1

;

*)<**

_L^ (V- (i - tr* fir-v-*" * dt m .ml J 7

v

tr

= i=J=.

I

V-

TT.WilJo

by

>

l

w partial

(1

_ tr d

"

[
'~!,^- tri] d«m

*

integrations.

It follows that

am

is

the coefficient of h m in the expansion of

I [V {t-ht(l - 0}-"-* {1 - + ht{l - t)}-*dt t

in ascending powers of h

this expansion is absolutely convergent

and

;

when

|A|<1. If

we

we

, t

write

=

w

(1 ,

- *) ,

—

,

find that

£ am hm = =

,«

-

f\"-* * JO 1

=I

f

T Jo

It is

now

M H"i (1

-— "^*

— 1)

2.4... (2n)

1

v+ft _ i /i (

f

Jo

1.3...(2w-l)r(i/

_1

and

and that

!• 3 ...(2n

2.4...(2w)

~tt'

ht)~i dt

_ u )-i (1 _ A2 M )-i dll.

=

evident that aan+1 « aan

- h (1 - 0}-"-* (1 - 0"* (1 +

{1

_

j

\_| '

+ rt + ^)r(^) + n + l)

T(i/

'

this formula at once gives Jollifies form of the expansion.

16*31.

The Neumann

series

summed

by Lommel.

The effects of transforming Neumann series by means of recurrence formulae have been studied systematically by Lommel*; and he has succeeded by this means in obtaining the sums of various series of the type

2an Jv+n (z) in

which an

a polynomial in

is

n.

Take the functions \s4 ¥

,

m (z) *= S

(v

+ 2w + l)/2ni (v + 2n +

= £

(v

+ 2n + 2)/

S8¥>m (z)

where fm (v)

is

1)

Jv+2n+1 (z), .

»=o 2)H

_1

(i>

+ 2w + 2) «/„ +!m+2 (*),

a function to be determined presently.

—

* Studien liber die Bessel'sehen Functional (Leipzig, 1868), pp. 46

49.


530

By

we have


- 64v m

(1)

z

(z)

,

= I

[CHAP. XVI

fnn, (v

+ 2n + 1) {J"„+2/l (z) + Jv+ pi+2 (z)}

»=o

+ l)Jv (Z) +

=/2m(V

2^ m

(z),

fm (v) satisfies the equation of mixed differences* + 2n + 3)+/m(»'+2n + l)=2(i/+2n + 2)/2w^ + 2n + 2).

provided that

/2m

A

(i/

solution of this equation

We

is

adopt this value offm (v) and then

- M*, m

(2)

fm^

(z)

z

Hence

(i;

1

l

{v)

it is

found by the same method that

J^ (z) + 2^„, m_

1

(z).

follows that

it

S4v m {z) = W*m ( v + !) Jy ( z ) ~ K/a»»-i (") J*+M + m and so £4v m (z) = «•" 2 f^ 2"/^ (v + 1)
Z2 S#v

,

m-i (z),

1

,

»=o

Therefore, since «#,,_, (z)

= 2

Jr,+2» +1

(*)

»=o

=s *>

/"*«/,

(0 *,

J

we have

- j„

w |j

o

and

similarly,

r(1 „_

from the expression

•

/r+l( *)

for

n+

j-)

&y m(z) t

(**)

}

>

t

t.-«r(^-» + *)

( ** )

}'

* Recent applications of Neumann series to the solution of equations of mixed differences are due to Bateman, Proc. Int. Congress of Math. 1. (Cambridge, 1912), pp. 291—294.

NEUMANN

16*32]

The

SERIES

531

potentialities of the other recurrence formula

were also investigated by Lommel, but the results are not so interesting. As examples of his expansions the reader may notice that

These results were given by Lommel, though his formulae contain numerical

The Neumann

16*32.

The sum

errors.

summed by Kapteyn.

series

of the series 00

2 nJn (z)Jn (a) n=l is

expressible as the integral r

iz 2

a

Jo

Mz-v) j(a _ v)dv

.

z-v

* the sums of the alternate terms of the series have been expressed by Kapteyn in the form of integrals

from which this integral may be deduced, and may be deduced from this integral.

conversely Kapteyn's formulae

We

proceed to establish this result by a simplified form of Kapteyn's

methods.

The

series

may be

written in the formf r (0+ >

\z

\z

2 {J„ (z) + J-n+1 (#)} Jn (a) = j^-j\&TT%)

n=0

x exp \\z

f tUh '

where the contours may be taken

Now,

Then,

if

ST

2 <«- +

+ £a

to

exp

('

I**

be the

r—

)

I*-"

(u

Hereto

" *)

circles \u\

+

(—s)} **

ia

= l,\t\ = A>1.

let

m = | (£ — 1/t),

+ m/ = 53

P

we have 3 (?

+

1)

*» {*" (« " D] da - * [J

°

te

(a)

"

^ (a)/

'>-

Archief voor Wiskunde, (2) vn. (1907), pp. 20—25 Proc. Section of ScL, K. Akad. Amsterdam, vn. (1905), pp. 494—500 Kapteyn has subsequently summed other series,

* A'jeuip

van Wet.

-J

Z

M= o

J

—

it

0+)

+

<0+>

~ (i»y /

J

f<

;

;

962—969. The interchange of summation and integration any points on the contours.

ibid. xiv. (1912), pp.

f are

is

permissible so long as

|

tu

|

> 1,

where

t,

u

[CHAP. XVI


532

Therefore, on integration,

I = Ce-^ + 1

f

&

where

C is

independent of

V"

By

a.

«•-"'

{Jo (v)

- J, (v)/t) dv,

o

.'

taking a

= 0, we

see that

C=l/t.

Hence we have

+1 =

-.

f

-a+

*exp fo (*

*) (*

- ^)

{tJ (v)

- J, (*>)} cfoj

dt

J

\[J (z-a + v) + Js(z - a + v)] J

(v) dv,

4 Jo

the majority of the terms having a zero residue at

t

= 0.

Consequently

I that

is

nJ.

(«)

J. (a)

! nJn (z)Jn (a) = l

*

n=l

we

select the

this equation,

we

aJ-li^!lj

(

(a-v)dv.

u

*

J

odd and even parts of the functions of z on each side of

find that

-5[ 4

is

when we

J

Mz ~

2n

Jm ( *> J™ (a) =

(

z-

;

and

\

one of Kapteyn's formulae

Ta „!,

4 \-T=-«

it

n=l is

+ ^^±^1 z+v

/. («

-

•)

*.

)

TUT].

4J

I

*-«

-*

4j

(

*+o

*-»

+

1V *>

i

'

I

follows that

I 2nJm (z)Jm (a) = l f "*

(3)

v)

integrate by parts.

Hence

which

*'

/o (U)

2 (2n+l)J2n+1 (z)Jm+1 (a)

(2)

which

}

to say

(1)

If

T ^"V"/

-|

4

J

+ f \^£P +V Jo[^ '

the other of Kapteyn's results.

^^\j.{t-v)dv, Z—V )

:

NEUMANN

16-4]

The reader should have no when R (v) > 0,

SERIES

proving by similar methods that,

difficulty in

a

S

(4)

{v

n=0

+ n)J^ n {z)Jv+n {ci)=^^\ L

Jv

'0

J

VZ f a

v

(z

Neumann

V) -f~ Z—V JM-v)dv

— v)

The Webb-Kapteyn theory of Neumann

16*4.

533

T

.,

,

series.

have been studied from the standpoint of the theory of by H. A. Webb*. His theory has been developed by Kapteynf and subsequently by Bateman|. The theory is not so important as it appears to be at first sight, because, as the reader will presently realise, it has to deal with functions which must not only behave in a prescribed manner as the variable tends to + oo but must also satisfy an series

functions of real variables

,

intricate integral equation.

In

fact,

the functions which are amenable to the

theory seem to be included in the functions to which the complex theory is applicable, and simple functions have been constructed to which the real variable theory

The

result

is

inapplicable.

on which the theory

Jo

so that, if

is

based

t

is

~

that (§ 13*42)

| 1/(4w

+ 2)

an odd function f{x) admits of an expansion of

(

the'

m=n

),

type

00

2

f(x)*=

a*n+i

JM+1 0),

n=0

and

if

term-by-term integration

/,o

We

permissible,

we have

/»/„„<= a=r2

are therefore led to consider the possibility that

/(«) =

(1)

and we

id

shall

2

(4»

+ 2) Jm+1 (*)

f"

J -^^f{t) dt

establish the truth of this expansion

conditions (I)

The integral Jo

exists

and

is absolutely convergent. * Messenger, xxxin. (1904), p. 55.

t Messenger, xxxv. (1906), pp. 122—125. X Messenger, xxxyi. (1907), pp. 31—37.

;

under the following

V [CHAP. XVI


534

The function f(t) has a continuous differential of the variable which do not exceed x.

(II)

coefficient

for

all

positive values

(III)

The junction f(t)

^ *&> {f(v +

-

2/' (0

(2) t

does not exceed

We

now proceed

first

+ f(y - 1)} dv

x.

to

sum

the series

(4n

+ 2)^+1

S= £ and we

t)

v

Jo

when

equation

satisfies the

(^r% ®/(0
interchange the order of summation and integration. It

is

evident

that 00

2 w=0

Jm+1 (x) [Jm (t) + J2n+2 (t)}

converges uniformly with respect to since

|

Jm (01^1

and

2 Jm +i (#) \

an absolutely convergent by §16-32,

t

I

we may

integral,

effect

+ 2) /zw+1 {x) J* n+1 (t)

(4w

l?o

/<>

=

}

/(0 Jo|"T=ir +

2.lo

1 f* f

I fI

- j J,

Jo (*

(t

- v)

- V)

J,

f*

^

(t

(x

f

-

v)

Jo

f(v - 1)

\ Jo

+ v)\

,

+ V) +f(t - V)}

f

^P dtdv. t

7 V (x - v)f(v ;

o

U

=

I" f Jo (u oJ o

1)

^P dtdv *

- t)f(x - u)^P-dtdu t

J

=

I

J± (u)f(x

— u) du

Jo

=f(x)— first

dtdv

transform the last integral by using §12*2, and then

Jo

The

T

-j^lMx-iddvdt

{f(t

V

+

We now

(unbounded) values of t, Hence, since f(t) possesses the interchange, and then,

for positive

convergent.

X/(0

8=

=

is

transformation

is effected

J^{u)f'(x — u)du. Jo by writing

v=x + 1 — u.

we have*

NEUMANN

16*4]

SERIES

535

Hence

I

(3)

(4i»

»=0

Now

+ 2) Jm+1 (x)

J

dt -^^-f{t) c

" f .0

write

Ji(0 Z so that

F(v)

is

{/(«+*)+/<*-»)}* 3

'

a continuous function of

m an absolutely

since Ji(t)/t has

v,

convergent integral.

we are to have S=f(x) when x has any value we must have

If then as (0, X),

in such an interval

X

f Jo

throughout this interval

;

J (x-v)F(v)dv = 0,

and, differentiating with respect to x,

F (x) = f^ (x-v)F (v) dv. Jo

Since \Ji(x—v)\^ 1/V2,

follows

it

by induction from

this equation, since

\F(x)\^±fjF(v)\dv,

F{xy

* hat

^W^V

\

where

A

is

the upper bound of

|

F(x)

|

in the interval

and n

is

any positive

integer.

If

we make n -»

x>

,

it is

clear that

F(x) =

and so the

0,

necessity of equation

(2) is established.

The from

sufficiency of

equation (2) for the truth of the expansion*

is

evident

(3).

been pointed out by Kapteyn that the function sin (x cosec a) is which equation (2) is not satisfied and Bateman has consequently endeavoured to determine general criteria for functions which satisfy equation (2); but no simple criteria have, as yet, been discovered. It has

one

for

;

[Note.

separately

If f{x) is not

;

and then

an odd function, we expand the two odd functions

it is

easy to prove, by rearranging the second expansion, that

f{x)=7, a^Jn

{x),

n=0

a° =

where

2 /

a»= n J

f(x)

Ji(\ x dx

_J(x) Jn (*)

\)

r^j

->

(«

>

provided that the appropriate integral equations are satisfied.] *

The

sufficiency (but not the necessity) of the equation

was proved by Kapteyn.

> 0),

[CHAP. XVI


536 16*41.

of reduced functions.

Cailler's theory

The Webb-Kapteyn theory of Neumann

series

which has just been ex-

pounded has several points of contact with a theory due to Cailler*. theory

is

based on Borel's integral connecting a pair of functions.

f(z)=X

cn z

Thus,

This if

n ,

then the function f(z)n defined by the series 00

f(z) B =

supposed convergent

for sufficiently

2 «=0

cn

.n\zn

,

small values of \z\ %

may be

represented

by the integral jo

The

function f(z)i,

If the

Neumann

maybe termed

series

the reduced function (la reduite) oif(z).

which represents f(z)

is

f(z)= 2 an Jn (z), tt=0

then we have, formally, f(z) R

= 2

an

n=0

e~*

\

*

1

2,

Now

Jn (tz) dt

-'

an

|V(l+^)-l) w

put

-t

z

and we see that 1

+?

.(

2£

Bf/G^).-i^Hence,

if

the

Neumann

series for

f(z)

is

2 an Jn (z),

then the generating

tt=0

function of

2 an £n

is

«=o

i-r / u-r/«' provided that this function

More

is

analytic near the origin.

generally, iff(z) has a branch-point near the origin of such a nature

that (1)

/(*)-S

an J,+n (z),

then (2 >

.l-^-rÂi^.),* Mini, de la Soc. de Phys. de Geneve, xxxit. (1902—1905), pp.

295—368.

.

LOMMEL'S FUNCTIONS

16*41, 16*5]

In like manner,

537

if

/(*)- 5 an z'+«Jv+n {z\

(3)

«=o

then

[Note.

ef"smbz=S, an Jn

If

I «„,»=

then

nil

This

which

result,

is

26z ( 1

{z),

+ g2

)

(l-2az-z*?+4b*z*-

immediately deducible from Gainer's theory, was set as a problem

in tlie Mathematical Tripos, 1896.]

Lommel's functions of two

16*5.

Two

variables.

which are of considerable importance in the theory of Diffraction and which are defined by simple series of Neumann's type, have been discussed exhaustively by Lommel * in his great memoirs on Diffraction at a Circular Aperture and Diffraction at a Straight Edge.

The

functions,

functions of integral order

Vn (w, z), are

denoted by the symbols

n,

Un

(to,

z)

and

defined by the equations /w»\ n+2*»

00

(1)

Un (W,z) = 2 (-W7)

Jn +*m(z),

(2)

Vn (W, Z)=X (-r (-) \Z I »n=0

J-n-»» (*>

It is easy to see from § 2"22 (3) that

Un (w,z)-V_l+2 (W,Z)= 2 (-r(-) \Z m=-<*>

(3)

Jn^n{z)

I

==C08

tfn+i (w, z)

(4)

(w {2

-

+

z2

mr\

2w-^-)^

- F_M+1 (w, z) = sin (^ +

~ -^

J

The last equation may be derived from the preceding equation by replacing n by n + 1. There

is

no

difficulty in

extending (1) to define functions of non-integral

order ; for unrestricted values of v

U.(w,z)= 2

(5)

„i=0 * Abh. der math. phys. Classe der

we

write

(-)**(

7) \zj

•/,+»»(*).

Akad. der Wist. (Miinchen), xv. (1886), pp. 229—328, and the definition of Vn {w, z) in it differs from that adopted subsequently by the factor ( - l) m Much of Lommel's work is reproduced by J. Walker, The Analytical Theory of Light (Cambridge, 1904). The occurrence of Lommel's functions in a different physical problem has been noticed by Pocklington, Nature, lxxi. (1905), pp. 607—608. 529

—664.

The

first

memoir

k. b.

deals with functions of integral order

;

.

W.

B. F.

18


538

[CHAP. XVI

The expression on the factor

w*

is

right is an integral function of removed) an integral function of w.

z,

and (when the

The corresponding generalisation of (2) gives a series which converges when v is an integer. And consequently it is convenient to define Vp (w, z) for unrestricted values of v by means of the natural generalisation of

only

(3),

namely F„ (w,

(6)

= cos

z)

+ (f + 1^

y ) + u-»+*

w

(

>

z )-

It is evident that (7)

U

(8)

V, (w, z)

As

9

+ Uv+i (w, *) 7

(w, z)

(z),

§ 2-22 that

V (z, £) = \

(9)

U. (z, z) -

(10)

U.iz, z) = -

and hence, by (7) and

v

+ Vv+2 (w, z) = (^y ,/_„ (z).

we deduce from

special formulae,

J

(™J

1

(z)

{

+ cos z},

(z,z) = I sin z;

(8), 1

U^{z,z)= Vm (z,z) = $(-)»\cosz- 2

(11)

(12)

M

Um+1 (z, z) = - V

provided that n

>1

in (11),

= \ (-)» {sin z -

(z, z)

and

(-)»^4,W}

»>0

"% (-)«•

I

Wl J^

+l (z)\

;

in (12).

It is also to be observed that, as a generalisation of these formulae,

Vn (w,z) = (-)»Un (z*/W> z).

(13)

The

2 m=0 CQS

functions

m0.Jm (z),

2

+ £) 6 Jm +

sin (to

.

in=0

1

(«),

which are closely associated with Lommel's functions, have been studied by Kapteyn, Proc. Section of Set., K. Akad. van Wet. te Amsterdam, vn. (1905), pp. 375 376, and by Hargreavcs, Phil. Mag. (6) xxxvi. (1918), pp. 191—199, respectively.

—

16*51.

The

differential equations for

It is evident

by differentiating

LommeVs functions of two that

§ 16'5(1)

Û (w,z) = --U

(1)

¥+1

v

(w

>

z),

and hence d2

^ and consequently

JJ V

(w, z)

z* = -

1

U. +2 (w,

z)

--

Uv+l (w, z),

variables.

lommel's FUNCTIONS

16*51, 16*52] It

is

now evident

U¥ (w, z)

that

539

a particular integral of the equation

is

Since the complementary function of this equation

is

2

Z* z 4- B sin ^— A cos s—

where is

A and B are

independent of

also a particular integral.

{6)

z, it is

Therefore

+

dz->-;dz-

,

clear from §

V

v

165 (6)

that

J-» +z(z)

w>-\w)

-

These equations are due to Lommel, Miinchener Abh. xv. (1886), pp. 561

16*52.

We

F_ K+2 (w, z)

(w, z) is a particular integral of

—563.

Recurrence formulae for LomnieVs functions of two variables.

have just obtained one recurrence formula

Û (w,z) = -Û

(1)

¥+1

y

for

U

y

(w, z),

namely

(w,z).

To obtain other formulae, we observe that

~ U.

OW

(w, z)

= 2

m -Q

(-)"• (v

+ 2m) («/*)*+*« J, +2m (z)/z

=4 S (-y*(w/sy^*{j +m* »»=0 w

and

(*)+j.4 Jlm¥l {M)},

l

so

2

(2)

L Uv

(w> z)

Again, by differentiating § 16*5 (6)

we deduce

*

u +i (w "

>

z)

-

that

^Mttf.sH-^F^M;,*),

(3)

2

(4)

If

ww)

= u~* {w> z)

AV

¥

now we take

(w, z)

w = cz,

=

F„+1 (w, z) + (z/wf F^» (w,

where

c is constant,

(5)

2Û (cz,z) = cU -

(6)

2

¥

y

1

we deduce

z).

that

(cz,z)-(l/c)U¥+1 (cz,z),

jz V„ {cz, z) = cVy+1 (cz, z)

- (1/c) F_, (cz, z).

Hence we get 4

d*

d?

U

'

cz,' Z CZ z) ^

= C* Uv^

+ 1 / c2> U"+* cz = c'«/„_, (z) + er*J, (z) -(c + 1/c)2 U, (cz, z) CZ Z)~ '

2U"

cz> *)

<

>

z)


540

Hence

it

Uv {cz, z),

follows that

[CHAP. XVI

and similarly V_ v+2 (cz,

z),

are particular

integrals of the equation

^+(c+iy^cj . (z)+c^Mz).

(7)

The

v

particular case in which z

U

(8)

is

= 2

{w, 0)

v

=

2

of

some K

*;/

V

and so £7,, (w, 0) and F_„ +2 («/, 0) are expressible of one variable by the equations

V. y+2 (w ,0)

HO) Of these The

results, (l}r— (8)

Ji w

1X

^

a

we have

;

,

in term's of

S

l{

interest

Lommel's functions

\

were given in Lommel's memoir.

following formulae, valid

when n

a positive integer (zero included),

is

should be noticed

Um {w, 0) = (-)»

(11)

#-„(w, 0)

(13) it

^

\ZZ

,

,

= cos(§u/ + |rwr).

follows that.

V

(14)

(w,0)

v.m (w,

(i5)

V^

(16)

16*53.

\w - 2

Um+l (», 0) = (-)» [sin iw J£fcgfc»gp]

(12)

Hence

cos

o)

{w, 0)

= l, Vn+1 {w,0) = 0, = (-)» m=Q

= (-)» 2

(

'

"7

,

\&Vn>)\

*

/J*"L

•

Integral representations of Lommel's functions.

The formulae (1)

tf,,

(w, z)

= -^\ Z

~ «> v

(2)

U^ (w, z) = *

«/„_ x {zt)

.

cos

[\w (1

- *')}

.

sin

{\w (1

- i*)}

.

t"dt,

.0 r1

J^

x

{zt)

.

*»
Jo

which are valid when R (v) > 0, may be verified immediately by expanding the integrands in powers of w and then using the result of § 12*11 (1) in

lommel's functions

16*53]

performing term-by-term integrations. replaced by the equations

^( w^) = - 27

(3)

_

r

_^_J

wp

which the phase of It is clear that,

—t

when

this formula

—

The

^(-*0.rin{i«(l-«-)}.(-^d«,

— tt

f

1

describes the contour.

J^ (zt) exp

we can obtain

w

and

± \iw (1 - **)} p dt. .

integral representations of

exp {+ \iw

- *>)}

(1

.

*•>

v

(w, z)

<&

w and

if

when

R (v) >

and at the upper

z are restricted to be positive.

evaluate the last integral, swing the contour round until

arg«= +

with the ray

V

Let us consider

z.

/„_! (««)

{

JO

when R(v)<§,

To

t

J

integral converges at the lower limit

limit

to ir as

R (v) > 0,

valid for positive values of

*

may be

Jî-MQ.cnllwil-ty.i-tYdt,

increases from

—

By modifying

they

r(0+)

Uv (w, z) ± i Uv+1 (w, z) = Zw"

(5)

v,

i

^M- g^^^

(4)

in

For other values of

_ I

541

it

coincides

ambiguity in sign being determined by the ambiguity in sign in the integral; such a modification in the contour is permissible by Jordan's lemma.

When we we

±tt, this

expand the new integral in ascending powers of

z,

as in § 13*3,

find that

— w"

f°°

«/,_, (zt)

exp ± \iw (1 {

- 1*}} V dt .

J o

^. mir ;+ m)

2

(

2"

that

is

(«)

^

Vw)

"exp(?

}M)A

'

to say

£ J/.-, (zt) exp {+ |«w (1 - *)}

;•

When we z

.to

2 * +2W

'

J,

> 0, and

.

combine the results contained in

< R (v) <

§,

r
(

± *? ±

this formula,

we

^ + ^)

see that, if

then

°5

(7)

(8)

^J

/^^)cosii W;(l-^}.^^ = cos(|

«

+ ^ -^), ;

^^(*«)«ntt«(l-lO}.ri«-rin(|+^-^).

w > 0,

:

,


542

from (1) and (2) combined with

It follows at once

V t_„(w,z) =

(9)

Fi_

(10)

(w, z)

-— z

J,-t (zt) sin

Since convergence at the origin

Vv (w, *) - -

(11)

V„

(12)

(to,

z)

we

is

now

{

\w (1 - 1*)} V dt. .

unnecessary, the theory of analytic

R (v) > 0.

see that

z v— 1 -

(1

- f))

—

«A_, (jf) sin {| W (1

- *>)}

£

J*

fao

w^

«/,_, {zt) cos

{\w

The

w

and

= - ^_

-2

,

ji

ji

provided that

that

1

=- ™

notation,

§ 16*5 (6)

J^(st)cM{tw{l-t)}.fdt, .'

continuation enables us to remove the restriction

Changing the

[CHAP. XVI

are positive

and

R (v) > \.

following special formulae are worth mention

Um (z,z) ^

(13)

-

^

p

j^

{zt) cos

f

«/*»-*

(*0 sin {|* (1

(1 __

<9n

<8)}

- <»)}

^

2n~ 1

.

«

.'o

ff»+i(*,*) (14)

=p

^

=

Jm-i (zt) sin ft* (1 - t*)}

f !

.

Again, from

(6),

we

(ft) C08

n, (1 _

.

|» «

.

*•»

^ eft.

o

see that

/f J«(rf>«p(T- r ).«'*- 1? r«^(± S5 T- r ).

(15)

and, in particular,

fV.w^WI ^). 8

(16)' v

The

sin \ 2 /

Jo last results should

be comparod with § 13 '3

wcos\2iW ;

see also Hardy, Trans. Cumb. Phil.

Soc. xxi. (1912), pp. 10, 11.

The formulae of this section (with the exception of the contour integrals) are found in one or other of Looomel's two memoirs. 16*54.

be

Lommel's reciprocation formulae.

It is evident from §

165 (13)

that functions of the type

closely connected with functions of the type integer.

all to

U

¥

U

¥

(z*/w, z) are

(w, z) provided that v is an

LOMMEL'S FUNCTIONS

16'54, 16-55]

To

appreciate the significance of such relations observe that

^ [cos (*«*) U

v

.

{±

,

zt)

+ sin (but*) Uv+1 (£ .

= sin (}«*) T- w * tf„ .

= - zJ, (zt) sin (\wt*) On (1)

543

integration

~

^

f

we

,

ztj

(^ *) wtU¥^ (£ .

ztj]

,

,

(wt/z) 1 -".

find that

JAzt) sin (Iwt^t'-'dt a

= — cos L.^(^,)-sinK.^ +1 (l

,,)

+

^g,

),

and, similarly, (2)

£=r

f\

x

(zt)

cos (\wt*)

*- dt

= smiw.Uv (^,z^-co8^wM¥+1 (^,zÛ

lf+1

Hence ( 3)

w

it

follows that

¥

f1

z==;

Ji- (^) cos [\w (1 - *

2

)}

.

t"

(^v

,oy

dt

j

"*^^(S'*) +8b * w -^-(S*°) +C08 *

tc '-

Cr

»-*(^

)'

and (4)

^

J

J,- r (zt) sin

and these integrals

differ

{£

w (1 - *»)] V dt .

from the corresponding integrals of the preceding

section only in the sign of the order of the Bessel function.

The reader will find some additional formulae concerning Lommel's functions in a paper by Schafbeitlin, Berliner Sitzungsberichte, vm. (1909), pp. 62—67. 16*55.

Some

Pseudo-addition formulae for functions of orders % and #.

very curious formulae have been obtained by Lommel, which connect

functions of the type TJV (w, z) with functions of the same type in which the second variable is zero, provided that v is equal to \ or

f

When we

write v

TJk (w, z)

-\

in § 16-53 (5),

± iU% (w,

z)

we

get

= (~) j

exp [±

i

(\w-zt-

+ (Â j

exp {±

i

($w+zt- $wt*)}

\wt*)} dt

dt.

U

'


544

Now

[CHAP. XVI

write

(W

z)2

4-

and we

(w

e

2w

— zf 2w

'

find that

U

h

n = eT* (-) f ,4

± iU%

(w, z)

(w, z)

- f )} d£

exp {±
eX P (± 8i + e±iz (-)* \7T/ Jf-v{z*/(2to8)>

(-

1

~ P)l d &

in the respective integrals, awci V°"> a/8 are *° oe interpreted by the conventions*

w+z = ~77o —\> V(2w)

.

V °"

/S>

v° v

— = w z

'

V(2w)

Hence we have

U

± i U$ (w, *)

(w, z)

= \e* (2a, 0) + %Ut (2
{

r

_eW-J

exp (± eri(l- £»)}<#

I

+ e ±fe (±)

exp {+

-

ft (1

£»)}

<*£

J

When we take o-f2 and Sf

2

as

new

variables in the last two integrals respectively,

these integrals are seen to cancel

;

and so we have the two

results

combined

in the formula

U

(1)

k

(w, z)

± iU§ (w, z) - te**{Ui(2*.

0)

±iUt {2*, 0)} ,

+ *«±*{17»(2S,O)±»0 I (2M)}, and, as a corollary,

U

(2)

k

{g,s)±xU% (s,s)-l*r*[Ui (4*,0)±iU% (4M

t

O)}.

These formulae are due to Lommel, Miinchener Abh. xv. (1886), pp. 601 —605; they are reproduced by Walker, The Analytical Theory of Light (Cambridge, 1904), pp. 401—402. 16*56.

Fresnel's integrals.

It is easy to see

from

0-h.i <».

0)

§

1653

(1)

= 2y_^

and

(y)

(2) that,

when R(v) > 0,

£ I-* sin ft w (1 -

I-))
so that (

X)

(2)

ovJrv

<2 " \

\*

« _1 v 2" i (v)Jo I

*

_1

f

t

2 " -1

cos

$ wt^ d ^^(w, 0) cos J w +

sin (A wt2 )

*

eft

= TJV (w,

0) sin

J7„ +1

(w, 0) sin | «/,

$w - Uy+l (w,

These are not the same as the conventions used by Lommel.

0) cos \w.

LOMMEL'S FUNCTIONS

16*56] If

we

take v

=£

545

and modify the notation by writing

\w = z — \m# we t

see that (3)

ôo.(iÂ-ij;Q*oo.«i«.yV^(«)A - [ Uk (2z, 0) cos z + U% (2z, 0) sin z]/
and (4)

^^(IÂ-l^^rinlA-l/V^)* = [ 0"j (2s,

= i ~ [ ^* (2*,

We

z-Ui (2s, 0) cos s]/V2 0) cos * - V% (2s, 0) sin s]/V2.

0) sin

thus obtain ascending series and asymptotic expansions for Fresnel's

integrals* I cos ($irta )dt,

(

fsin(A7r*2 ) dt.

J0

(2)

Jo

The ascending series, originally given by Knockenhauer, Ann. der Physik und Chemie, xu. (1837), p. 104, are readily derived from the ^-series, namely ,, ( ,,„)=

<»>

(^f _ ^. + _^__....} 1

>

i

while the asymptotic expansions, due to Cauchy, Comptes Rendu*, xv. (1842), pp. 554, 573, are derived with equal ease from the K-series, namely

W,--(^-L-

(8)

+ ...}.

Tables of Fresnel's integrals were constructed by Gilbert, Mem. couronnees de I' Acad. pp 1—52, and Lindstedt, Ann. der Physik und Chemie, xvn. (1882), p. 720; and by Lommel in his second memoir.

R. des Set. de BruxeUes, xxxi. (1863), (3)

Lommel has given

various representations of Fresnel's integrals by series

which are special cases of the formulae f

(V («)«ft-2

(9)

S Jy+m+1 (z)

r

JO

n=0

=

Z n+l

N

n~o(vi-l)(»

2 ('JAW" »=o

(10)

("

+

l)(»

+

3).,.( y

Mim.

de I'Acad. des Sei. v. (1818), p. 339.

t It is supposed in (9) that

R (v) > - 1.

l)'

+ 2»-l) jr ¥+n (*).

zn

Jz *

J^ n (z)

+ 'd)...(v+2n +

[Oeuvreg,

i.

(1866), p. 176.]


546

These are readily

Lommel (11)

by

verified

[CHAP. XVI

Other formulae

differentiation.

also

due

to

are

f'j-i(t)di

=

JO

X (-)»J n + i(i*)]

2*COsi* [

t

\_n=0

J

+ » 8 ini#[j (-)»^ +| (i,)]

i

o

(12)

JV (*)cft-2i«n|#^2(--)»^ t

+l (i#)]

-2looBt#[2 (-)-^ +| (J*)]. o

These may also be 16*57.

The

verified

by

differentiation.

Hardy' 8 integrals for Lommel's functions.

fact that the integrals f

•

(

cos

\

(

Jo

.

at

V

f*

dt

— tb\ = t/l±P'

/

.

,

Bin [at

/

,

Jo

\

are expressible in terms of elementary functions*

b\

—

tdt tdt

— r) * tjl±

when a and

b are positive

suggested to Hardy f the consideration of the integrals /••

/

cos j

and he found them

.

r*

dt

b\

r + *-ir±F ,

.

/

b\

{

t)T±t»

, at+

ain ]

tdt

be expressible in terms of Lommel's functions of two and unity respectively. This discovery is important because the majority of the integrals representing such functions contain Bessel functions under the integral sign. to

variables of orders zero

If 1/t be written in place of

<> ( 2)

and

t,

it is

i>(-+Di!*-*J>Mr!». f>(- + »)

since,

by

§

*|L- ± j;* (- +

-. ) i

6*13 (3),

it is sufficient to

We now

»)*-/xtt+?

l

i

l-fjsin (at

+

^

=J

{2
confine our attention to the case in which b

< a.

write c

*

seen that

= V(6/a),

x

=

2
Hardy, Quarterly Journal, xxxn. (1901), p. 374.

that the integrals have their principal values. t Me»$enger, xxxvm. (1909), pp. 129 132.

—

6

= $(l-c*)/c,

When

the lower sign

is

taken

it is

supposed

LOMMEL'S FUNCTIONS

16*57]

and then the substitutions /

f"

and cosh u =» r shew that

f" cos (x cosh w) rfw

dt

b\

.

— ceu

t

547

= /," °" <* °° 8h M)

{<»«+l/(<«">

+

rfa

5=+17J5^j(

_ a 4- 6 P° cos (at) t<2t ~~x~Ji (^ + t )V(t»-1)* .

2

Now

consider

e^rcfr 27^^J ^ (^+T•)^/(Ta -l) 1

where it,

T

,

a contour consisting of the real axis and a large semicircle above

is

the real axis being indented at t =* ±

The only

1.

pole of the integrand inside the contour

T (0*

2-iriJ

it

f

+ t*)V(t*-1)

is

at

and so

id,

2tV(^+l)'

As the radius of the large semicircle tends to infinity, the integral round tends to zero by Jordan's lemma, and hence /"•

_ 7re~*' 2J_ 1 (^ + T )V(l-T«)~2V(^ + l)' 1 f1

cos(arr).TdT

.

rdr

a

(^+t«)V(t»-1)

ii

sin (arr)

Thus we have f* ° 08

/

Jo

\

cos

so

&\

«yi

+
c&

tre~

(

°~ 6)

a+b

[* sin

(a;

2^" Jo

2

cos 0»

)

.

cos

+ cosa

d '

tf>

4*/*

^rn^ = rT7^ ccos ^" c8cos3* + c5cos5 ^~'-^'

But and

.

we

find that

cos(a*

(3)

Jo

Similarly

it is

+

-)— =-^

found that*

\ tdt F Bn /(* + h-)_,

f±\ (4)

wl^Jî*).

•

*

.

Jo

™r<»-6

>

g

2 _ r TTt^J^z), .

.

and °°

(5)

/ b\ dt Pjr°° cos^ -^=j7rsin(a + 6)-7r 2 (-^(J—V^*), + -J r fl

(6)

Pj sin^ + o

*

The

-J I

^

:

i =-j7rcos(a+6)-7r 2 (-)mi

details of the analysis will be

1

csw «/am

found in Hardy's paper.

(a;).

[CHAP. XVI


548

The

last

two results may be written in the form

^

(7)

1

( Wi

*)

+F

1

(«,«)

— -Pj

00.(^+5)^,

e

0;< w ,«)+f.( W

(8)

provided that 0-<

An

2«)

rr7

2

>

w < «. LommeVs functions.

Integrals of Gilbert's type for

16*58.

(^; +

« in

,*)---pJ o

obvious method of representing U,(w, z) and

V

v

(w, z)

by

integrals

is

-

to substitute the Bessel-Schlafli integral of § 6 2 for each Bessel function in

the appropriate

series.

u. <«,

When which 1 and get J

Now

1

*>

=

I

\

,

thus get

1 5 gi, s o

the contour

=\ w

We

, w,i..^™f (0+) .__/'. -
so chosen that it lies wholly outside the circle on

is

we may change the

order of summation and integration

the residues of the integrand at

iw

f

,

and

"

± \iw are

iz*

_

vrri)

so ir

Making a /o\ (2)

slight

x

i

(0+)

1

f

(WO"

(4

*\dt

change in the notation, we deduce that

m F^*)-Ssi.. x

X

(

and, in this integral, the points

° +)

^/w >"

/<

*V*

f

iTW^ exp V2-2l)T'

+ iw

lie

outside the contour.

In general it is impossible to modify the contour in (2) into the negative half of the real axis taken twice, in consequence of the essential singularity of the integrand at the origin. The exception occurs when z = 0, because then the essential singularity disappears, and /q\

vt

m

(w, 0)

=

*

[

{0+ Ht/v>ye* dt

and hence (4)

V

v

—w

sinvTr fêxp^w-ie-iwtr

provided that R(i/)>0 and a

-

is

Jo

1

2

du,

an acute angle (positive or negative) such

that Ja

+w

+ argw|< J-t.

LOMMEL'S FUNCTIONS

16-58, 16*59]

If v is equal to \ or § , the integral

Formula

(4)

on the right

549

in (4) is called Gilbert's integral*.

was obtained by Lommelt from the formula of

§ 16*53 (11)

by a transforma-

tion of infinite integrals.

From is

(4) it is clear that,

when

w are

and

v

positive,

Vv (w, 0)

has the same sign

as,

and

numerically less than

C uv-\

sinyTr

«

A similar but less exact inequality The reader will when w is positive. 16*59.

From

Vv (w, 0)

c e

-kuw du _

was obtained by Lommel.

Vv (w,

also observe that

*

-r(i- v ).(fror

Jo

0)/sin vir is

w

a positive decreasing function of

Asymptotic expansions of Lommel's functions of two variables.

deduce asymptotic expansions of

Gilbert's integrals it is easy to

and

Uv (w,

0) for large values of \w\; thus, from §16*5(8),

we

have

/_ \m

p-l

where p is any positive integer. We choose and then, by § 16*58 (4), we have

p to be so large

—

=—

(~) p y^+tp (w, 0)

-

7T oo

1+U

/ Jq

that

8

exp ia

*)

M F+2p-lg-J«W^M

-{/;

R (v + 2p) >

l

= 0(w-'-«'), when

\w\

is

large and, as in the similar analysis of § 7*2,

Hence

^

(i) for the values of

w

and

is

»~i, r(i-,->:).(t

<

tt.

i

are both positive, (— ) p

Vv+2p (w,

0) has the same sign

numerically less than sin vtt

r°° 7" w »+2P-lg-i««'^ = ,

lt

7T

Jo

(— )p r(l-i/-2p).(|w;) +^' ,'

so that the remainder after

numerically less than the (p

may be

It

^

|

under consideration.

When v+ 1p and w as,

<

arg w

|

proved in like manner from

V

(2)

p terms in + l)th term.

9

(1) has the

§

same sign

as,

and

is

16*53 (11) that

{»,s)~ 2 (-nzjivy+^J-îz) TO=0

when w

and z are fixed but it is not easy to obtain a simple expression which gives the magnitude and sign of the remainder. |

|

is

large while v

;

* Mint. courorm€e» de VAcad. R. des Sci. de Bruxelles, xxxi. t MUnchener Abh. xv. (1886), pp. 582—585.

(1863), pp. 1

—52.


550

[CHAP. XVI

It is evident from § 16*5 (6) that the corresponding formulae for

Uv (w,

z)

are

U

(3)

v

0)~cos(£w-£i/7r)+ 2

{w,

r=

U

(4)

p

r^-i- 2m) (^w)sw-'+s

(-^(^^-^J^^^).

z)~cos($w + y*/w-$inr)+ I

(w,

'

These results were given by Lommel*, but he did not investigate them in any detail.

Vv {ex, x), when

The asymptotic expansion of x

large

is

and

The dominant term

--

{ex, jc)

N ow, if c> 1 as

,

—

|

/

or 1 and c

while

is fixed,

-^ j

than

—\

is

which shews that

cos

(«t+Jwr- }*-) sin

j

J c* (1

- * )} 2

—

.

the functions £ or (1 — £9 ) ± (a*

increases from 1 to oo

t

is

for general (real) values of v greater

readily derived from § 16*53 (12)

K

v

been investigated by Mayallf.

positive, has

,

and hence

it

+ £ vir — \ ir) vary monotonically may be verified by partial integra-

tions that

V

(5)

p

/

""" 2 \* c2

(cx,x)~{—\ -~_^co8{x +

\vtr-\Tr),

the next term in the asymptotic expansion being If,

has a it

(x~l).

-t2 ) + (xt+ \vir — \tr\ qua function of t, and hence, by the principle of stationary phase (§ 8*2),

however, c
maximum at

1/c;

follows that

(6)

V, (ex, x)

~ -—^ cos

\%x (c

+ - ) + \vtr \

when c—1, the maximum-point

is at one end of the range of and so the expression on the right in (6) must be halved. We consequently have

Finally,

integration,

Vv (x,

(7)

x)

~ \ cos (x + \mr).

This equation, like (5) and (6), has been established on the hypothesis v> — J; the three equations may now be proved for all real values of v by using the recurrence formula § 16'5 (8). that

»

MUnchener Abh.

xv. (1886), pp. 540,

+ Proc. Comb. Phil. Soe.

ix. (1898),

572—573.

pp. 259—269.

CHAPTER XVII KAPTEYN SERIES 17*1.

Any

Definition

of Kapteyn

series.

series of the type oo

2 in

which *

Such

and the

u

+ n)*}, a Kapteyn

coefficients an are constants, is called

series owe, their

investigated,

«nJ*+»{(»

name

to the fact that

they were

qua functions of the complex variable

z,

first

series.

systematically

by Kapteyn f

in

an

important memoir published in 1893. In this memoir Kapteyn examined the question of the possibility of expanding an arbitrary analytic function into such a series, and generally he endeavoured to put the theory of such series into a position similar to that

which was then occupied by

Neumann

series.

Although the properties of Kapteyn series are, in general, of a more recondite character than properties of Neumann series, yet Kapteyn series are of more practical importance; they first made their appearance in the solution of Kepler's problem which

was discovered by Lagrange £ and rediscovered half a century later by Bessel§; and related series are of general occurrence in a class of problems concerning elliptic motion under the inverse square law, of which Kepler's problem may be taken as typical. More recently, in the

hands of Schott|| they have proved to be of frequent occurrence in the

modern theory of Electromagnetic Radiation. The astronomical problems, in which all the variables concerned are real, are of a much more simple analytical character than the problems investigated by Kapteyn; and in order to develop the theory of Kapteyn series in a simple manner, it seems advisable to begin with a description of the series which occur in connexion with elliptic motion.

17 '2. Kepler's problem and allied problems discussed by

The

Bessel.

notation which will be used in this section in the discussion of the

motion in an

ellipse of

a particle under the action of a centre of force at the

focus, attracting the particle according to the inverse square law, is as follows

The semi-major are denoted

by

* It will, for the

t Ann.

tei.

axis,

a, b,

most

and

semi-minor e.

part, be

axis,

and the eccentricity of the

The axes of the assumed that

de VEcole norm. sup.

(8) x.

ellipse are

r is zero.

(1893), pp. 91

—

120.

J Hist, de VAcad. R. de$ Sci. de Berlin, xxv. (1769) [1770], pp. 204—233. pp. 113—138.] § Berliner il

ellipse

taken as coordinate

Abh. 1816—7 [1819], pp. 49—55.

Electromagnetic Radiation (Cambridge, 1912).

[Oeuvrei, in. (1869).

THEORY OP BBSSEL FUNCTIONS

552

axes, the direction of the axis of

The

centre of force.

x being from the centre of the

[CHAP. XVII ellipse to the

taken as origin of polar coordinates, the radius vector to the particle being r, and the true anomaly, namely the angle between the radius vector and the axis of x, being w. The eccentric centre of force

is

anomaly, namely the eccentric angle of the particle on the ellipse, is denoted by E. The time which has elapsed from an instant when the particle was at the positive end of the major axis

M

The mean anomaly

vector would turn in time

way as

to

called

t.

defined as the angle through which the radius

is

if

t

is

the radius vector rotated uniformly in such a

perform complete revolutions in the time

it

actually takes to perform

complete revolutions.

The geometrical

properties of the ellipse supply the equations *

r

(1)

=

a(1 ~ 1

6

T = a (1 - coos E\

+ e cos w

'

from which the equations (2)

tan \w

(3)

sinw

= a/(jz^) tan \E,

= -^1

are deducible

;

sin# = -^j g,— — e cos E 1 + € cos w -

,

and an integrated form of the equations of motion (the anaSecond Law) supplies the equation

lytical expression of Kepler's

M = E-e8mE.

(4)

Kepler's problem

is that of expressing the various coordinates r, w, E, which determine the position of the particle f, in terms of the time t, that is, effectively, in terms of M. It is of course supposed that the variables are real and, since the motion is elliptic (or parabolic, as a limiting case), < et. 1.

The solution of the problem which was effected by Lagrange was of an approximate character, because he calculated only the first few terms in the expansions of

E and

r.

The more complete defines

E as

solution given by Bessel depends on the fact that (4) a continuous increasing function of such that the effect of

M by

M

E by 2ir. It follows that any function of E with limited

increasing

of

2-n- is

to increase

total fluctuation is

a function

M with limited total fluctuation, and so such functions of 2? are expansible

in Fourier series, *

The

qua functions of M.

construction of these equations will be found in any text-book on Astronomy or

Particle. See, e.g. Plummer, Dynamical Astronomy (Cambridge, 1918), Ch. f Kepler himself was concerned with the expression of E in terms of M.

Dynamics of a

m.

KAPTEYN SERIES

17-2]

In particular values of E,

E is

e sin

553

an odd periodic function of M, and

so, for all real

expansible into the Fourier sine-series

it is

go

e sin

E = 2 ^ n sin wi/, n=

An = j

where

—

r

2 " / esin ttJo

E sin nMdM E cos nl/>

2e sin

f

2 H

M7T

L

= _

l

f*

cos

,

v

«^*

W7T/o

Jo

,. d (e sin E) —i-rfjf., nM 71#

—dM—

p cos nil ir dE-dM ,,, »-«*

2

tvt

n7rj

=—

/'"

2

nM.dE

cos

mr

'

= - Jn (ne). n

Hence

it

follows that

E~M+ n2= W-Jn (ne) sin nM,

(5)

l

and

this result gives the complete analytical solution of Kepler's

problem con-

The series on the right is a Kapteyn series which converges rapidly when e < 1, and it is still convergent when e = 1 cf. §§ 8*4, 8*42. cerning the eccentric anomaly.

;

The

radius vector

similarly expansible as a cosine series, thus

is

/v%

GO

-

=B + 2

1 B = TT-

where

=-

/*"

(*

Bn cos nM, — e cos£) dM

Jo

[' (1-e cos

E)*dE

it J o

= l + £e*, while,

when n ± 0, 2 f*

Bn = — IT

i

(1

— e cos E) cos nM dM

Jo

_ r»(i-««.j)..«if[»7T

L

=

2c f*

mr

sin

Jo

_2f- sin nJf + wiry o

Esin (nE - ne sin ^) d-E

J©

— *A'(~X so that (6)

O,

= 1 + ie - 2 8

n-l

— ft

«/»'(n
cos

nM.

rf(^g) dJf <3

"o


554

[CHAP. XVII

The expansion

w - M is

of the true anomaly is derived from the consideration that an odd periodic function of M, and so 00

w — M=*% Cn sin nM, n=l

where

w

2

[

it

Jo

Cn = -

(w — M) sin nMdM

,[_ 8(«,-JQc,Un nrr

L

r-

_>

.(*»

nnJa

Jo

\d>M

^ /

-j-fidE = --[o cosnJf da .

=

-e

2 V(l

2

W7r

Bn

.

The

- nc sin E ) „ — ecos.E'

cos (nff 1

J

,

not such a simple transcendent as the coefficients A n most effective method of evaluating it is due to Bessel*, who

This expression

and

f»

)

is

used the expansion

-e ) 1 — € COS i? 2

V(l

+ 2fco8E y

1

/i 1

the substitution,

71

17*21.

+•...,

•

where

On making

2f* ^ cos2ff + 2/»cos3ff

we

+

V(l

-

«)

find at once that

»=i

L

J

Expansions associated with the Kepler-Bessel expansions.

A large class of expressions associated with the radius vector, true anomaly and eccentric anomaly, are expansible in series of much the same type as those just discussed. Such series have been investigated in a systematic manner by Herzf, and we shall now state a few of the more important of them; they are all obtainable by Fourier's rule, and it seems unnecessary to write out in detail the analysis, which the reader will easily construct for himself. First,

we have r cos

w «= x — ae =

a(l-(?)-r €

so that

rcosw =

/i\

(1)

a

- fa3 e + ?. i

w=1

v 2 ' - r// n (ne) cos

n

%g nM,

J

and next /aN (2)

rsinw a

2 r / \ b r = V(l— vr — «"> v2 -J = -smJs n (ne)smnM, •

.

a

e

• Berliner

n =i

Abh. 1824 [1826], p. 42. Nneh. era. (1884), col. 17 28. Various expansions bad also been given by Plana, Mem. delta R. Acead. delle Sei. di Torino, (2) x. (1849), pp. 249—332. In connexion with their convergence, see Cauchy, Compte$ Rendu*, xviu. (1844), pp. 625—643. [Oeuvres, (1) vin. (1893), pp. 168—188.]

t

A$tr.

—

KAPTEYN SERIES

17-21, 17*22]

while cos

(3)

Next,

m is

if

mE = m

2 -

n

*=i sin

(5)

2

- «/»'(ne) cos nM.

„«i

n

{«/»_»» (ne)

—

any positive integer*,

cos

(4)

—

E— —ae = - \e +

565

" 1 »«a n

mE — m 2

The expansion

of a/r

The expansions

(ne)

+ «/»+« (ne)} sin nM.

= 1+2 2 Jn (ne)coanM.

of cos w and sin w are

w=—e+

1

— 6*

*

2 2Jn (ne) cos wif,

(7)

cos

(8)

sinw^V^-e*) 2 2/n'(ne)sinnJlf.

The expansions

nM,

(ne)} cos

particularly simple, namely,

is

-

(6)

{«/"„_«»

t7"w+OT

of cos w/r*, sin wjr* are of a simple form,

(9)

-cosw« 2

(10)

^ s nM;= ^lzi!} X 2nJn (ne)smnM.

2n«/n'(ne) cos

namely

nM,

i

[Note. It is pointed out by Plummer, Dynamical Astronomy (Cambridge, 1918), p. 39, that these are readily derived from the Cartesian equations of motion in the form

d*x dJt*

combined with

(1)

and

+

a3 cos to r*

cPy

3¥~> +

"**

a3 sin y.

w

-Ot

(2).]

17*22. Sums of special Kcupteyn series. The reader will observe that, in the case of the expansions of even functions of M, the results simplify when we take the particle to be at one of the ends

of the major axis, because then the three anomalies are while the radius vector of the last section

Herz

is

we thus obtain the

— e) or to

a (1

all

equal to

+ e). From

or to

ir,

the results

following formulae, which were given by

in the paper already quoted:

n

n=i

5- = 2 I* 1 ~ * n=l

(2)

(3)

(T

* It is seen

insertion of col. 69.

equal to a (1

i^ - £»

from

(3) that,

a constant

[Gen.

Jn (ne),

term.

jr.' (ne),

n

n=i

(-)«-> Jn i^-S + 1

e

(ne),

n=i

^L = J

(-)-» « J.' (ne).

m is equal to 1, the expansion (4) has to be modified by the These two formulae were given by Jaoobi, Attr. Naeh. xxvni. (1849),

when

Math. Werke, tu.

(1891), p. 149.]


556

More generally we

(_)»«-i r r d? d* m

i

__^

j-j-j

dM

=

oo

» {ne)

(5)

Since

~j

J

lM^TZ7^\ M-_rnl nimJ

2

=_

if

= 2

-r^, the expressions r

l-ecos# dE*

we regard

e

and

M as

>

x

~i

(

- )"-

1

on the

be calculated for any positive integral value of m, with Again,

XVn

differentiating § 17*21 (6) that

\m r (Ism

L (

< 4)

by

find

[CHAP.

n«» J» (we).

left in (4) v /

and

(5) v / can

sufficient labour.

the independent variables,

it

is

easily

seen that 1

^_ ~~

E (1 - e cos E))

de\isin

E (1 — e cos 2?)

sin

#(1 -ecosE? 911

(1

by

_a_

therefore, if

°°

1

{

)

we

we

'

as the lower limit,

1

=-

=rp:

differentiate with respect to

cosi£

,_>

v

w= i

integrate with e =

1

If

°°

%

~~

~;~a jit sin'ilf

n

sin

nM

M, we

[' .

j

Jo

„=i

cosJlf

„;«a t? /i ~TZZrW\*a sin»i£(l-ecos#)

/

Jn (nx) doc.

find that €

"*"

n (l-ecos#)» ft

OB

=2 2

na cos n3f

.

last

/

J» (nx) doc.

Jo

»=i

The

two expansions do not appear to have been published previously.

Expressions resembling those on the right of

(6)

and

(7)

have occurred in the researches

of Schott, Electromagnetic Radiation (Cambridge, 1912) passim.

Thus, as cases of (8)

The

E " =9e

i

— = - -— = - 2 sm.£(l-ecos2?) sinif -.

'

sin °x "

-ecos^'

oe (sin 2£ (1—6 cos E))

(6) v

4- c e

§ 17'21 (6)

d

and

Z? cos w ° " ^

-ecos^)3

9ilf 1

so that,

87

esini?

_ =

dE

cosE sin 8

(4)

JiW2j^ last of these

and (2ne)

may be

(5),

Schott proved

(ibid. p.

110) that

= ^lL±^4j J^JV^Sii*)**.-^^. obtained by taking if equal to

and w

in (7).

KAPTEYN SEBIES

17 23]

Meissel's expansions of Kapteyn's type.

17*23.

Two

extremely interesting 9

(1)

n*

2

series,

namely

a

Z JmVne) n ~i

V

557

e

+?

+ (!*

l"+f

g «f— •»K2»-1)«) ^ nti (2«-l^ + f

€*?

+ ?)(* + £)

e'f

€ l«

f

+ £»

"(l»

+

f)(8«

+ p)

ft

+

+

have been stated by Meissel * who deduced various consequences from them it is to be supposed at present f that < e $ 1, and f; is real.

The simplest method pansions

of procedure to adopt in establishing these ex-

to take the Fourier series J

is

» cos2yiif _ 2 wti n +

(which to

is

valid

when

M^

^

7r

cosh

~

£*

it),

(ir

— 1M) £

replace

1 2|"a

2£sinh7r£

'

MbyE—e sin E, and integrate from


ir.

*

^

(2ne)

__ 1

f v cosh (tt - 2E + 2e sin E) f

l),™

fsinh^ tt cosh (20 + 2ecos $)£

r}^

[

.Vtf+F~Wo

I

__!(**

(

-TrJ.^t 2

=

/"**

"H

IsinhTr^

cosh 2gfl cosh (2eg cos .

_ IV

fl)

[

wJo

Now

tt

1)

the last expression

|sinh7r|

J is

£*J

an even integral function of

e,

and hence

it is

expansible in the form§ 2>m+ifim-t6»m rj» CQS2m

«o

TO =i

(2w)!

Jo

COsh 2 jff

sinh7r|

r

,c

by a formula due

to

mtir(m + i + if)r(i» + i-»f) Cauchy||; and the truth of Meissel's first formula

'

»

is

evident.

The second formula

| w=l

follows in like

cos (2n

-1)M

manner from the Fourier

= (2n-l)*+f ~

tt

- M)% 4fcosh^

series

sinh (|tt

'

—

* Astr.

Nach. cxxx. (1892), col. 363 368. t The extension to complex variables is made in § 17*31. % See Legendre, Exercices de Gale. Int. n. (Paris, 1817), p. 166. § It is easy to see that the term independent of e vanishes. ||

M€m.sur

les

integrates difiniet (Paris, 1825), p. 40.

Cf.

Modern Analysis,

p. 263.

now

THBOEY OF BBSSBL FUNCTIONS

568

[CHAP.

XVn

Now, since the series obtained from (1) and (2) by differentiations with respect to £* are uniformly convergent throughout any bounded domain of real values of

We

£,

we may

differentiate

any number of times and then make

thus deduce that

•

Jm (2ne)

„-i

»,ro

| Jt^. {(2n-1) € n% (2n-l)»» 1

'

}

is an even polynomial of degree 2m, and an odd polynomial of degree 2m — 1.

are polynomials* ine; the former

the latter

The

is

The

by Meissel in the cases

values of the former polynomial were given

the values for to— 1,

2,

!?

i'

2.'

2

« 1, 2, 3, 4, 5

*-** + *

'1

~8»

2

72"

32

m— 1, 2, 3 are

values of the latter polynomial for

5« s

«

«

2'

2~18' 2~

e

<£_

8i

«*

+ 460'

Meissel also gave the values of the latter polynomial for

Conversely,

to

3 are

it is

m=4,

5.

evident that every .even polynomial of degree

2m

is

expressible in the form oc

2

«*»«/» (2ne),

and that every odd polynomial, of degree

2m — 1,

is


2 6„J»_1 K2n-l)€}, where o» and bn are even polynomials in 1/n and l/(2n — 1) respectively, of degree 2m. 17*3. It

Simple Kapteyn

was stated in

§

series with

171

recondite character than

Kapteyn series are of a more and we shall now explain one of between the two types of series.

that, in general,

Neumann

the characteristic differences

In the case of

Neumann

complex variables.

series,

series it

is,

in general, possible to expand each of

the Bessel functions in the form of a power series in the variable, and then to rearrange the resulting double series as a power series whose domain of con-

vergence

is

that of the original

Neumann

* It is to be noted that the coefficients of c**

sero; they are

t-^ 2.(m!)«

and

series.

and c**-1

<-

in the respective polynomials are not

r"

2.1».8»...(2m-l)«

.

KAPTEYN SERIES

17-3]

The corresponding property Kapteyn

of

series is quite different

for the

Jr+n {(v + n)z}

convergent and represents an analytic function

domain

;

series

2,a n is

Kapteyn

559

in

(cf. §

87) throughout the

which

*exp^(l-*')

l-W(l-*«)

1

,.

•

piiîânl'

while the double series obtained by expanding each Bessel function in powers of z is absolutely convergent only throughout the domain in which

[#|.expV(l-|«| 1 )

and the the

1

< Um

domain is smaller than the former thus, when the limit is 1, domain is the interior of the curve shewn in Fig. 24 of § 8*7, in

latter

first

;

which the longest diameter joins the points i 1, while the shortest joins the points itx 0*6627434 while the second domain* is only the interior of the circle jr - 0C627434. ;

j

|

Hence, when we are dealing with Kapteyn expansion into double series

we

series, if

we use the method of

succeed, at best, in proving theorems only for

a portion of the domain of their validity; and the proof for the remainder of the domain either has to take the form of an appeal to the theory- of analytic continuation or else it has to be effected by a completely different method.

As an example

of the methods which have to be employed,

we

shall give

Kapteyn'sf proof of the theorem that

,— = 1 + 2

(1)

provided that z

lies in

1 Jn (nz),

the open domain in which z exp V(l

l

-

+ vXl-**)

<1.

This domain occurs so frequently in the following analysis that it is conit as the domain K; it is the interior of the curve shewn in Fig. 24 of §87.

venient to describe

Formula

To

(1)

is,

of course, suggested

by formula

establish the truth of the expansion, 1

and then Since

it

we

(2) of § 17'22.

write

+ 2 2 Jn (nz) = 8(z),

has to be proved that 8(z)

= 1/(1 - z).

J^^f^^pmj^

* For an investigation of the magnitude of this domain, see Paiieoz, Journal it Math. ziv.

33—89, 942—246. t Nieuw Areme/voor Wiikunde, xx. (1898), pp. 96—102.

(1849), pp.

z.

(1898), pp.

128—126; Ann. $d. de YficoU norm,

tup, (8)


560

we

see that, if

that on

we can

all

L

!

then

W where

«*P {*«(«- 1/Pi

I

= J_

«/.v

/ox

To

with centre at the origin of such a radius

the inequality

it

(2) is true,

T

find a circle

[CHAP. XVII

V

2wt

'

we

investigate (2),

values of

}

eft *

*

the analysis of § 8*7. If z = pe**, t = eu+ie and m being positive), then (2) is satisfied for

recall

,

are all real

p, u, a,

+ r'exp{jjr («-!/<) ~ r* exp {£* (* - 1/*)}

l ( J(r+) 1

(/>

if

+ sin

a p V(sinh u

and when u

is

8

a)

-u<

;

chosen so that the last expression on the

value, this value is (§

left

has

its least

87) °g

1+VU-*

8

'

)

which is negative when z lies in the domain K. Hence, when z lies in the domain K, we can find a positive value of u such that the inequality (2) is satisfied

when

t

|

= eu

.

1

Again,

if

we

W

write 1/t in place of

w

where 7

is

the circle

1 1

When we

so

2S(z)

1

+ * exp {-£*(*-

1/0}

-*exp{-J« (*-

1/*)}

t

=

e~u

proved

this,

is the

—

27ri; (r+>y _)*-exp{^(r-l/0}

sum of

the residues

prove that there

V and

we

'

«

of the integrand at

its

poles which

lie

7.

only one pole inside the annvlus*, and, having notice that this pole is obviously t = \.

For the number of poles

—

'

.

inside the aunulus bounded by

We next

find that

combine (3) and (4) we find that « + «xp { *»(«- 1/0} d* 28 (si = t

w

and

j

we

2iri J (y+) 1

S(z>>=J-[

(4i\

in (3)

t

<*

log

is

equal to

is

[l-r»exp{fr (*-!/*))]

[

2m'J (r+>y _ )

dt

dt

—Lf

W

2inJ (r+)

+

{

dlog[l-texp{-lz(t-l/t)})

27riJ(r+ )

= J.

T

mJ

dt

dt dt

dt

d\og[l-tr>ex V

(fjj.)

+-L.f liri J (T+) i T+ )

\$z(t-l/t)}-\

dt

dt

dlos[-t^{-¥(t-i/t)}]

dt

dt

* The corresponding part of Kapteyn's investigation does not seem to be quite so convincing as the investigation given in the text.

KAPTEYN SERIES

17-31]

Now

then

|

the

first

U <1

of these integrals vanishes ;

561

for, if

we

write

on T, and so the expression under consideration

|

may be

written

in the form

ti J r +) U=o <

)

dU

j J

dt

and the integral of each term of the uniformly convergent

series involved is

zero.

Hence the number of

zeros of 1

- f exp [\z (t 1

1/t)}

in the annulus is

equal to

>W

2iriJir+ It follows that

28 (z)

is

NK)>-

equal to the residue of t

+ exy{$z(t-l/t)}

t-exp{fr(t-Lft)\ at

t

—1

;

and

this residue is easily calculated to

been shewn that S (z) is equal the throughout the whole of the open domain in which the series

It has therefore

domain K,

i.e.

— z). to 1/(1 — z) throughout

be 2/(1

defining S(z) is convergent. [Note. It

is

of K, except at

type;

S («)

sum 1/(1 - z) on the boundary made to theorems of an Abelian

converges to the

proof requires an appeal to be

§ 17'8.]

cf.

17*31.

We

possible to prove that

«= 1, but the

The extension ofMeissel's expansions to the case of complex variables.

shall

(1) 7

(2)

now shew how

g v „=i

o

«M2n*)^ n* + ?

to obtain the expansions

* l«

+ £»

% J*»-i\&ri-l) Z _ )

z*?

z*?

(l'+^M^ + l?)

(1"

o

+ (?) (* + £») (3» + + ...,

z>?

z

+

f!£

_

(i»+^)(3»+r)(5'+r)

+

K

which are valid when z lies in the domain and £ is a complex variable which is unrestricted apart from the obvious condition that ft must not be an integer in (1) nor an odd integer in (2). These results are the obvious extensions of Meissel's formulae of § 17*23. [Note. The expansions when £ is a pure imaginary have to be established by a limiting process by making £ approach the imaginary axis ; since the functions involved in and (1)

(2) are all

even functions of

{,

no generality

is lost

by assuming that

R (f) is positive.]


562

In order to establish these formulae,

it is first

[CHAP. XVII

convenient to effect the

generalisation to complex variables of the expansion of the reciprocal of the

radius vector given by § 17*21 (6). That 1

we take the expansion

is to say,

+ 2 2 Jn (nz) cos n,

which we denote by the symbol 8 (z, ), and proceed to sum it by Kapteyn's method (explained in § 17*3), on the hypotheses that is a real variable and that z lies in the domain K. We define a complex variable yjr by the equation

=

The

singularities of

yjr

— z sin ^r.

qua function of

yfr,

,

are given

by cos^r =*

\jz,

that

is

= arc sec z — VC^ — !)• 2

None

of these values of

<£ is

real* if z lies in the domain

to oo through real values,

creases from

ifr

K

;

and, as

in-

describes an undulating curve

which can be reconciled with the real axis in the ^r-plane without passing over any singular points. It follows that

if,

we

for brevity,

write

Usr*exp[is(t-l/t)}, then

S ( *'

l-tfa

>-»/„(T+) l-2Ucoa(j> + U* By

with the notation of § 17*3.

and

so 2S(z,

its^ poles

We

)

which

is

lie

equal

to the

shall

'

t

the methods of that section

sum of

inside the annulus

now shew that having proved this, we then

the residues

of the integrand at those of

T and y.

bounded by

notice that these poles are obviously is

equal to

dlog(l-2U cos <(>+lP) 2iri

'

dt

o*+,y->

nog(l-2ffcos
<

_lf

= _!.(

dlogjPexpi-zjt-l/t)}] dt

(r+>

J

= 2, is

easy to shew that such values of

satisfy the equation

±t»,* exp N/(l-z»)

^ r

so that

«=-=** !

|

<1.

^

\% Un co6(n + l)<^\~dt-r2 a

•/nj
* It

,

dt

(r+)

+ 2irij

we have

there are only two poles inside the annulus, and,

Cauchy's theorem, the number of poles

By

dt

}

~

l+*/(l

-*•«)

'

*

t

= e***.

KAPTEYN SERIES

17-31]

563

the integral of each term of the uniformly convergent series vanishing, just as in §173.

Now

the residues of

1-ff' at t=* e*** are both equal to 1/(1 =

(3) K

r 1—^COS^r

'

1

-2U cos +U*'t

1

— z cos ^r);

and therefore we have proved that

= 1 + 22 Jn (nz)coBn, H=1

and the series on the right is a periodic which converges uniformly in the unbounded range of real

in the circumstances postulated;

function of

values of

<£.

R (£) > 0, we may multiply by e~<* and integrate thus

Hence, when /

That

is

Jn (m)

tr& dd> + 2 2

Jo

e~<* cosnd>

d$ =

Jo

«=i

|

Jo

dd>. T 1-^cos^rr

to say, ,

w

f r*i^' Jo

(4)

where the path of integration

l

"^-J+2i^ »+ f

)

£

is

the undulatory curve in the -^-plane which

corresponds to the real axis in the ^-plane

undulatory curve

may be

l

»=i

;

and,

by Cauchy's theorem,

this

reconciled with the real axis.

Now, when the path of integration is the real axis, the integral on the left an integral function of z\ and this function may be expanded in the

in (4) is

form

m =o ml Jo

By changing v

'

the sign of z throughout the work we infer the two formulae

»=i4n»+f»

K)

»-i

(2m)! J

m=1

raY

«=i(2m-l)!j*

(fr-lf+p

which are now established on the hypotheses that z

lies in

the domain

'

K and

that £(?)><>.

By

dividing the paths of integration into the intervals (0,

and writing \ir I

Jo

+ 0, f w + 0,

. . .

e~** sin"* ^frcM

1

for

=

-ty

.

.

ir),

in the respective intervals,

—

t

£

I

cosh £0 cos"* Odd .

sinh$7r£./o 1

we

(w, 27r),

...

infer that

[CHAP. XVII


564

and that

—

r^ (~e-& sin2™- y!rd<4r I "cosh f r T = cosh^Tr^Jo 1

Jo

"r

(r +

2

8 }

ir

+

a }

...

.

6&d

cos"*" 1

{^+ (2m- i^j

and writing 2^ for f in (5) we at once infer the and the mode of extending the results to explained. The required generalisations already been other values of £ has

By substitution

in (5)

and

when

truth of (1) and (2) all

z.

(6)

R (£) >

;

of Meissel's expansions are therefore completely established.

17*32.

The expansion of z*

a Kapteyn

into

series.

With the aid of Meissel's generalised formula it is easy to obtain the expansion of any integral power of z in the form of a Kapteyn series. It is convenient to consider even powers and odd powers separately. In the case of an even power, zm we take the equation given by § 17*31 (1) in the form ,

X

n\ {l)

f

27rtJ

2r(n + l + tflr(n + l-ig) | J„*(2mz) , a<3

£*»-ir(l+iOr(l-i£)

m=1 m'+f* ipr(n + l-»fl _ J_ [ % r(n + + "2w»J»tir(.m + l+»0r(m + l-t0 l.

«—«#

where the contour of integration is the circle f| = n+ $. Since both series converge uniformly on the circle, when z lies in the domain K, term-by-term |

integrations are permissible.

Consider

now the value

of

f—

2«J,f|-. + »

1

(•»+£)

m

When «$ w, there are no poles outside the contour, and so the contour may be deformed into an infinitely great circle, and the expression is seen to be equal to unity; but when m > n, the poles ± im are outside the circle and the expression

is

equal to unity minus the

at these two poles,

i.e.

— The

sum

of the residues of the integrand

to

(m + w)! (m — n—

m»»+r

expression on the left of (1)

.

is

1)!'

therefore equal to

Next we evaluate 27rtj|£ _ n+i (

r(n + l+it)T(n + l-iZ) r (m + 1 %o r (»» + i - #)

dl;

£*»-*»+*

•

KAPTEYN SERIES

17*32]

When m^ri,

the origin

is

the only pole of the integrand, and,

contour to be an infinitely great But,

when

expression

m > n, there

circle,

If

if

we take the

seen to be equal to

is

are no poles inside the circle j

f

j

= n + \,

£^fig-^

»w-<*-)-«J H we

the expression

1.

and the

is zero.

Hence we have

o)

565

replace

n by n -

and subtract the

1

+ ...^.

result so obtained from (2),

we

find that 2W

= „ S (m + n-l)\Jam (2m*)

(m + n)\J2m (2rm)

|

w= » +1 m^.(m-»-

m?»-\{m-n)\

»t.

1)!'

and so **»

(3)' V

If

= 2n* 2 (w + n- !)»./„ (2m* )

m^.(m-M)!

mZ n

n=

equation (3)

1,

'

at once deducible from equation (2), without the

is

intervening analysis.

have to deal with an odd power, a2" -1 we take the equation 17*31 (2) in the form

When we given by §

,

2.{l' + ra H38

J_f

+f }...{(2n-l)* + f*} a

X Zi {i -aisL ATI I J |f|=2W

2

+-r

2

H3 +r2 }-..{(2n-i)

x

2

2

+?»}

Ji {i +r U3 +n--K2m-i) +r 8

a

dZ

? + (2m-iy

2

a

^

i

}

and we deduce in a similar manner that (5)

2

i JÛfc, _!),,_ 2

m=i

-K^- 1)^1

(" + "-

2

7 (m-^^m-ra-l)!

«.=,»+i

Hence

1 *»>->-2fn-±Y> -2(n tfJS^

(6} (b)

The formulae

(3)

m

and

(6)

may

5 a*V»-»» (**) -»

(7)

t

jn

which

is

values

1, 2, 3,

w -___ (m _

(m + n-2)!^ _1 {(2m-l)^} .

w)i

be combined into the single formula (n +

w - 1) (»

!

«/"«»— {(n

+ 2m) z\

+ 2mr.m!

obviously valid throughout the domain

—

'

K when

n has any

of the

This formula was discovered by Kapteyn*; the proof of it which has somewhat artificial, seems rather less so than Kapteyn's

just been given, though proof. *

Ann.

sci.


(3) x. (1893), p.

103.


566

The

17*33.

Kapteyn

investigation of the

series

[CHAP. XVII

for zn by the method of

induction.

We

now give an

shall

of zn as a

Kapteyn

alternative

method*

of investigating the expansion

which has the advantage of using no result more

series,

abstruse than the equations

JL=l + 2

(1)

1 ~~

z

which were proved § 17'3

it is,

;

_L- = l + 2 I (-rJm (mz), L+Z „,„!

2 Jm (mz),

m-l

for real variables in § 17*22

of course, supposed that, if z

complex, then z

lies in

and

for

then

is real,

complex variables in

— 1 < z<

1,

and,

if

z is

the domain K.

The induction which

will

be used depends on the

fact that

when the sum,

ao

f(z), of the

W&8

r»=l

—

2 am Jm (mz)

series

is

known, then the sum F(z) of the

»M=1

m WV

%

series

Kapteyn -

can be obtained by two quadratures,

if

the former series

converges uniformly. To establish this result, observe that, by term- by-term differentiations,

*

T^ +

^

d Z

= }- 1

* CLm [Z

'

Jm" {mz) + (z/m) Jm

{mz)]

= (1-* ) 2 2

m=1

(z£)F(z) = (l-z')f(z);

so that it

follows at once that

Now, from

(1),

F(z) can be determined in terms off(z) by quadratures.

we have

I J^(%nz) = and

F(z)

so, if

2

==

^-

2

^^

,

Jfm

,

(z^)F{z) = )sz\

then Therefore, in the

where

a m Jm (mz),

A

and

B

domain K,

are constants of integration.

A=B =

If

we make

0.

Consequently

J"»< 2w *)

jP-2 I

(2)

m=i *

Wataon, Messenger,

rn*

xi/vi. (1917),

pp. 150—157.

z-*~0,

we

see that

KAPTEYN SERIES

17-33]

In

like

567

manner, we deduce from (1) that

I Jw+1 |(2m + l) *} = -%,, l—z'

t»=o

and hence that

The expansions

Now

assume

of sn

when n

or 2 are therefore constructed.

is 1

some particular value of

that, for

n,

zn

is

expansible in the

form oo

sn = n 2

and consider the function

By

(z)

2

(s) defined

= (n + 2f X

m-i

Jm (mz),

by the equation

"HL-JL bm>n Jm (mz). fnr

the process of differentiation already used,

=<» +2

s

integration

we have

>^S+*SS-< b+2

= (n + 2) On

bmn

>'*" ,

*n+a.

we deduce that

It is obvious that

A' = B'

{z)

=

= zn+* + A

'

log z

+ R.

from a consideration of the behaviour of

(z)

near the origin.

Hence the expansion

of zn+* is

*«+*

It follows at once

n+a

=

6WjB+a JTO (mz),

——*— bm

,

n-

by induction that

fan' i

and so is

a

V

where

That

= ( W + 2) 1

—:r™—n=n T (m -n +TT\ 1)

7=

1 »=i (2wi)^-

•

.?n a

to say 2n

=2

a

J

r(m + n)J„(2iiu)

ntnm^rim-n + l)'

~ and this is equation (3) of §17'32. The expansion of zm is obtained same way from the expansion of z the analysis in this case is left 1

;

reader.

in the to the


568

We

therefore obtain the expansion

<**)»

f 4)

which

[CHAP. XVII

(K

+ 2OTr,, m

> ,

the expansion obtained by other methods in

is

pansion

£

= »-

is

§1732; and the

ex-

valid throughout the domain K.

Since the series

mZo(n + 2m) is

n+1

.m\

absolutely convergent (being comparable with

% l/m ),

K and its boundary.

converges uniformly throughout

a

the expansion (4)

The expansion

is

there-

on the boundary of K, and in as well as throughout the domain K.

fore valid (from considerations of continuity)

particular at the points z

17*34.

=±

1,

The expansion of lf(t

— z)

in a Kapteyn series.

the expansion of zn , obtained in the

From

deduce, after Kapteyn*, the expansion of l/(t

K and

t

outside a certain

lies

sections,

we can

z lies in the

domain

two preceding

— z) when

domain whose extent

will

be defined later in

this section.

Assuming that

1

>

\

1

j

z

we have

|,

_1__1« JL =!+ + n+1

t-z Now,

t

^t

t

?!!i w+1 *

nZ i

m-

exp V(l - g 2 )

z if

| r (n + to) Jn+zn {(n + 2m) z\ (n + 2m) n+1 .ml

2

|

.

1+V(1-* )

j

= jr

2

the repeated series

is

expressible as

'

•

an absolutely convergent double

series if

the double series

*

*

2 wrc2

„-i«-o(w is

convergent.

But the terms

r (n + m) Fn+2W

+ 2m)"+ .m!|<|^ 1

in this series are less than the terms of the

double series

2*rw+2m _ 2Fex pF 2 .-i*-om!|«r ~|*Ki*l-2iT U|>2F. *

*

i

provided that

Hence, when

t\>2

2;

exp V(l

—2

f2

)

1+V(1-^)

rearrangement of the repeated series for \/(t — z) is permissible, and, when we arrange it as a Kapteyn series, we obtain the formula

= S <& {t)Jn(nz), r— — Z <£«,(*) + 2 n «i n

(1)

t

*

4nn.

sci.

de VAcole norm. sup.

(3) x.

(1893), pp.

113—120.

KAPTEYN SERIES

17-34, 17*35]

569

where * «*.(*) -1/fc

(2)

<© n (0

(3)

From

it

and

w (Mn_2m+1

-

•

,

9

we may deduce a very remarkable theorem

the last formula

by Kapteyn

2

1

discovered

we have

;

—¥

therefore,

by

K

- S* §

91

(

w-m-1)! _ 1

(rt-2m)2 .(w-m-l)

<*»

!

(2), 2

$nt<® n (t) = i

by §9-12(1), <£ n (*) = n (1

so that, (4)

when n =

Kapteyn's polynomial

It is

by

0„ (nt)

^)

{|n*

n (n*)},

+ sin £mr + 1 cos 2

2

|n7r

1, 2, 3, ....

polynomial

for,

- £2 )

(*

now

§ 8*7

therefore expressible in terms of

possible to extend the

combined with

domain of

validity of the expansion (1)

the series on the right of

§ 9*17, it follows that

(1) is a uniformly convergent series of analytic functions of z

and

t lie

in

Neumann's

O n (nt).

and

t

when

z

domains such that

n(*)
(5)

n(*)
—^-^ - &) ^ W = 1+V(1-^) .

,

where

x

I

ft (z)

z exp V(l

The expansion (1)

is

I

-^

|

•

!

therefore valid throughout the domains in which both

of the inequalities (5) are satisfied. [Note. This result gives a somewhat more extensive domain of values of t than was contemplated by Kapteyn he ignored the theorem proved in § 9*17, and observed that (since the coefficients in the series for n (t) are positive) when t ^ 1, ;

®

j

|

|«MOI<*.(l'l)<®»(i)=i. by

(4)

;

so that

Kapteyn proved that

(1) is valid

when

0(*)<0(1),. |*|^1.] 17 '35.

Now

A Uemative proofs of the

L

t

it is

expansion of

l/(t

- z)

into

a Kapteyn

—Z

=
(()

+2 | n=l

©n

(t)

Jn (nz\

possible to verify this expansion in various ways. Thus, if <©„

n (1 - *2 ) the reader will find

it

±^S 2 + T1— S* +

* Cf. Kapteyn, B. F.

2 n (nt) +sin \mr

(

1

-«2) *

(t)

be defined as

+t cos2 \nn,

an interesting analysis to take the 7 t

W.

series.

that explicit expressions have been obtained for the coefficients in the expansion

series

nOn (nt)Jn (nz),

n =i

Nieuw Archief voor Wiskunde,

xx. (1893), p. 122.

19


570

substitute suitable integrals for the Bessel coefficients and reduce the result to l/(t—z) after the manner of § 9'14.

Or

again, if

we

then, dividing

(<

2

by

-l)(<-2)3

= whence the

I

+ .(*r^} " (1 -*2)

i

—z\ and making use of §

1

(P-\?{t-Z? e

nW

(nz)\®

differential equation for

it follows

e» n2

17'3

®» W * (n2)

(1),

we

we

polynomials, and

find that

'

find that

(?-\f(t-zf

C+ft'+l)™'*** (t)

©n

(t) is

* (*»+!) corf jn»r

easily constructed in the

-n JW.W and hence

Neumann

differentiate the expansion twice with respect to 2

{{T^f and

[CHAP. XVII

form

^-^3

^-^3

A n and

17*4.

2?n are

J,

that

••(<)«» (1 - <2 ) 0» (»«) + sin 2 \nir + < cos2 \nn +t~! {A n Jn (nt)+Bn where

)

independent of

t

;

but

it

Yn (nt)},

does not seem easy to prove that

The expansion of an arbitrary analytic function

into

A n —Bn =0.

a Kapteyn

series.

We

shall

now prove the

following expansion-theorem

Let f(z) be a function which ^ a, where a^.1.

is analytic

throughout the region in which

fl (z)

Then, at all points z inside the region,

f(z) = «o + 2

(1)

2 OnJn (nz),

where

*.-%z JG.V)f(t) *,

(2)

and

the

path of integration

This result

is

is the

obvious

curve on which fi

when we

(£)

= a.

substitute the uniformly convergent

expansion

©

(<)

+2 £

<® n (t)Jn (nz)

n—l

for l/(t— z) in the equation

since D, is

(t)

=a

on the contour, while both fl(s) <

inside the contour.

This theorem

is

due to Kapteyn.

1

and

D, (z)

< fl (*) when

s

KAPTEYN SERIES

17*4, 17*5]

571

It is easy to deduce that, if the Maclaurin series for f(z) is

/(*)- 2 an zn

,

n=0

then

Oo=Oo,

(3)

-

a

(4)

17*5.

Kapteyn

The theory

of

1

~ 2m>2

'** (n

which v

series in

Kapteyn

is

-m"

(w •

1)!

a"-*»

not zero.

series of the

type

oo

% am Jr+m{(v + m)z\ in which v

is

pansion of zv

not zero or an integer, can he

The

.

i

made

to

result of § 17*33 suggests that it

depend on the ex-

may be

possible to

prove that

M-'L i.Zw.mS"-*'***™-

(1)

throughout the domain K.

enough to establish

expansion* when \z\< 0*6627434 but no direct proof of the validity of the expansion throughout the remainder of It is easy

K

the domain

this

;

known, and the expansion has

is

to

be inferred by the theory

of analytic continuation.

To

obtain the expansion throughout the interior of the specified

expand the

series

on the right in powers of z. The

T(p + m) w =o(i'+2m)" +1 .m!'2

L

(-Y~ ,'

+2r

When

r>

known

to vanish identically

the last series

identically for all values of (inside the circle)

m (v + 2m)" +3r

V(v+ r + m + 1) + 2m)gr~ r (v + m) F (v + 2r + 1 m\(r-m)\ T{v) T(v + r + m+lj' a polynomial in v of degree 3r — 1 which is

(r-TO)!

t» £ 2"+2Tc> + 2r+l) to OT I

1,

circle,

coefficient of z v+2r is

(-)'-'" v {

is

1

whenever v v.

is

an integer.

The expansion

by a comparison of the

It therefore vanishes

(1) is therefore established

coefficient of z 9

on each side of the

equation.

From this result, we can prove that, under the conditions specified

j£- = S

(2)

where

n\ W * This

was done when

1

7*4,

(0 Jh- {(" + n ) z \>

+ n-2my>r(v + n-m) z^- 1 ^ W " 2 wto (tv + ny+n-^.mlt"-™^ (v

cJ

«**,-,

pp. 42—46.

64*, „

in §

|

z

|

< 0*659 by

Nielsen, Ann.

set.

'

de VEcole norm. sup. (3) xvin. (1901),


572

It is not difficult* to express

An,»(nt+% vt), defined

And

[CHAP. XVII

in terms of Gegenbauer's polynomial

£4n ,v(t)

in § 9*2.

the reader will easily prove that if/ (2) satisfies the conditions specified

in § 17-4, then

*"/(*)

(4)

=

Sv

+

J*+n {(v

n)z],

n=0

where (5)

in

a„,

,

= Û j/0)

a

n

,

r

(0 dt,

which the contour of integration surrounds the origin

W

(fo

n

=1 '

where a

,

a lf

...

(v

X*

t

2 m

"

and hence

;

+ n - 2w ) T (y + ~ w) ttn,^ (%v + n)»+»-™" m 2

to

!

.

are the coefficients in the Maclaurin series for f{z).

—

[2sote. Jacobi in one of his later papers, Aatr. Nach. xxvm. (1849), col. 257 270 [Ges. Math. Werke, vn. (1891), pp. 175 188] has criticised Carlini for stating that certain expansions are valid only when \z\ <0'663.... But Carlini had some excuse for his state-

—

ment because the expansions are obtained by rearrangements

of repeated series which are

permissible only in this domain, although the expansions are actually valid throughout the

domain

K.~\

Kapteyn

17'6.

series

of the second kind.

Series of the type

2 A./„. {(*±-» + ») M } /,„ {(*£ + n) have been studied in some detail by Nielsen f.

.

But the only

series of this

type which have, as yet, proved to be of practical importance J, are some special series with

fi

— v, and

with simple

coefficients.

The

results required in

the applications just specified are obtainable by integrating Meissel's expansion of § 1731 (1) after replacing z the domain K, 2

by

It is thus found that,

z sin#.

«?+? "»=i(i +r)(2 2 +H...(« +r )-Wo 2

n=i

2

2

sin

throughout

""'

so that

m yy)

g

I JnHnz} nti

n*+?

I

and hence we deduce that while the

sum

t

2

1

"

"2.4 ,(P ,

tm is

'S,Jn 2 (nz)/iv

+ ^)(2 + t ) 2

2

a polynomial in z2 of degree m;

l n^n Jn2 (nz) may be found corresponding expansion 2n2m J 2n (2n2).

of series of the type

manner from the

*g»

1.3

z*

21 2 +

t

in a similar

* Cf. Nielsen, Ann. sci. de VEcole norm. sup. (3) xvin. (1901), p. 60. sci. de VEcole norm. sup. (3) xvm. (1901), pp. 39—75. % Cf. Schott, Electromagnetic Radiation (Cambridge, 1912), Chapter vin.

t Ann.

,

KAPTEYN SERIES

17-6, 17-7]

573

Thus Schott* has shewn that

l/^-sTobr^

(2)

j^M-^,.

(3)

A

general theory resembling that of § 16 14 -

deducible from the ex-

is

pansion <*«>

(4)

»-

r( 2Ui)

V

io(. +

^^ +w>^

J

+ '-m!

which is easily derived from § 175 (1) and is valid throughout K\ but it seems unnecessary to go into details which the reader should have no difficulty in constructing, in the unlikely event of his requiring them.

17*7.

Kapteyn

series

which converge outside the domain K. lim

If

|

y/an

\

=

1,

n-»oo

we have seen

that the

Kapteyn

%an Jn (nz)

series

function throughout the domain K.

But

represents an analytic

when x

is real, \Jn (nx)\< although when \z\ > the series does not converge at points which are not on the real axis.

the series

may converge along the whole of the real

The behaviour that

it

since,

of such a

Kapteyn

series

axis,

may

be

summed

resembles a power-series throughout the domain

up-f-

K and

1,

1

by saying

that

it re-

sembles a Fourier series on the real axis outside K.

As an example,

let

us consider the series

Jn (fix) s= £ It is evident that, if

since the Fourier series

is

=

yfr

— x sin

yjr,

then

uniformly convergent.

Now, when x > 1, <£ decreases as -\jr increases from to arccos(l/#) and then increases to ir as yjr increases from arc cos(l/#) to it. If m be the integer such that the minimum value of <£ lies between — 2mir and — 2 (m -f 1) it, let the values of yjr corresponding to the values

-

0,

of

be

7„, 7j, ...

ym

2tt, ,

-4tt,

8 m> &m-i,

ir{ r=0 Jyr

...,

-

••• $i>

Jym

S

2rmr, -2mir, ,

...,

-

2tt,

and then

r^Jtr+r

J 8

)

n =l

n

* Electromagnetic Radiation (Cambridge, 1912), p. 120.

t

The

suggestion of these analogies was

made by

Professor Hardy.


574

Now when series

lies in

yjr

the intervals

under the integral sign 2

£<£

(7,.,

7r+ i) and

(8 r

[CHAP. XVII

8r+1 ) the

,

sum

of the

is

- %Tr + %ir* + r (r + 1) tt + (r + 1) 2

ir,

and, since l(yfr

— x sin yfr) dty = £i/r + x cos

j(yfr

— x sin 1/r)

2

may be shewn

it

2

d^r

= £\Jr

without

3

4-

will see

— sin

+ £#

2

-\Jr)

(-^

— sin y(r cos ^),

difficulty that

m

2

r-0

The reader

2# (>|r cos yfr

much

m

S = ^x* + %x + 2

this

sjr,

{£ (S r

- 7,. ) + x (cos 8r - cos 7,.)} + 2tt 2

r(S r ~7r ).

2

r-1

that a large class of Kapteyn series

may be summed by

method*. The convergence of Kapteyn

17*8.

series

on the boundary of the domain K.

K

With the exception of the points + 1, the boundary of presents no means of Debye's asymptotic expansion the consideration of the convergence of the Kapteyn series 2an «/r+M {(1/ + n)z] features of special interest; because, by

reducible to that of the power series

is

v «•

f

*expV(l-

8

.s

)

)*

*V»t 1+V(1-* )) 2

and that of two similar

The

points

expansions

fail.

series f

with

*Jn*, \/n

6

'

written for

+ 1 But the lacuna thereby produced

>Jn.

present more interest, because the ordinary asymptotic is filled, for real

values of

v,

by the following theorem of an Abelian type: The convergence of

i ft

n-l » J is sufficient to

ensure both the convergence of 2 OnJv +n (v + n) and the continuity throughout the interval^ < x ^ 1.

2 an J,+n {(p + n) x)

of

Since So„/w8 converges and {n/(p + w)p

is

monotonic, with a limit as n -*

00

,

+ n)* converges; and since, by § 8 54, (y + ny Jr+n (v + n)

it

follows§ that 2a»/(i/

is

monotonic, with a limit as n -»• 00

-

,

it follows

that ^,anJv+n (y

+ n)

converges.

connexion the researches by Nielsen, Overtigt K. Danske Viderukabernes Selskabs, 127—146, should be consulted. t If a^/^/n does not tend to zero the series cannot converge; and if it does tend to zero Sc^/^/re 5 is absolutely convergent, and so, if we replace each Bessel function by the first two terms of the asymptotic expansion with a remainder term, the series of remainder terms is absolutely * In this

1901, pp.

convergent.

J Due allowance has to be made for the origin § Bromwich, Theory of Infinite Series, § 19.

if

»»<0.

KAPTEYN SERIES

17*8]

J^\{v + «)x) «/„+„ (v + n)

Again,since is

575

a function of n which does not increase as n increases, for

in the interval

j$#^ 1,

it

all

values of

x

follows from the test of Abel's type for uniformity

of convergence* that

% an Jv+n is

{(p

+ n) x\

uniformly convergent (and therefore continuous) throughout the interval ^ x % 1 and this proves the theorem. ;

J

By it may be shewn that if "2a n v+n (v + n) converges, so does Sotn/n^, so that the convergence of Xcin/n* is both necessary reversing the reasoning,

and sufficient for the theorem theorem of its kindf.

to be true; the

theorem

is

therefore the best

*

Bromwich, Theory of Infinite Series, § 44. t This was pointed out by Professor Hardy. (1917), pp.

171—174.

Cf.

Watson, Proc. London Math. Soe.

(2) xvi.

CHAPTER

XVIII AND

SERIES OF FOURIER-BESSEL

DINI

Fourier s formal expansion of an arbitrary function.

18*1.

In his researches on the Theory of Conduction of Heat, Fourier* was led to consider the expansion of an arbitrary function f{x) of a real variable of x in the

form

/(#)= 2

(1)

am J

(jm x),

where jlt jif js , ... denote the positive zeros of order of magnitude.

The

J

(z)

arranged in ascending

necessity of expanding an arbitrary function in this

manner

arises

problem of a chain oscillating under gravity and in Euler's problem of the vibrations of a circular membrane with an initial arbitrary symmetrical displacement (§§ 1'3, 1*5).

also in Daniel Bernoulli's

In order to determine the plied both sides of (l)by

xJ

coefficients a m in the expansion, Fourier multi-

(jm x) and integrated between the limits

and

].

It follows from § 5*11 that

xJ J Jo

{jm x)J (jn x)dx=\

™+n,

'

KtJi(jm),

m = n,

and hence Fourier inferred that am

(2)

If

we now change

- j—j £

*/(*) J„ (jm t)

dt

the significance of the symbols Jm, so thatf^,^,^,,

...

denote the positive zeros of the function J„ (z), arranged in ascending order of magnitude, then

/(«)= 2

(3)

amJ,(im*)>

TO=1

where

am =

(4)

2 *

¥+1

f tf(t) J v (jm t) dt

\Jm)J

This more general result was stated by LommelJ; but, of course, neither = does the procedure which has been indicated establish the validity of the expansion; it merely indicates how in the general case nor in the special case v

the coefficients are to be determined on the hypothesis that the expansion exists

and

is


* La ThtorieAnalytique de la Chaleur (Paris, 1822), §§316—319. t The omission of the suffix y, associated with jj , jj, js, ..., should cause no confusion, and considerably improves the appearance of the formulae. % Studien iiber die Bessel'schen Functionen (Leipzig, 1868), pp. 69 73.

—

it

J

FOURIER-BESSEL SERIES

18-1]

In fact the simplicity of the procedure reader might anticipate that, restrictions, the

if

is

somewhat deceptive;

the function f(x)

expansion would be valid for

577

is

all

for the

subjected to appropriate

values of v for which the

integral r\

tJ f Jo isvconvergent,

i.e.

when

Dini, however,

v

>—

v

(jmt)Jv (jn t)dt

1.

remarked that he was unable to deal with the range

- \
Several subsequent writers, while proving theorems for the latter range, asserted that the extension to the former range was merely a matter of detail; but it was not until after 1922 that

anyone took the trouble to supply the detail which is tedious and of no great In the exposition given here, it will be supposed that v> — \.

interest.

The first attempt at a rigorous proof of the expansions (1) and (3) i& contained in some notes compiled by Hankel* in 1»69 and published post-

A

humously.

more complete investigation was given by Schlaflif a year and an important paper by Harnack* contains an investigation of the expansion (3) by methods which differed after the publication of Hankel's work;

appreciably from those of earlier writers.

A few

years after the appearance of the researches of Hankel and Schlafli.

the more general expansion 00

/(#)= 2

(5)

m=l

where \, \2 \3 ,

,

...

bm J v (X m x),

denote the positive zeros (in ascending order of magnitude)

of the function

z-»{zJ v '(z)

when

v

The (6)

> — \ and

H

is

coefficients in the

{(xw3

+ HJ

v

(z)},

any given constant, wa3 investigated by Dini§. expansion are given by the formula

- v*) J* (x,„) + x^j;* (\ m )j bm =

2\,B2 [ tf (t)

v

Jo

(\ m t)

dt.

The mode of determination of the numbers Xm subjects f{x) to what is known as a mixed boundary condition,' namely that /' (x) + Hf (x) should formally vanish at x — 1. '

The expansion

(5)

was examined by Fourier (when

of the propagation of heat in a circular cylinder

v

= 0)

when heat

in the

is

problem

radiated from

H

is the ratio of the the cylinder; in this problem the physical significance of external conductivity of the cylinder to the internal conductivity.

—

* Math. Ann. vin. (1875), pp. 471 494. In the coarse of this paper, Hankel obtained the integral formula of §14*4 as a limiting case of (3).

t Math. Ann.

x. (1876), pp. 137—142. % Leipziger Berichte, xxxix. (1887), pp. 191—214; Math. Ann. xxxv. (1889), pp. § Serie di Fourier (Pisa, 1880), pp. 190—277.

41—62.


578

was pointed out by Dini that the expansion

It

the insertion of an initial term

when

H+v=

;

(5)

name

must be modified* by

and, although Dini's analysis

contains a numerical error, this discovery seems to associate Dini's

[CHAP. XVIII

make

it

advisable to

rather than Fourier's with the expansion.

The researches which have now been described depend ultimately on a set lemmas which are proved by Cauchy's theory of residues. The use of complex variables has, however, been abandoned, so far as possible, by Kneser f and HobsonJ, who have constructed the expansion by using the theory of of

integral equations as a basis.

On

aesthetic grounds there

is

a great deal to be said for this procedure,

seems somewhat unnatural to use complex variables in proving theorems which are essentially theorems concerning functions of real variables. On the other hand, researches based on the theory of integral equations are liable to give rise to uneasy feelings of suspicion in the mind of the ultrabecause

it

orthodox mathematician.

The theory has recently been made distinctly more complete by the important memoir of W. H. Young§, who has thrown new light on many parts of the subject by using modern knowledge of the theory of functions of real variables in conjunction with the calculus of residues. An earlier paper by Filon|| which makes some parts of the analysis appreciably must also be mentioned here.

less synthetic

The question of the permissibility of term-by-term differentiation of the expansion which represents a function as a series of Bessel functions has been discussed by FordU, who has obtained important results with the help of quite simple analysis

(cf. §

184).

More recondite investigations are due to C. N. Moore**, who, after studying the summability of the expansion by Cesaro's means, has investigated the uniformity of the convergence of the expansion in the neighbourhood of the origin,

and

also the uniformity of the

summability of the expansion (when

not necessarily convergent) in this neighbourhood.

The reason why the uniformity of the convergence (or summability) of the expansion in the neighbourhood of the origin needs rather special consideration

are valid

m,

for

is

that

when

it is

XmSc

necessary to use asymptotic formulae for

is large;

and, as

x approaches

which the asymptotic formulae are

* Details of necessary modifications

when

J v (Xmx) which

zero, the smallest value of

significant, is continually increasing.

H + v <0 will be given in § 18-3.

—

The modification

was also noticed by Kirchhoff, Berliner Sitzungsberichte, 1883, pp. 519 524. t Archiv derMath.undPkys. (3)vn. (1903), pp. 123 133; Math. Ann: lxiii. (1907), pp. 477—524% Proc. London Math. Soc. (2) vn. (1909), pp. 359—388. § Ibid. (2) xvm. (1920), pp. 163—200. Ibid. (2) iv. (1906), pp. 396—430. Cf. §§ 19 21— 19 24. IT Trans. American Math. Soc. iv. (1903), pp. 178—184. ** Ibid. x. (1909), pp. 391—435; xu. (1911), pp. 181—206; xxi. (1920), pp. 107—156.

—

||

18-11]


579

In the exposition which will be given in this chapter, the methods of the calculus of residues will be used to a far greater extent than has been usual in recent researches; this is a reversion to the practice of Hankel and Schlafli and (in the special case of Fourier series) of Cauchy. The advantage of this procedure is that it results in a great simplification in the general appearance of the analysis throughout the whole theory. And, although it

seems impracticable to prove certain theorems (notably those* relating to fractional orders of summability) with the help of complex variables, the gain in simplicity is so

very

marked that

it

has been possible to include in this chapter

many more theorems than would have been

possible if the methods of the theory of functions of real variables had been used more exclusively.

As an example may be mentioned

where

c x is

of the simplicity produced

by using complex

variables, it

that comparatively crude inequalities, such as

a constant, independent of z, when v is given and exceeds — |,are prove all the requisite theorems concerning convergence at a point

sufficient to

(or summability at a point) and they are also sufficient to prove theorems concerning uniformity of summability throughout an interval of which the origin

may be an end point. Direct proofs of theorems concerning uniformity of convergence throughout such an interval require more elaborate inequalities, but in this work the use of such inequalities is evaded by deducing uniformity of convergence from uniformity of summability by an application of Hardy's

convergence theoremf.

may be stated

It

here that the theorems of this chapter correspond exactly theorems concerning Fourier series which are given in Modern Analysis.

to the

In addition to the memoirs which have already been cited, the following may be menBeltrami, R. 1st. Lombardo Rendiconti, (2) xm. (1880), pp. 327—337 Gegenbauer, Wiener Sitzungsberichte, lxxxviii. (2) (1884), pp. 975 1003 Alexander, Trans. Edinburgh

tioned

:

;

—

Royal Soc. xxxm. pp.

223—260;

(1888),

Volterra,

pp.

;

313—320; Sheppard, Quarterly Journal, xxm.

Ann. di Mat.

(2)

xxv. (1897),

p.

(1889),

145; Stephenson, Phil. Mug.

(6)

xiv. (1907), pp. 547—549; Messenger, xxxm. (1904), pp. 70—77, 178—182; Rutgers, Nieuw Archie/, (2) vin. (1909), pp. 375—380; Orr, Proc. R. Irish Acad. xxvn. A, (1910),

pp.

233—248; and Dinnik, Kief Polyt.

[Jahrbuch


Inst.

{Engineering Section), 1911, no.

1,

pp.

83—85.

der Math. 1911, p. 492.]

The investigations by Alexander are mainly based on operational methods, while Orr dealt with expansions in which functions of the second kind are involved. 18*11.

The various types of series.

In the special case of series of circular functions, it is necessary, as the make a distinction! between any trigonometrical series

reader will remember, to

00

Ia *

+ £ m=l

(am cos

mx + bm sin mx),

Such theorems have been investigated by Moore and Young, % Cf. Modern Analysis,

t Modern Analysis, § 8-5.

§ 9-1.

—

f


580

and a Fourier

series in

am It

which

which the

= —If' I

7T J — v

[CHAP. XVIII

coefficients are expressed as integrals,

f(t) cos mtdt, bm

If".

=-

IT JJ

f(t) sin mtdt.

-.jf

make a similar distinction* between the types of series be dealt with in this chapter; any series of the type

necessary to

is

will

00

2 am J v (jm x),

m=l in which the coefficients

am merely form a given sequence of constants,

will

be called a series of Bessel functions. If,

however, the coefficients in this series are expressible by the formula

a™

=

2 75 TT~\ v v+iKjm)

f

1

f (*) J "

f

J

(

^ dt

J™

>

the series will be called the Fourier-Bessel series associated withf(x).

And

further, the series converges to the

if,

the interval

(0, 1),

sum f(x)

for

any point x of

the series will be described as the Fourier-Bessel expansion

off(*\ In like manner, the series

2 where X 1}

A-,,

X3)

...

bm

Jv (\ m x)

t

are the positive zeros of

zJ v '(z) + HJ v {z), will

Dims

be called


series

If the coefficients b m are determined

by the formula^ l

{(Xm2 - * 2 ) J.2 (x»)

+ xm J/ 2

2

(xm)} bm

= 2K*

j

tf(t)

Jo

J v {\m t) dt,

the series will be called the Dini series associated withf(x).

And

if,

interval (0,

further, the series converges to the 1),

sum/(#)

for

any point x of the

the series will be described as the Dini expansion of fix).

Some writers have been inclined to regard Fourier-Bessel expansions as -* oo merely a special case of Dini expansions, obtainable by making but

H

there are certain distinctions between the two expansions which

view somewhat misleading 18*12.

(cf.

*

The

Special cases of Fourier-Bessel and Dini expansions.

is

coefficients

function whose expansion has simple coefficients has already been greater part of the terminology

(1920), pp.

t It

this

§§ 18*26, 18*34, 18"35).

There are very few expansions of simple functions in which the assume a simple form.

One

;

make

is

due

to

Young, Proc. London. Math. Soc.

167—168.

supposed that the integral

is

convergent for

all positive integral

J It is supposed that the series is modified, as in § 18*34,

values of m.

when H + v ^ 0.

(2)

xvm.


18-12, 18*2]

Another is x', which gives

investigated in § 1542.

2Jv{j x ? \ m=\Jm" v+i KJm)

(i)

x"= £

/a\

r

m=l It will

0$a;
(*-m

rise to

,

-

S

be seen subsequently that (1) is valid when + v>0. Cf. §§ 1822, 18-35.

if

the formal expansions

2\m «/y {\m X) J v+1 {Xm) 2 2 V*) J v - {X m ) + A. m «/„' (Xm)

V

_.

i»

581

'

^x <

1,

and

(2)

when

H

t

The reduction formula 1

\n2

r + 2» + Jv (X m <) dt = i

J

is easily established,

(v

+ 2n) Jv (X m )-Xm Jv

so that the Dini expansion of of x v+2n + l

The Dini expansion

positive integer.

case the general coefficient

is

'

(X m )

j?"

t

v

/: In order to calculate this when

v is

an

J

V

+ *-> J" r (X m «)

may be determined when similarly be determined;

known

expressible in terms of "i

+ 2n

may

- 4» (»+n)

i>

is

rf<

any

in this

functions and

Jv (\ m t)dt. integer,

McMahon*

has proposed to tabulate the

function ('

|

which

is

K

J

(t)

dt=*Jx

(x)

+ Js {x) + Jb (.r) + ...,

a special form of one of Lommel's functions of two variables (§§

The methods of Hankel and

18*2.

The

16*5, 16*56).

Schlafli.

which were described in § 181 are based on the by Dirichletf in his researches on trigonometrical series of Fourier's type; this method of proceeding is obviously suggested by the earlier investigations

analysis used

fact that the trigonometrical series are special cases of the Fourier-Bessel

expansion, obtained by giving v the values

±

\.

In the case of Fourier's theorem, to prove that 00

f(x) = $a

where

am

it is sufficient

f(x) = lim ,.e.

that

=-

+ %

(«,» cos

f(t) cos mtdt,

IT J

bm

—n

mx + b m sin mx),

=

\

7r J

f{t) sin mtdt, -v

to prove that

—1

/" ir

/

*,

{%

n

+ co8(x-t) + cos2(x-t) + ...+cosn(x-t)\f(t)dt,.

/(.)-

t

too

1

± i) (x — t) *,..,. * Liil-t) '<*>

f* sin (n

g^ —

American Assoc. 1900, pp. 42 43. The tabulation is most simply conjunction with Table I. (pp. 666—697) ; see Table VIII. t Journal fUr Math. nr. (1829), pp. 157—169. * Proc.

§ 10-74(3) in

effected

by using


582

[CHAP. XVIII

In the case of the general- Fourier-Bessel expansion, the corresponding limit to be evaluated is

n -*

and so

it is

when n

is

residues

is

oc

m=l "

v+1

\Jm)

J

necessary to investigate the behaviour of the

sum

and it is in this investigation that the use of the calculus more than desirable.

large;

In the case of Dini's expansion, the corresponding

examination

of

sum which needs

is

2\ n*J ¥ (\ m x) J {\n t) v

An

application of the calculus of residues which will be described in §§ 18*3

1833 shews that the difference of the two sums is readily amenable cussion, and so we are spared the necessity of repeating the whole

to dis-

of the analysis of the Fourier-Bessel expansion with the modifications appropriate to the

more general case of the Dini expansion.

18*21.

by

The Hankel-Schldfli contour

integral.

We shall now begin the attack on the problem of Fourier-Bessel expansions discussing properties of the function Tn x), defined by the equation (t,

(\\

rn

/.

x

x

*"* v \.lrn ) Jy \Jmt) -j-2 -r^p° H-i \Jm)

_ v ~

4 n [l,X)-

K**)

m=l

where

0
O^t^.1, and the order v v

The method which many of the details

will

be used

is real

and

is

,

subject to the condition

+ \> 0.

is

due

to

Hankel* and

Schlaflif,

though

of the analysis are suggested by Young's J recent memoir.

function Tn (t, x) is obviously as fundamental in the theory of FourierBessel expansions as is the function

The

sin (n

+ ^) (x — t) — t)

sin £ (x

'

in the special theory of Fourier series.

In order to obtain the formulae connected with sequently required

Tn (t,

it is

necessary to express the

x) as the residue at

x) which are sub-

,

When

this

of the

sum

for

complex variable w, has been done, we express

of a function, of the

jm which has poles at jlt j2 js ...jn * ,

Tn (t,

mth term

* Math. Ann. vm. (1875), pp. 471—494. t Ibid. x. (1876), pp. 137—142. J Proc. London Math. Soc. (2) xvrn. (1920), pp. 163—200.

;


18-21]

Tn (t, x) as

583

the integral of this function round a rectangle of which one of the

sides lies along the imaginary axis while the opposite side passes

jn and jn+1

The

.

sides parallel to the real axis are then

moved

between

off to infinity

in opposite directions, so that, in order to secure the convergence of the integral,

necessary to prescribe the behaviour of the integrand as

it is

There are three integrands which we

I(w)

|

-* oo

(3) (4)

2

2

of these was the integrand studied by Schlafli; the other two are

first

suggested by the work of Kneser and Carslaw which was described in

A study of the asymptotic (4)

We

^x< t <

when

x±t and

tv=jm

Tn the case of

.

and then,

if

w = jm +

0,

2

;

(3)

is

< t< x < 1

when*

(2),

all

have the same residue, namely

2

Jy (jm t)jJ y +1 (jm ), we define the function f g(w) by the formula

p_ xi {tJy (aw) J*+i (tw) - xJ

g (w) =

(5)

15 42.

1.

proceed to verify that the integrands 2«/„ ( jm x)

at

§

values of these integrands indicates that (2)

suitable for discussions in which

and

.

namely

J v+1 (tw) - xJ ¥ (tw) J„+1 (xw)}/{(t - x*)J v (to)}, ttw {«/„ (w) Yv (xw) — J, (xw) Yv (w)} Jv (tw)jJy (w), it w [Jy (w)Yy(tw)-J v (tw) Y v (w)} J v (xw)jJ v (w).

2 {tJv (xw)

(2)

The

shall study,

j

where

is small,

Jy (W)=0jy' (jm)

v

(tw) J„ +1 (xw)},

we have

+ 1& Jy" (jm ) +

.-,

so that

WJy>(w)

= 6*jm Jy'*(jm) +

It is easy to verify,

cient of 0* on the right vanishes;

at jm

>

Jy' ( jm ) { J y" (jm) + Jy' (jm )} + .... using differential equation, that the Bessel's by 3

coeffi-

and hence the residue of g (w)/{wJv2 (w)}

is

9'(jm)l{jm Jy*(jm)},

and

this is easily reduced to

2 J„ (j m x)Jy (jm t)/J\ +1 (jm )

by using recurrence formulae. In the case of

(3),

the residue at j m

is

Yv (jm x) - Jy (jm x) Yv (jm )} Jv (jm t)/Jy = - IT jmJy (jmX) Jy (jm t) Yy (jm)/Jy (jm) = 2 Jy (jm x) J, (jm t)/Jy (jm),

TTJm {J, (jm)

(jm )

2

by

and this same way.

§ 3*63,

in the *

This

is

most

is

the expression required

easily seen

by writing the integrand

\iriw {HyV) (w) HyV) (xw)

- HvP) (xw)

f The results obtainable by using the integrand

;

the integrand (4)

in the

is

dealt with

form

HrW (w) } Jv (tw)\o v (w).

(2) are

discussed in great detail by Graf and

Gubler, Einleitung in die Theorie der BesseVschen Funktionen,

i.

(Bern, 1898), pp. 131

— 139.

'


584

We at

±

[CHAP. XVIII

next take the contour of integration to be a rectangle with vertices A n ± Bi, where B will be made to tend to ao and A n is chosen so

Bi,

,

that jn < A n
A n>

.

j

n

is sufficiently

Now

large (§ 1553).

easy to verify that the three integrands are odd functions of w,

it is

and so the three integrals along the Again,

w = u + iv,

if

it

positive or negative, while

so, for

of the rectangles vanish*.

left sides

may be verified that when v is large, and either w>0, then the three integrands are respectively

(e-fe-s-t'M),

and

we when

(e-<*-'>l t'i),

any assigned value of

An

,

(e~ (t

-x) W),

the integrals along the upper and lower

sides of the rectangle tend to zero as

B -* oo when

x and

t

have the relative

values which have already been specified.

We

thus obtain the three formulae

_ j^

Tn (t,

(7)

x)

-I

rA»+»i

An+ \

ri

" w \J

2

V

6wJ v (x0w) J v { tdw) d0dw

(w)

Y

w

(xw)

(0
2;

x±t)

J^lp

- J v (xw) Yv («,)}

(0
(8)

x)

=I

An+CCt

f

Appp

y

1)

J

w {J, (w) F„ (tw) - J„ (tw) Yv («,)}

m

(0
From

j

jv

(9)

when

w

is

m\ ^ exp l!f ^,

on the

(

line joining

A n — ccitoA n + aoi

exceeds a value which depends on

|<*-<>r.ft.)|«

iJ,(^

v.

c'

exp

and

£

F,

^

^ 0,

provided that n

(

Hence

;s? ^J_«p{-!.||*

i

so that

(10)

\Tn (t,x)\^ '

**' 2

ttc 2

1* 1

- #* 1

(2

- x - t) *J(xt)

This inequality gives the upper bound in question. * It is necessary to make an indentation at the origin, but the integral round the indentation tends to zero with the radius of the indentation.

:


18*22]

585

It is also easy to see that

f Jo

t"

+l

Tn (t,a;)(t*-a?)dt J

fv+l rAn+aai

= "^7 7n

1*

J An-ooi

J" (xw

/ "+ 2

•

">

(^ ~ xJ

>-» ( tw )

J*+i ( aw)} „,.>-/,,

V)J,-(W)x

,

and hence (11)

f t"

+1

Tn (t, x) (JP -x*)dt\$

Ijo

I

ircÂ n (2

— x — t) *J{xt)

[Note. Theorems obtained by a consideration of integrals involving Bessel functions first kind only can usually be made to cover the origin, in view of the fact that the constant Cx in equation (9) is independent of t in the interval 0
of the

written

valid when 0^#^1, 0î
and (8)

it is

ry (tw)\^ Cl

{\tw\-"\oS \tw\ + \tw\-i}exio{\ l(tw)\},

'

a somewhat tedious matter

to obtain

a simple upper bound

to the integrand in

from this inequality.]

Equation (6) was used by Schlafli to prove that, when n

^n(g-a?) TnV(tx\~ ' (
l

sin

f

'

is large,

then

_ *™A n (t + x)l sm$-jr(t + oc)j'

but, since the order of

we

evident,

which

shall

difficulties

18*22.

magnitude of the error in this approximation is not next evaluate some integrals involving Tn (t, x) by means of caused by the unknown error may be evaded.

Integrals involving

Tn (t, x).

The two fundamental formulae which we lim t

(1)

t'

+1

n-»ooJo

shall

Tn (t, x)dt = a?

t

obtain are as follows

(0
1)

= $x*.

(0
Tn (t,x)dt = lx».

(0
lim rt» +1 Tn (t,x)dt

(2)

now

n-^co J

From

these

it is

obvious that 1

lim

(3)

f\''+

In the course of proving (1) a* [

it will

t'

+1

Jo

uniformly as n

-*• oo

when x

Tn

lies in

(t,

A

is

any positive number.

w) dt -* x* + l

the interval

^x^ where

be apparent that

1

— A,

'


586

We

[CHAP. XVIII

boundedness of

shall also investigate the rt

r t' +1 Tn (t,x)dt

Jo in the interval in

< t ^ 1.

which

was given by Young, Proc. London Math. Soc. (2) xvin. (1920), pp. 173 174, and the proof of it, which will now be given, is his. Formula (2) seems to be new, though it is contained implicitly in Hobson's memoir.

Of these

—

results, (1)

It is evident that

3^%^r.

\\»«Tn (t,x)dt = Z JO

When we

tn=\jm*>v+\\Jm)

transform the

manner of §

18'21, 1

we f

sum on

the right into a contour integral after the

find that it is equal to

00 *

2J, (xw)

dw

1 [

wJ v {w)

tiai-vi

A n+«>i 2 J, (xw)

dw

wJ„\w)

îrijAâoi

In the former of these two integrals, the origin has to be avoided by an indentation on the right of the imaginary axis.

Since the integrand reduces to

iri

is

an odd function of w, the value of the

integral

first

times the residue of the integrand at the origin, so that

fW

f^

(tx)dt=x»--±-

/:

2

^(«™>^

Now WJy (w)

!/An-*>i

CÂ n \/x

J

-

c

4ci

c2

and, from this result, (1)

is

A n (1 — x) six

evident;

a* f

t>

+1

evident that

it is also

Tn (t,x)dt-x» + l

Jo

tends uniformly to zero as n -* It will

oo so

^#^ 1

long as

— A.

be observed that the important expansion

2J - (

**+»-«* I

(4

which was formally obtained in

Formula

(2) can

§

y>

(0
m=\ JmJ v+i \Jm) 1812, is an immediate consequence of (1).

be proved in a somewhat similar manner (though the more elaborate) by using an integrand involving

details of the proof are rather

functions of the second kind. +

It is easy to see that

/W. .)*- i ^ 'J>Uf)f.

JO

ft

»=1

= hm

-^

u~*)

+l Jm*f v+i\Jm)

{

J¥ (w)

F„ (xw)

- Jy (xw) Y¥ (to)}

V,

.

.

A

~

.

'


18*22]

587

Now take < x ^ 1 in the last integral and substitute for the Bessel functions the dominant terms of their respective asymptotic expansions, valid when w |

large (§7-21).

is

j

The

error produced thereby in the integrand ; and, as n ôo, we have

0<#<1

0(l/w*) when

v?

J Jn-ooi

Now

-

—

lim

[

.

An+m C0S w ~ OTr ~ ( * I*)

b-*.» 2iriJ

-*«fshall

type; so

is

\™

.

=

at mosti

\n)

the result of substituting these dominant terms A * +m s iD w 0-- x - i*") 7 ) sin (**» ~ [

—

lim

We

An

i

is,

cos ( 2*w -w-\inr-\ir) w cos (w — £ vir — ^tt)

H -Bi

lim I*'*" B-âaJ^-Bi

008(2^-^-^^-^) wcos(tt/-£w — £tt)

have to

discuss, almost immediately, several integrals of this general convenient at this stage to prove a lemma concerning their

it is

boundedness as n -*Lemma. The

oo

integral

lim

f^**

«»(*«• -*»»-* *)

9-+-<*> B-**>JA J n -BiV>coa(v>-$vv—\ir) j.

is

(l/»),

-1
as »-*.» , if

If we put

,

consideration

may be

written in the form

r "£%'*

2;Ucos(A-l)A.

tA cosh v /o 04»*+^)

L

When — 1 < X < 1,

— first

Again,

if

part of the

Â < 1

v

.

n

1 »_|

the modulus of this does not exceed /""

-4»

and the

(A *+v*) cosh 'jof"J2^*

-sin(A-lM n

J•

coshAg.cfo cosh »

2 ^4 n *

/""

J

t'lsinhAi;!^ ' cosh v

Lemma is obvious. (1 - A) =£, we have* £v (1 —A) sinh Xv=£sinh (fl— £) ^ cosh v

and v

t

so the integral to be considered does not exceed (in absolute value)

9j

and the second part of the

dv

["

Lemma

is

t"

+1

dv

Lemma

that

proved.

It follows immediately from the

f

{">

iOA

Tn (t, x)dt

= \x* +

(1/n),

o

when

< x < 1 and ;

this is equivalent to (2).

The function £sinh(t;-£) has one maximum, at & sinh* (v - &)/cosh (v- f ) which is less than sinh (t> &)• *

say,

and

its

value there is equal to

V


588 Moreover,

we

J

.

we

if

from the

infer

close the range of values of

Lemma +1

f

Lastly

we

n

-*-

1,

P f^ Tn (t, x) dt

Tn (t, x) dt,
when

oo

x on the right so that

that the integrals

Jo

Jo are bounded as

[CHAP. XVIII

1.

shall consider

t»

I

+1

Tn (t,x)dt,

Jn

and we

shall prove that,

ofn,x and

function

— x + 1 ^ 1,

f

n -*

oo

1

and

1,

to integral is a bounded

.

shew by the methods which have just been used when t^x, then

It is easy to 1

as

t,

< t£

when

i.e.

that,

when

f +1 Tn (t,x)dt

Jo %t» +1

= I

J v (jm x)J v+l (jm t)

£ f^V. <«>*<«>- A (~)r.<«))^¥*»

- im l

*l

B-*-<*>

J

'>v\'")

A n -Bi

sm w(l—x). SJn (frW — ^ 1/7T — \ IT ) ~ w cos (w - £*7r - £tt) VZj J,™* TTiat J An-Bi r^n+Bi cos ^^ + frw — w — \vtt — \tc) / i'+i ~ \AJ ~ jbÎo 27m? _ w cos (w - \vir - \ir) 4n p+k fAn+Bi C0S(lV + tW — XW — hviT — \ir) + hm ^—— t J / dw. £"+*

1 \

/

[An+Bi

M

,

fl j

—

These integrals are of the type examined in the section

and

;

and sO the

To prove that the f

t

v+1

i.e.

when

integral

is

Lemma

given earlier in this

— l<#+£— 1^1

bounded when

original integral is

— 1 < 1 + £ — x^l,

,

1

< t^. x ^ 1. bounded when 0
first

shew that

Tn (t,x)dt

Jo

= hm

(J„ (w) Fh-i (tw)

-^t-

- J v+l (tw)

Y

v

(w)\

—

\

,

and then apply the arguments just used in order to approximate to the integral on the right the details of the analysis are left to the reader. ;

It has therefore been proved that, if i

A

be

an arbitrary positive number, then

ct

V +x Tn (t, x) dt
where

JO

U is independent of n,x

and

t

when

A ^ x 4 1, A ^ t ^ 1.

,


18*23]

589

These results constitute the necessary preliminary theorems concerning and we are now in a position to discuss integrals, involving T (t, x), n which occur in the investigation of the Fourier-Bessel expansion associated with an arbitrary function f(x).

Tn (t, x),

The analogue of the Riemann-Lebesgue Lemma*.

18*23.

We

shall

now prove

that, if (a, b) is any part of the closed interval (0, 1), not an internal point or an end point of (a, b), then the existence and the absolute convergence of

such that x

is

f**f(t)dt J

are sufficient to ensure that, as n

a

-*-

oo

6

[

wherej

tf(t)Tn (t,x)dt

= o(l),

0<#<1.

The reader

will observe that this

theorem asserts that the only part of the

path of integration in

F tf(t)Tn {t,x)dt

Jo

which

is

of any significance, as n -* oo

,

is

the part in the immediate vicinity of

the point x.

convenient to prove the theorem in three stages. It is first supposed is bounded and that the origin is not an end point of (a, b). In the second stage we remove the restriction of boundedness, and in the third It

is

that t*/(0

stage

we remove the Let

(I)

and

restriction concerning the origin.

t-"f{t)

= F{t){*-<*),

the upper bound of \F{t)\ in (a, b) be K. Divide (a, b) into p equal parts by the points t1 t*, ...*,_, (t = a,t = b); and, after choosing an arbitrary p positive number e, take p to be so large that let

,

,

p

2 ( Um - Lm ) (tm —
where

e,

Um and Lm are the upper and lower bounds of F(t) in

^

so that

^<0 = *(«m-0 + |

a>

m (t) $ |

Um - Lm in (tm_

1

,

m_

(t

x

,

m ).

t

«-(*)•

m ).

t

It is then evident that Jf«

tf(t)Tn (t,x)dt =

|

*(*„_,)

m=l

f"

Jt m-i

+ 1 * Cf.

t If

Modern Analysis,

x=l,

it ie,

t"+*Tn (t,x)(t*-x?)dt

/

t^Tn (t,x)(t*-a*) a)m (t)dt

§ 9-41.

of course, supposed that

b
1

'


590

and hence, by the inequalities (10) and (11) of §

[CHAP. XVIII

18*21,

Sc^Kp \\f{t)Tn {t,x)dt

— x — b) \/x

TTC
J a

4c

that

'/(*)r.('.«)'»|<

the choice of e fixes

are at liberty to choose

p

;

An

1wtf( g

when

(and therefore

e

8 Cl

2

has been chosen,

That

is

left less

to say,

than

6

Consequently the integral iso(l)as^l n theorem to be proved.

arbitrarily small.

this is the

When F (t)

choose r intervals

fi,

is

we

by a

— X — b) *Jx

2

(II)

^>)

A n > 2Kp/e.

the integral on the

,

7TC2 (2 is

_,. t) vg[^f+'J-

so large that

A n we may make

suitable choice of

which

Jtm -i

a;

to say

is

/>

Now

fm

p

2

(2 - - 6) V#m=l

2 7rc 2

not bounded throughout

such that F(t)

is

(a, b), let it

-*-

oo

,

and

be possible to

bounded outside these intervals and

such that

2

When

t

lies in

!

\F(t)\dt
one of the intervals

/u.

we use the 4c

inequality

2

l^ (P -^)r„(^)l^ W(2 _;_ 6)-^, and hence, intervals

If

If is the upper bound of \F(t)\ in the parts of (a, 6) outside the by applying (I) to each of these parts, we have

if

/*,

we take

e sufficiently

sufficiently large,

the expression on the to

small (thus fixing

and

left) arbitrarily small,

A

K) and

we can make the expression on the

then take n to be right (and therefore alsc

this is the result

which had

be proved. rb

(III)

If

I

&f{t)dt

exists

and

is


we can choose

Jo so small that

p ]<*/(<)

\dt

<

e,

and then, since we have 4 Cl2

n

\i tf(t)Tn (t,x)dt

2

Trc 2

(2

— x — b)
2 «

-a-2

eft.

i

:


18*24] it

591

follows from (II) that

\\f(t)Tn (t,x)dt j

K

where

7rc 2

o

2

8d2 —x—

(2

is

the upper bound of \F(t)\ in

it

follows that the expression

\r + l)Kp

3e

An

2

b) >Jx (v, b)

when the

intervals

^ are

omitted.

Hence

small by taking

Lebesgue

Lemma

sufficiently large, is

on the left can be made arbitrarily and so the analogue of the Riemann-

completely proved.

The Fourier- Bessel expansion.

18*24.

We

n

now prove the following theorem*, by means of which the sum of the Fourier-Bessel expansion associated with a given function is determined shall

Let f(t) be a function defined arbitrarily in the interval (0 <*/(*) dt exist J

and (if it

is

"m =

Let

an improper integral)

/8

**

where v

v+i

,

.

,

\jm)

~

let it

1)-

and

let

be absolutely convergent.

f tf(t)J„(jmt) dt,

+ § > 0.

Let x be any internal point of an interval (a, b) such that such thatf(t) has limited total fluctuation in (a, b).

< a < b < 1 and

00

Then

2

the series

m=l is

convergent and.

We

its

sum

observe that,

first

a m J v (jm x)

% \f(x + 0) +f(x — 0)}. by §§ 1'8-21, 1822,

is

2 am Jv (jm x) =

m-\ J {/(*

f

Jo

tf(t)Tn (t,x)dt,

~ 0) +/(* + 0)} = lim x~»f(x - 0) f n-*oo

VT 1

n

(t,

x) dt

J o

+ lim x~'f(x + 0) [\^Tn (t, x) dt. Hence,

if

8* (*) s

V*

j

{IT' f(t)

- x-»f(x - 0)} Tn (t, x) dt +

it

is sufficient

to prove that

*" +1

f

{«-/(«)

Sn (x)-+0

as

- «r*/(* + 0)} Tn (t, x) dt,

n-oc

in order to establish the

convergence of *00

2 to the

0>mJv (jmX)

sum \ {f(x + 0) +f(x - 0)}. *

Hobson, Proe. London Math. Soc.

(2)

vn. (1909), pp. 387—388.


592

We now

[CHAP. XVIII

discuss 1

V+

1

.

- xr»f(x + 0)1 Tn (t, x) dt

[tr"f(t)

J iX

and the reader can then investigate the other integral involved same manner.

in detail,

Sn (x)

in

in precisely the

The function t~ v f(t) — x~ v f(x+0) has limited and so we may write*

total fluctuation in {a,

(t) t~"f (t) - or* f(x + 0) = X* (*)» functions of positive increasing bounded are an d Xi (0 X2 (0

b),

x

where

t

in (x,

b),

such that

Xi(*+0)-#(* + 0)«0. Hence, when an arbitrary positive number positive number 8 not exceeding b — x, such that

whenever x < t < x + We then have

-/x+S p+i r

v

|

+ 0)} Tn (t, x) dt

f ( t ) _ a;—/ (a; + <" +1

'

J x

0)}

Tn (t, x) dt X+S fa;+a

™

ra;+8

We now

chosen, there exists a

8.

'V+i it- f(t) - ar*f{a

+

e is

X! (0

Tn (t, x)dt-

1

>

*** X

x, (t) Tn (t, x) dt.

obtain inequalities satisfied by the three integrals on the right.

from the analogue of the Riemann-Lebesgue lemma that the modulus of the first can be made less than e by taking n sufficiently large. It follows

Next, from the second mean- value theorem and 8 such that between £

r

+S t»+>

Xl

(t)

Tn (t, x) dt =

it

follows that there is a

('*'

xx (*

J X

+ 8) J

*" +1

Tn <*» *> dt

number

>

*+f

and, by § 1822, the modulus of this does not exceed 2Ue; and similarly the modulus of the third integral does not exceed 2Ue. By treating the integral and *ina similar manner, we deduce that, by taking n between the limits sufficiently large,

we can make the

difference

S am J v {jm x) and } {/(* + numerically less than (8Z7

+ 2)

e;

and

between 0) +/(*-<>)}

this is arbitrarily small.

Hence, by the definition of an infinite

series,

we have proved

that, in the

00

circumstances postulated,

2 m-l

and

this is the

am J v (jm x)

is

convergent and

i {/(* + <>)+/<* -0)}; theorem to be proved. * Cf.

Modern Analysis,

§ 3-64.

its

sum

is


18-25]

593

The uniformity of tlie convergence of the Fourier-Bessel expansion.

18*25.

Let f(t) satisfy the conditions enunciated in §1824, and also let/(<) be continuous (in addition to having limited total fluctuation) in the interval (a, b).

Then

the Fourier-Bessel expansion associated withf(t) converges

the sumf(x) throughout the interval (a

uniformly to

+ A, b — A) where A is any positive number.

This theorem is analogous to the usual theorem concerning uniformity of convergence of Fourier series *; the discussion of the uniformity of the convergence of the Fourier-Bessel expansion near x = 1 and near x = requires rather more careful consideration, in the

untrue when x to

=

first

place because formula §

and in the second place because examine the bounds of 1,

it is

1822 (1) is

not practicable

t

P^Tnit^dt, Jo

when x and

t

are small, without using approximations for Bessel functions of

the second kind.

The difficulties in the case of the neighbourhood of x = 1 are easy to overcome (cf. § 18-26); but the difficulties in the case of the neighbourhood of the origin are of a graver character and the discussion of them is deferred ;

to § 18-55.

We shall prove the theorem concerning uniformity of convergence throughout

(a+ A, 6- A) by a recapitulation of the arguments In the

choice of 8 which (a

of the preceding section.

since continuity involves uniformity of continuity f, the was made in § 18*24 is independent of x when x lies in

first place,

+ A, b -

A).

Next we

discuss such an integral as

fx J

v+1

~ *"/(*)} Tn (t, x) dt.

{'""/CO

Since 8 is independent of Lebesgue lemma (§ 18-23) that

follows from the proof of the

x, it

Riemann-

this integral tends to zero uniformly as n -» oc

,

provided that

F

t»+i{t-»f(t)-x-''f(x)}dt

J x-\-6 X

is

a bounded function of x.

Now l

11*1

l^r^^f^-^'f^)} dt U I

and this is bounded in (a bounded in this interval. * Cf.

Modern Analysis,

+ A,

b

\o

\t h f(t)\dt+\x-*f{x)\

•/

- A)

since f(x)

is

[\»+idt, Jo

continuous and therefore

§ 9-44.

t Cf. Modern Analysis, § 3 61. It is now convenient to place an additional (trivial) restriction on s, namely that it should be less than A, in order that the interval Ix- d, x + d) may lie inside the interval (a, b).


594

[CHAP. XVIII

Similarly the other integrals introduced in § 18*24 tend to zero uniformly,

and so n

2 am J„(jm x)-f(x)

m=l

tends to zero uniformly as w-»oo

and

this proves the

the convergence

The uniformity of

18*26.

near

,

theorem

stated.

of the Fourier-Bessel expansion

x=l.

It is evident that all the terms of the Fourier-Bessel expansion vanish at

the point x

= 1,

so that, at that point, the

sum

of the terms of the expansion

is zero.

Since uniformity of convergence of a series of continuous functions involves the continuity of the sum,

it is

evident that the condition

/(l-0) = is

necessary in order that the convergence of the Fourier-Bessel expansion

associated with/(j5)

We

shall

The

analysis

may be uniform near x = l.

now prove

is to be continuous in conditions stated* in § 18*24, the combined with (a, 1) and that /(l) is zero, throughout (a + A, 1). are sufficient for the convergence to be uniform is

that the conditions that f(x)

almost identical with that of the preceding section;

V+i

\t-"f(t)

-x-»f(x)}

we take

Tn (t,x)dt,

/,

just as before, and (0,

(x

x

— 8,

we then

divide the interval (0, 1) either into three parts

— 8,x + 8), (a* + 8, 1), if x ^ 1 — 8, or x ^ 1 — 8. And we then prove that the 1),. if

— 8),

(x

integrals, as the case

Again,

may

1

—

when /(l) = 0, we can choose

o\

$x$

x

— 8),

be) tend uniformly to zero. 8t so that

\x~>f(x)\<6,

when

into two parts (0,

three integrals (or the two

\f(x)\<€

1.

Then the expression

f{x)-x-f{x){\'^Tn {t,x) dt Jo

tends uniformly to zerof as «-*oo expression does not exceed

(1-8,, *

lies in (a + A, 1— 8J, and the any value of n when x lies in

when x

(U+l)e

for

1).

The

interval (a, b)

is,

of course, to be replaced by the interval

(a, 1).

t Because the integral involved tends to x" uniformly throughout (A, 1

- S t ), by

§ 18*21.

.


18-26, 18*27]

595

Hence we can make

f(x)-x-f{x)\\^Tn (t,x) dt .'o

arbitrarily small for all values of

a; in (a + A, 1) by a choice of n which is independent of x\ and this establishes the uniformity of the convergence of

\\f(t)Tn (t,x)dt

Jo to the

sum/(ar) in (a

+ A,

1) in the postulated circumstances.

The order of magnitude of the terms

18-27.

in the Fourier-Bessel series.

It is easy to prove that, if t*f(t) has limited total fluctuation in (a, b), any part (or the whole) of the interval (0, 1), then

tvhere (a, b) is

fjf(t)J v (\t)dt = 0(^j as \-*-oo

From

this

theorem we at once obtain Sheppard's result* that

when 0<#$1;

this equation, of course, has a

theory of Fourier

well-known parallel in the

series.

We first observe that, as a consequence of the asymptotic expansion of § 721, I

r*

Mo where

c is

Now

| '

/

J

a

a constant, independent of

write **/(*)

in (a, 6);

t*Jv (t)dt < t

when

c,

t

lies in

the interval

= >M0 ~ ">M0>

where fr(t) and and then a number £ exists such that

= ^(a)

*i (0 **
*

f

<*

J a

^

%

(t)

(0, oo

).

are monotonic

Jv (\t) dt + ^ (b) f V «/, (Xt) dt Jf

<2c{|+i(a)|

+ |*i(6)|}X-»

= 0(\-*).

A similar result If

it is

holds for

known merely

yfr2 (t),

and hence the theorem stated

is

evident.

that rh

f t*f(t)dt J a

and is absolutely convergent, then theorem that exists

f

J a

*/(0 Jv (M) dt

* Quarterly Journal,

all

that can be proved

= o (1/V\).

xxm.

(1889), p. 247.

is

the


596 This theorem

is

due to W. H. Young*, and

it

the same manner as the theorem of § 18'23.

when

|

t*f(t)

is

struct the proof,

Divide

and

(a, b)

[CHAP. XVIII

may be proved

We

in precisely

out the proof

shall write

bounded, with upper bound K, and leave the reader to confunction is unbounded, on the lines of § 1823.

when the into

p

equal parts by the points

tx

t2 , ...

,

,

=

t -, 1 -(t

p

«,
= b),

the parts be so numerous that

let

2

{t

m — £m _i) ( Um — L m) <

€,

r»=l

where

Um

Next

and

Lm

are the upper and lower bounds of t*f{t) in

=

tlf(t)

let

F(t) = F(tm^) +

F(t),

oy

(t

m -i, tm ).

m (t),

and then I

\\f{t)Jv (\t)dt\
\Ja

I

where

c

is

m

I

m = l>

tU

P

the upper bound of \t*J v

The theorems of

(M)dt\+ £

(t)\ in

1-8-23,

this section can be

tm

m=l J['m-i

I

reasoning resembling that used in § and this is the theorem to be proved.

(0^#<

v

J tm-i

the interval

\tiJ v (\t)a} m (t)\dt

(0, oo

the integral on the

made

).

Hence, by (Ar J ),

left is o

to cover the closed interval

1) in the forms

This

is

Hence

evident

when

the general

it is

remembered that

term in the Fourier-Bessel series associated with fix)

tends to zero {after multiplication by »Jx) throughout the interval (0 ^

x^

1)

if x*f(x) has an integral which is absolutely convergent ; and, if this function has limited total fluctuation, the general term tends to zero as rapidly as l/jm

.

18*3.

Th? application of the Hankel-Schlafli methods

to Dini's expansion.

We shall now consider a class of contour integrals by means of which we can obtain theorems concerning Dini's expansion, analogous to those which have been proved for Fourier-Bessel expansions, either in a direct manner or by means of the corresponding theorems

for

The Dini expansion associated with/(#)

Fourier-Bessel expansions. is

00

2

bmJrCkmX),

where \i,X 2 \» ...are the positive zeros (arranged in ascending order of magnitude) of the function *Jw'(z)+HJ.(z), >

* Proc.

London Math.

Soc. (2)

rem.

(1920), pp.

169—171.

;

18 * 3 ]

DINI SERIES

where

H and v are real constants, and v

The

597

coefficients bm are to

bm .

f Jo

+ iÔ.

be determined by the formula

tJ* (Xnt) dt = [\f(t) Jv (\J) dt Jo

so that l

2\m*j tf(t)J„(Km t)dt

Before proceeding further, we shall explain a phenomenon, peculiar to certain Dini expansions, which has no analogue in the theory of Fourier-Bessel expansions.

The investigation of Dini expansions which has poles at the zeros of

is

based on properties of a function

z-{zj;{z) + HJv (z)};

when

and,

H+v= H+ v

Further,

0, this last function

if

is

has a zero at the

origin.

negative, the function has two purely imaginary zeros.

only to be expected that these zeros should contribute to the terms of the series, and such a contribution in fact is made. It

is

If

H+ v = 0, an initial term 2( J

(1)

/

+ 1)#"

[\»+if(t)dt Jo

has to be inserted on account of the zero at the origin. If initial

H+ v

is

negative and the purely imaginary zeros are ftX,, then an

term

K

(V + P»)/.

1

'

/,

(5l.)-V/r (^)io

must be inserted on account of the

zeros + i\

/()y, (V)Ctt '

.

These initial terms in the respective cases will be denoted by the common symbol <%(#), so that the series which will actually be considered is

# (*)+ m—2 bm J (K zero when U + v positive and v

n x),

1

where <%(#>

is

(1) or (2) in the respective eases

is

is

denned as the expression

H+v = 0, H + v<0.

[Note. The fact that an initial term must be inserted when ff+v=0 was noticed by Dini, Serie di Fourier (Pisa, 1880), p. 268, but Dini gave its value incorrectly, the factor xv being omitted. Dini's formula was misquoted by Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), p. 354. For corrections of these errors, see Bridgeman, Mag. (6) xvi. (1908), pp. 947—948; Chree, Phil. Mag. t (6) xvn. (1909), pp. 329—331 and C. N. Moore, Trans. American Math. Soc. x. (1909), pp. 419—420.] Phil.

j

' '


598

We now

[CHAP. XVIII

consider the function

2wJy (xw) Jv (tw)

Jv (w) {wJJ (w) + HJV («/)} This function has poles at ji,j2 ,j3

The residue

,

at^

of the function

Xj, Xa, X,,

...,

...,

(0 or

±

i\o).

is

2Jy(jm x)Jy{jm t)

The residue

J

\ m is 2XTO J, (Xmiv) Jv (\mt)

at

(Xm ) \\n JJ' (Xw)

v

+ JJ (Xm ) + H JJ (\m)} 2 \ m2 J y (\fnOC) J p (Xmt)

X^ - VZ) JS (Xm ) + X„» Jy'^ (X^) when H+v = is

*

(

The residue

at the origin

-4>(u + l)x"tv

The

residues at

i\ when H+j/is

±

negative are both equal to

2X 2 /y (X (x„a

Now

let

Dn be

a?)/y (XoQ

+ ir-) /,* (x ) - x 2 //« (x

a number, which

D

Sn (t,x; H) = 2

•

)

between X„ and X„ +1 so chosen that and let jjf be the greatest of the

lies

,

not equal to any of the numbers jm numbers jm which does not exceed n it is

Let

.

;

.

2J

^^~'^-S4

a

m=l

«»

i>+i

»

(x,t)

\Jm)

2\^Jy(\m X)Jy(\m t) + KfJJ'iKd'

2 2 m=l (Xw -I^)^ (^m)

where £4

(x, t) is

defined to be

0,

2 •(*/.+ l)xv t* or

2X 2 /p (A ar)/> (X

(V + v ) L 2

according as

H+

i>

is positive,

2

CK)

~X

2

Q

//" (,Xo)

zero or negative.

Then, evidently,

2 am Jy ( m x) - JT

m-1

We

shall

< x<

1,

(x)

- t bm J

v

m=l

(Xnx) = f\f(f) Sn (t, x

now prove a number of theorems leading up the existence and absolute convergence of

Jo are sufficient to ensure that, as n -* oo

,

tf{t)Sn (t,x;H)dt = o(l). /:

;

H) dt.

JO

to the result that,

when

DINI SERIES

18-31, 18*32]

599

This equation enables us to deduce the properties of Dini's series in respect of convergence* from the corresponding properties of the Fourier-Bessel series.

18*31.

8n (t, x

The contour integral for

;

H).

It is evident from Cauchy's theory of residues that

2wJ (xw)J (tw)dw H\-— [ 2^]^^ Jv wy]w j; {w) + HJw Dn+zoi

v

(t,x;ll)-

n it

v

.

(t0)

(

p

1_ 27rt

wi

2wJv (xw)Jv (tw)dw

f J -ooi

Jv (w) \wj; (w) + HJV (w)\

'

where the symbol P denotes Cauchy's principal value.' The integrand being an odd function of w, the second integral vanishes, and so we have '

m K)

«

nK(t v '

An

J_ [»»+"* 2wJ (xw)J,(tw)dw m _~2TriJ J (w){wJ '(w) + HJ v

)

'

J)n - aDi

immediate consequence of

,(w)}-

r

l>

this formula (cf. §

l

1821)

is

that

c3

|iS;<-U;10|*. (2-X-t)y/(xt)'

(2)

where

.

cs is

independent of

n,

x and

t.

Also

and hence ,

t"+*S»v. n (t,x;H)dt\^

where

c4 is

18*32.

/-j

•

!/.o

independent of n, x and

The analogue for

We shall

now prove

of the interval

Sn (t,

4

(2-x-t)Dn ^/x'

t.

H)

x;

of the Riemann-Lebesgue lemma.

the theorem that, if

(0, 1), then the existence

and

(a, b) is

any part (or

the whole)

absolute convergence of

"fif{t)dt

J a I

are sufficient

to

ensure that, as n -*- oo

,

b

f tf(t)8n (t,x;H)dt = o(l), J a

< x < 1. And, if b< 1, the theorem is valid when < x $ 1. The proof has to be divided into three stages just as in the corresponding theorem (§ 1823) for Tn (t, x). We shall now give the proof of the first stage, when it is supposed that t*f(t) is bounded and a > 0. The proofs of the re-

provided that

maining stages should be constructed by the reader without *

Except

at the point

x=l.

difficulty.

.


600 Let

and

t-"f(t)

the upper bound of

let

Divide

(a, 6)

|

^(0

(

a

>

and, after choosing an arbitrary positive p

1

Um and L m are

where

(

a»

m (t) ^ j

be

if.

number

Um - Lm) (tm -

t2

,

...

,

tp^. (t

Um — L m

in

= a,tp = b),

take p to be so large that

e,

m ~!) <

t

e,

(<„*_!, £ TO ).

= F («„_,) +
F(t) |

°)

the upper and lower bounds of F(t) in

Let so that

F(t),

by the points t u

parts

p equal

into

m

!

=

[CHAP. XVIII

(£ OT_i, £ TO ).

Then tf(t)Sn (t,x:H)dt /,

= I Fit^f**

"Sn (t,x;H)dt + I f

W

J

t»

t»+*a> m

(t)Sn (t,x;H)dt.

Hence, by §1831, /"

and

if

^7

6

we now take n

|

J

so large that

*«/(«)

left

Dn ec >2Kpc 3

s» ft * ff ) ;

*

< |

and the expression on the right the

tends to zero as

[2Kpc<

1

(g

i

,

we have

_*.V»

arbitrarily small.

is

1

Hence the

integral on

n -*- oo

the reader has removed the restrictions concerning boundedness and the magnitude of a by the method of § 18'23, the theorem is completely

When

proved.

As a

corollary, it should

be observed that Cb

a* j

tends uniformly to zero as w if b

**

1,

where

18*33.

A

Dim 8

is

an

—

tf(t)Sn (t,x;H)dt

a

oo

when

^x^

1 if 6

<

1,

and when

^x

*S

1

-A

arbitrary positive number.

expansion of an arbitrary function.

immediate consequence of the result of the preceding section the existence and absolute convergence of the integral

An

is

that

\*f(t)dt /,o

with/(#) behaves are sufficient to ensure that the Dini expansion associated (or summability), as the Fourierin the same manner, as regards convergence Bessel expansion throughout the interval (0 < x < 1).

DINI SERIES

18-33]

For

it is

evident that

M

(x)

N n + 2 &m «/„(Xm#)- 2 am Jr(jm x) m— 1

tn—1

when < a; < 1 when ^ x ^ 1 — A.

tends to zero as n -*-oo

uniformly to zero

Now, (§

1523),

since the it

601

n

this

sum

(multiplied

by

>Jx)

tends

numbers \ m and jm which exceed v are interlaced has the same value that Dw may be chosen so that n — |

|

N

follows

for all values of

and

;

a certain stage.

after

Therefore, since n

uniformly throughout

(0, 1),

am Jv (jm x)~*0,

2

x*

we have proved that n

x

2 xi{bm J (\m x)-amJ (jm M (x) + »=1

tends to zero, as w-*-oo

That

is

,

x)}

v

v

uniformly throughout

(0, 1

— A).

to say, the series 00

xi&

(x)+ 2 x*{bm Jv (\m x)-am Jv (jm x)) m=l

is

uniformly convergent throughout

(0, 1

— A)

and

its

sum

is zero.

It follows from the 'consistency theorems' concerning convergent series* that, is

when the

(uniformly)

series is 'summed' by Cesaro's means, or any summable and its sum is zero.

similar method,

it

'

'

Hence, if for any particular value of x in the interval

(0, 1

— A),

the series

00

2 xla m Jv(jm x)>

m=l

associated with f(x), is convergent (or is

summable by some method), then

the

series

X*M

(x)

+ 2 xtbnJ.Q^x) m=l

is

convergent (or is

same

summable by

the

same method) and

the two series have the

'sum.'

And if, further, the Fourier-Bessel series (multiplied by \/x) is uniformly convergent (or uniformly summable) throughout an interval (a, b), where

Qâ
1,

then also the Dini series (multiplied by \/x) is uniformly convergent (or uni-

formly summable) throughout

(a, b).

In particular, iif(x) has limited total fluctuation in

(a, b)

where

0â<6< 1, * Cf.

w.

b. p.

Bromwich, Theory of

Infinite Series, % 100.

20


602

[CHAP. XVIII

then the series

m=l converges to the

sum * {/(* + <>)+/(• -0)}

at all points

x such

a + A^x^b — A, where

that

the convergence is uniform, if

The value of Dini's

18*34.

We

f(x)

shall

now complete

A

is arbitrarily

continuous in (a,

is

= 1.

x

series at

small ; and

6).

the investigation of the value of the

sum

of Dini's

by considering the point x = 1 and we shall prove the theorem, due to Hobson*, that, if/(a) has limited total fluctuation in the interval (a, 1), the sum of the Dini expansion at x = 1 is /(l — 0).

series

;

We first write Tn {t,x; H)=Tn (t,x)-Sn {t,x; H) =

A 4-

2\wt2 J y (\m x) J v {\m t)

*?

and then we have CB»+"i W (W, X) J, (tw) dw

1

27rt'Ji>n _aoi

Tn {t,x;H) = \

1

f

]D»

2iri J Dn -9>i

k

wJJ(w) + HJv (w)'

+ " i w(w,t)J v (xw)dw

w Jv

(w)

+ HJV (w)

'

where

(w, x)

=

7T

[{w j; (w)

+

#«/, («/)} 7, (#w)

The former representation latter when 0< x< t^ 1. [Note. tions of

Tn

(t,

Tn

of

Tn

by

and

x) given

x H) analogous ;

§ 18*21 (7) -

to § 18 21 (6) is

Y: (w) + HYV (v>)} J, (xw)]. valid

when 0<

t

'

< t < 1.

We

t

;

series.]

v+1

Tn (t,l;H)dt

have

/V T(tlH)dt- lim

* Proc.

< x^

1,

the

x; H) are strictly analogous to the representa1821 (8) the fact that there is no formula for the reason why Dini series were discussed in § 18-33 §

consider the value of

f Jo when

is

(t,

with the help of the theory of Fourier-Bessel

Now

{w

Tn (t,x\ H)

These representations of (t,

-

'—

Dn+m [

JJ-ôc

Tl J Dn-Bi

Bôo

-JTl

London Math.

J

»(»M)^, («*)<** WJ W

(W) +

HJ

Dn - Bi \wJ„+i(w)

Soc. (2) vii. (1909), p. 388.

V

(W)

\WVJ

DINI SERIES

18-34]

For any given positive value of bounded function of t in the interval

And when t =

follows from §

(S, 1).

When

when n-*oo

has the limit 1

1, it

8, it

603

8

** t *S

1821 that 1

— h, it

this is

a

(1/Dn).

is

.

It follows that

I bm J S (1) +«=i

(\ m )

v

-/(l -

0)

=

I

V' {r"/(0 -/(l - 0)} Tn

(*,

1

;

H) dt.

Jo

Since t~"f(t) — /(l — 0) has limited total fluctuation in (a, 1) we may it in the form X\ (0 ~~ X* (0> where X\ (0 and X* (0 are bounded positive

write

decreasing functions of

such that

t

Xi(i-o)=x*(i-<>)=o. Hence, given an arbitrary positive number B, not exceeding 1 — a, such that

e,

we can choose

a positive

number

whenever

We f

1

— 8 **

£ •$ 1.

then have

V+

1

[t-"f(t)

-/(l - 0)}

Tn {t,l;H) dt

Jo

= P~V> {*-7(0-/(l

-

0)} T»(*,l;

5)*

Jo

+ P <" +1 X* (0 T* (*• l\H)dt-C t " +1 x, (0 TM (*, 1 #) cfc. Ji-s Ji-« ;

By arguments right

is

similar to those used in § 18*24, the first integral on the

o(l) as w-*oo

;

and neither the second nor the third exceeds

2elim in absolute value

(cf.

§ 18*24),

|

f +1 Tn (t,l; H)dt

and this expression

is arbitrarily

small.

It follows that

lim f

V

+1

{f'f(t) -/(l

- 0)} Tn (t, l;H)dt = 0,

n-ât>J

and so we have proved

that, in the circumstances postulated at the

beginning

of this section,

m=l converges to the

sum /(l — 0).

This discrepancy between the behaviours of Dini series and of Fourier Bessel series (§ 18*26) is somewhat remarkable.

»


604

The uniformity of the convergence of Din%s expansion in an interval

18*35.

extending

[CHAP. XVIII

x= 1.

to

Because Dini

series

do not vanish identically at x —

that /the condition that f(x) is continuous* in existence and absolute convergence of

seems not unlikely combined with the

1, it

(a, 1),

\\*f(t)dt, Jo

and the condition that f(x) has limited in (a

+ A,

We

may

total fluctuation in (a, 1),

be

ensure the uniformity of the convergence of the Dini expansion

sufficient to

1).

shall prove that this

The reason

for

in fact, the case.

is,

the failure in the uniformity of the convergence of the

Fourier- Bessel expansion

(§

x

18*26) near

—1

was the

fact that

v+ *T (t,x)dt \ t n

Jo

does not converge uniformly to xv in (A,

1),

as

was seen in

§

1822.

We

shall

prove that, on the contrary, t

t"

+l

Jo

Tn (t,x;H)dt

does converge uniformly to xv in (A,

and the cause of the

1),

failure is

removed.

A

1826 should then enable the reader to see without Dini expansion converges uniformly in (a + A, 1).

consideration of §

difficulty that the

It is easy to see, from § 18*34, that

(t,x\H)dt

\ t*+*Tn

Jo is

the

sum

v+l 7rt

at \ lt

of the residues of

- {w YJ (w) + HYV (w)} J„+1 (tw)] x J„ (xw)l{wJJ (w) + HJ V («/)}, or + iko if H + v < 0. at

[{wj; (w) + HJ V (w)} F„+1 (too)

Xjj, ...,

\„, plus half the residues

Hence

t

f

v+1

Tn

(t,

x;

H) dt

Jo is

the

sum

of the residues of

- (2/w) (H + v) J v (xw)l{ivJ v and hence, when

n t>

+1

'

(w)

+ HJV (w)),

< x ^ 1,

m r„ v T n (t,x; H)dt = x ,

*

Without

H +r-\ v

restriction

f

z,» +0°*' f

Jw (xw) dw r//\ Lu r

on the value of /(l - 0).

/

\l

DINI SERIES

18-35, 18-4]

and the integrand on the right*

is

605

of the order of

magnitude of

eip{-(l-«)|J(t0)l] and so the integral on the right converges uniformly when A ^ x $ 1. That is to say

to zero like l/(Dn «Jx)

l t"^Tn {t,x;H)dt converges uniformly to xv in (A,

1); and we have just seen that this is a sufthe uniformity of the convergence of the Dini series associated with/(<) to the sum f{x) in (a + A, 1) under the conditions postu-

ficient condition

for

lated concerning f{t).

The

18*4.

differentiability

of Fourier-Bessel expansions.

In the earlier part of this chapter we obtained an expansion which, when written in full, assumes the form

We is

now study

shall

the circumstances in which, given this expansion,

it

permissible to deduce that (2)

/'(#)

= 2 am jmtV Jv'(jm>v x). m=l

This problem was examined by Fordf, and his investigation to Stokes' researches on the differentiability of Fourier series^. Ford also investigated the

method

is

differentiability of Dini's expansion

when

is

analogous

H=-v,

but his

not applicable to other values of H.

It is evident that

we can prove the

truth of (2)

if

we can succeed

in

proving that

= - 2 am jm>y J / (*) - ^/(*) x m-l

( 3)

v+1

and the numbers jm>v are the

positive zeros of

JT~* l#JV+, (*) + (*

Now we know thatf(x)-(v/x)f(x)

inside

(jmtV x);

+ 1) Jr+i (*)}.

admits of the Dini expansion

any interval in which the function has limited

fluctuation, provided that

fy{f'(t)-j/(t)\dt exists *

and

is

The term

is infinite; this

absolutely convergent.

wJv'(w) is more important than the term in Jv (w) except in the Uviit when shews clearly the reason for the difference in the behaviour of the Dini expansion

in

from that of the Fourier-Bessel expansion (cf. § 18-26). t Trans. American Math. Soc. vr. (1908), pp. 178—184.

H

% Cf. Modern Analysis, § 9-31.


606

The

bm

coefficients 6m are given

= {

=

[CHAP. XVIII

by the formula

*iV* Jo I V' (*) " "/(*)} J*+i ijm.pt) dt Wi (>,,) +j*m,*J'\+1 (i»,r) JV» - (" +

W

7v^)/o S r *

= j-

8

/•

^

r+V- <^<>*

x

J

Jm,v

I" jm,v

*/(*) Jv (jm,yt) dt\ jo

wu

Q»,

tf{t) Jv+1 (jm

provided that

Sufficient conditions that this

,

may be

v

t)\

= 0.

the case are

t***f(t)~0 as t+0,

(i)

/(l-0) = 0,

(ii)

f(t)

(iii)

is

continuous in the open interval in which

< t<

1.

These conditions combined with the existence and absolute convergence of

are sufficient to ensure the truth of (2) in any interval in which

has limited total fluctuation.

The summahility of Fourier-Bessel

18*5.

A

series.

consideration of the values of the coefficients in the Fourier-Bessel

combined with the expression of Tn (t, a) as a contour no easy matter to discuss by direct methods the question of the summability, by Cesaro's means, of the Fourier-Bessel expansion.

series associated with f(

integral, suggests that it is

It

is,

however, very easy to investigate the summability when the method

of Riesz*

is

used to 'sum' the

series,

and then the summability (01) can be

inferred with the help of quite elementary analysis.

The expression which method of Riesz is lira

and when mable (R).

will

be taken as the 'sum' of the series by the

I (l-if)am Jv {jm xy,

this limit exists, the Fourier-Bessel series will

be said to be sum-

It is evident that l

(1)

I (l-i?)a m J,(jm x)=f tf(t)Tn (t,x\R)dt, * Cf.

Hardy, Proc. London Math. Soc.

(2)

ym.

(1910), p. 309.


18*5, 18*51]

607

where

Tn (t,x\R)= i

(2)

- ip) -^ (}?*>/: [* m l)

(i

,

and so it will be convenient to discuss the properties of Tn (t, x R) after the manner of § 1822 before we make further progress with the main problem. j

Theorems concerning

18*51.

When Tn (t, x\R) function of

and

t

x,

Tn (t, x

\

R).

defined by equation (2) of § 18'5, it is a symmetric shall proceed to establish the properties of the

is

and so we

^ t^ x ^ 1, and we can then write down the <
function on the hypothesis that

corresponding properties when results already obtained.

We

first

observe that

™

(l

-

Tn (t, x R) |

~) [J*{w) Y

¥

is

the

sum

of the residues of

(xw)-Jv {xw)

Y„(w))

J

j^

For brevity we write

w {J, (w) Yp (xw) — Jv (xw) Yp (w)) = 4> (w, x), and then

obvious that,

it is

when* t < x,

AJ

ZlJAn-«>i\

since 4> (w, x)

We

shall

Jp (tw)/Jp (w)

now

is

Jp (w)

V

an odd function of vk

obtain some upper bounds for

Jp (tw)jjy (w) on the line joining A n — oo i to A n + oo |

4> (w, x)

|

both when w is imaginary axis

and

^t%

1,

;

i,

and when

the formulae which will be discussed are valid

w

when

is

on the ^x^ 1

the sign of x — t being immaterial.

To obtain these inequalities, we shall use series of ascending powers of w when w\ is not large, and inequalities derived from the formulae of Chapter vn when w is not small. |

|

*

as

|

When

B -»• ao

.

the origin.

t

$.

.r,

There

± iB to A n ± ill do not tend to zero indentation at the origin, because $ (w, x) is analytic at

the integrals taken along the lines joining is

no need

to

make an

THEORY OF BESS EL FUNCTIONS

608

We first

J, (tw) (1)

<êxp{-
Jy{W)

w

when

Jv (tw)/Jy (w). We

deal with the factor

|

|CHAP.

observe that*

I(w)\}

on either contour; this follows from inequalities of the type

is

§ 18*21 (9) when w is not small, and from the ascending series not large (i.e. less than jm). |

|

We

next consider 4> (w, x), which

make two investigations concerning this when — \ ^ v ^ \, the second when v ^ \.

former being valid

The

(I)

investigation

first

|

w

|

is

equal to

function, the

convenient to

is

§ 7

is

when

H® (xw) - H.® (xw) H® (w)}

\iw \HJ» (w) it

XVm

is

quite simple.

It follows from §3*6

and

33 that h

pixw

J.

I

o—ism

^.-(^K^J.i^-^K^g-jjJ

(2)

for all the values of

w

I

j

and x under consideration when

I

—\^v^\

Hence

.

|*<» *)|<êxp{(l-*)|/(«)|}.

(3)

f

When

(II)

v

>

w is not large, it is easy to deduce J¥ (w), Yv (w), Jv (xw) and Yv (xw) that

\ and

ascending series for

|

|

from the

\^(w,x)\
(4)

If

|

w

\

is

we use the

not small,

inequalities (deduced from § 7*33)

iW'WK^-

(5)

1

,

|ff,.(„)|<*gp.

together with the inequalities

<\H.n (aw)\}\ei™>\, \\H®(xw)\\.

W

and

It follows from § 3'6 |

xw

is

§ 7*33 that the inequalities (6) are true

whether

Hence,

large or not.

I

(7)

1

when v>\ and

|

w

|

|

< k2 kt {*"* + <*r'\w '} exp {(1 - x) I (w) I*

is large,

\

1

4> (w, *)

|

(3), (4)

< h (ar* + x—) exp {(1 - x) / (w)\ j

and (7) we

},

numbers Jtj, k^, is , ... are positive and independent of w, x and may, however, depend on the value of v.

* It is supposed that the

their values

},

whatever be the magnitude! of \xw\.

If we now combine the results contained in formulae deduce that, whether — ^%v^\ or v^^, (8)

j

f Provided of course that

£x ^ 1.

t;


18*51]

when

w

any point of either contour and

is

609

< x ^ 1. Hence, by

(1), it follows

that

J

^\
\^{w,x)

(9)

O v \W)

I

when

i

^ x ^ 1 and

Q^.t^.1.

We now return A n ±iv

and +

Tn (t, x R). If we replace w by and second contour integrals respectively, we

to the integral formula for

iv in

the

first

^t
deduce that, when

1,

ir.ft.|ji)i<^(^ +

We

^)j;^«.*-*g+^.

have consequently proved the two inequalities |r.(*,«|iJ)i< -j£ _ (x

(io)

(ii)

be remembered that k6

make x — t tend |

|

One

to zero, if

|

x and

r-< t "' B when

we

^ £ < a? ^ 1

vertices

+ iA n

,

;

t

is

is

a:

t,

so that

we may

so.

required in order to discuss the behaviour

are nearly equal.

To obtain them, we

write

1 )-si( -2;)*<" '>-^(5r' i

in this integral the contour

A n ± iA n

t

independent of

desire to do

other pair of inequalities

Tn (t, x R) when

(o«t<*«i),

yj |
|j..

It is to

of

\

taken to be a rectangle with

is

.

easy to see that (9) is satisfied whether w be on the horizontal sides or on the vertical sides of this rectangle and the factor 1 — (iv/A n ) does not It

is

;

exceed

V2

in absolute value at

any point of the contour.

Consequently the modulus of the integrand does not exceed k,1r* (x-l

and since the length of the contour

similarly,

when

$ x $ t^

(13) \

The

6A n we ,

;

infer that,

\^MB)\< ZA " k^f->

(12)

and

is

+ x~») V2

when O^t-^x^.

1,

.

1,

Tn (t,x\R)\<^!^±^.

last four inequalities are sufficient to

enable us to discuss adequately

the summability (R) of Fourier- Bessel series.

The reader will observe that the consideration of small values of x has increased the length of the analysis to an appreciable but not to an undue extent.


610

The analogue of Fejer's theorem.

18*52.

We

[CHAP. XVIII

can now prove that the existence and the absolute convergence of l

\

ftf{t)dt

Jo

are

ensure that the Fourier-Bessel series associated with f(t) is

sufficient to

summable (R) /(# ± 0) exist.

x of the open

at all points

And

sum (R) of the

the

interval (0, 1) at which the two limits

series is

*{/<* + Q)+/(*-0)}. This theorem

is

obviously the analogue of Fejer's theorem* concerning Fourier

series.

Since f a series which that,

when

is

convergent is

summable

(R), it follows from § 18'35

1,

X

[

lira

Hence

it

t>

follows that,

Tn (t,x\R)dt=

+l

when the

lim

V*

f

+ 0)

limits /(a:

1

Tn (t,

exist,

R) dt |

then

X +i

lim

n -»• oo J(o

We

t''

Tn (t,x\R)x-'f(x-0)dt+

are

now

Tn (t,x\R)x-'f(x + 0)dt

sum Sn (x

in a position to consider the

I 1 \ m=l (

- 2p) a m Jv (jm x) -"»»/

shall prove that it can

VT 1

!

JO f

J

and we

lim (\'+ 1 n-*>oo J x

t*+ l

n (t,

Tn

(t,

\

R), defined as

x R) x~"f(x - 0) dt |

x R) x~"f(x + 0) dt, |

x

be made arbitrarily small by taking n

sufficiently

large. .

The sum 8n (x R) \

is

equal to

V» {t~'f(t) - x->f(x - 0)} Tn

f jo

(t,

x\R)dt

l

+

t

f

v+l

{tr"f(t)

- ar'f(x + 0)} Tn (t, x\R)dt.

J X

Now, on the hypothesis that the limits f(x ± 0) exist, if we choose an arbitrary number e, there exists a positive number J 8 such that - *-v f{x + 0) < e, (x^t^x + f *-7(0

positive

We

now choose a

t- v f(f)

* Cf.

8,

~ «r*f{x - 0) < e, |

(x

> t> x -

positive function of n, say

sufficiently large values of n,

the points x ±

8),

j

I

ll

x ± a (n), X It

is

o- (n), which is less than 8 for and divide the interval (0, 1) into six parts by

x.

Modern Analysis,

S).

§ 9*4. t Cf. Modern Analysis, § 8-43. convenient to take 8 less than x and 1 - x.


18*52]

In the intervals (x

(0,

x—

(x

8),

+
— 8,

x — a (n)) and also in the intervals

use inequalities of the form given in

and (11); and in the intervals (x —
(x,

x+

we use

§

1851 (10)

inequalities

\Sn (x\R)\ does not exceed


^ît^ sr

611

{ •

* /(t) - tv+ * x~ v/{x

2k6€ (xri + x-») r f*-'

Zk6

*

+ -, A-*V_ &*Jx}f

(*'

+*

+

W

° )} dt >

V*dt

{*"/(*> 1

3A^

+

f*

tv+i dt

- *-/(* + 0)}

dt. I

x +*

n

For any given value of e (and therefore of 8), the first and last terms in this expression can be made arbitrarily small by taking n sufficiently large, on account of the convergence of

n \t*f(t)\dl /;

The remaining terms do not exceed 2ke e(3x-* + x-')

An and,

we take
if

,

small as

We |

1

this is

+

3A n*a(n)) V2

independent of

e sufficiently

small

J' n,

and

it

can be made as

initially.

can therefore make the intermediate terms in the expression for as small as we please by taking e sufficiently small, and when this

Sn (x R) |

f

U(n)

|

has been done, the

first

and

last

terms can be made as small as we please by

taking n sufficiently large.

That

is

to say, \Sn (.x\R)\ can be

made

arbitrarily small

by taking n

sufficiently large, so that

lim

Sn (x\R) = Q.

Hence lim

i(l-2p) a m Jv (jm x) = x->f(x - 0)

n—~oom = l\

-n-nl

+ x~"f(x + 0)

lim »—>»

lim n-»-

since the limits on the right exist.

oo

1

Tn (t, x\R)dt

f"H

Tn (t,x\R)dt,

f J

f Jx

THEOKY OF

612

BESSE.L FUNCTIONS

Since each of the limits on the right proved that

is

equal to

\xv

[CHAP. XVIII ,

it

has

now been

2 am Jv (jm x) summable (R) with sum l\f(x + 0) +f(x - 0)} provided that the f{x ± 0) exist and this is the theorem to be established. is

limits

;

As a

corollary, the reader should

continuous in

(a, b),

be able to prove without difficulty that, if f(t) is the summability (R) is uniform throughout the interval in which

a + A^x^.b-A, where A 18*53.

is

any positive number.

Cf. § 18*25.

Uniformity of summability of the Fourier- Bessel series near the

origin.

We shall now examine the uniformity of the summability (R) of the Fourier-Bessel expansion throughout an interval of which the origin is an end-point. tiplied

It will be supposed that the expansion is modified by being multhroughout by
tinuous in the interval (0, 6), then the modified expansion is uniformly throughout (0, b — A), where A is any positive number.

Given

whenever

e,

we can now choose

x-8^t^x+8 and

8 (less than

> 0, provided

t

A)

summable

so that

x

that

lies in (0, b

- A).

Since continuity involves uniformity of continuity, this choice of 8

may be

taken to be independent of x.

We

now

write

Sn (x\R) =

J

V> \r*f{t) - x~"f(x)} Tn

and then examine x*Sn (x R) |

We

\

express x*Sn (x

be omitted when x <

•* (

^

A n 8*

+

\

8),

R)

\

manner

after the

as the

sum

(t,

x\R)dt

of § 18'52.

of six integrals (some of which are to

and we see that x*Sn (x R) does not exceed {

j

)

'>rnrvw--/(.)|i* Jo 2kt e (x*+* + x)

["

f* -<-<*>

dt

3A n*

+ An (t-*y V2 U.-« x+° (n) +s r sAj dt 4^5 [ dt + r

A n LV 2 /Li.

+

xi

J»

I

r*

J.. rW

dt

\

1

!* + ,in)(t-xYJ

r\

\{t- y f(t)-*-f(*)}\dt.

In this formula any of the limits of integration which are negative are supposed to be replaced by zero.

FOUBIER-BESSEL SERIES

18*53, 18*54]

Now

this

upper bound

for

613

x*Sn (x R) does not exceed

|

|

j

is bounded (because it is continuous,), this can be made by a choice of n which is independent of x.

and, since x~"f(x) arbitrarily small

Consequently x*Sn (x R) tends to zero uniformly as n -*j

Now

it

has already been shewn at

18 22) that

(§

F+

f

oo

-

1

Tn (t,x)dt

Jo is

uniformly convergent in

— A), and

(0, 1

so,

since uniformity of convergence

involves uniformity of summability,

x*-»f(x) f t v+l Tn (t, JO tends uniformly to xi f(x) in

x\R)dt

— A).

(0, b

Hence, since xi Sn {x\R) tends to zero uniformly, xi [ tf(t)Tn

(t,x\R)dt

Jo

tends uniformly to x*-"f(x)

t"

Tn (t, x

+1

J

\

R)dt,

i.e.

to x*f(x) in (0, b

- AY

Jo It has therefore

been proved that OO

2 am x*Jv (jm x)

TO = 1

is

uniformly summable (R) in

(0, b

— A)

with

sum

x*f{x), provided that

fV
,

.'o

exists

and

18*54.

We

is


and that

t~ v f(t)

Methods of summing Fourier- Bessel '

shall

'

is

continuous in (0,6).

series.

now investigate various methods of summing the Fourier- Bessel

series*

2 m=0

on the hypotheses

(i)

a m xlJv (j m x)

that the limits /(a,- + 0) exist,

(ii)

that

j\if(t)dt •

exists

and

is

absolutely convergent, and

It conduces to brevity to write

tends uniformly to zero *

The

(§

fm {x)

(iii)

that the series

in place

1827) as m-ôo when x

summable

(R).

that/,,, (x)

lies in (0, 1).

factor x^ is inserted merely in order that the discussion

uniformity of summability near the origin.

is

oia m x*Jv {j w x), so

may

cover the investigation of


614 Consider

first

the limit

(l-k)fm (x)

I

lim

m

n -» qo

•»

Jn/

\

1

which gives the most natural method Since (jH /A n )

—

and

(of Riesz' type) for

summing

the series.

evident that

l, it is

I (4lLZh\ fm(x)

lim exists

[CHAP. XVIII

equal to

is

I

lim

Again, since/n (a?) = o(l),

it is

(l-Mfm

(

x).

easy to see that n

2 fm(x) = o(n),

m=\ so that

Jn

w-»-ac \

/

m=l

and therefore

I (l-k)fm{x)=

lim

li

m | (i_is) /m(a ); .

the limit on the right exists in consequence of the hypotheses made at the beginning of the section.

Again, since

whether

m be

o{n) or 0(n), lim

and

it

W

n

Jn

follows that

2,(f-™)/».W=o.

so

lim

5

(l-^/M (*)=lim lllJf)fm {x).

Consequently the hypotheses that the limits f(x ± 0) exist (0

1)

and

that the integral

Jo exists

and

is

absolutely convergent are sufficient to ensure that 00

2 am xljv (jm x)

m=l is

summable (C

By the same is

1) with

sum £#* {f(x + 0) +/(.# — 0)}.

reasoning, if /(a?)

uniform in (a + A, 6

summability (C 1)

is

— A)

;

is

continuous in

and, if a

uniform in

(0, b

—

(a, 6),

the summability (C 1) £ -*• 0, the

and i~*f{t) has a limit as

— A).

FOTTBIER-BESSEL SERIES

18-55, 18-56] 18*55.

615

Uniformity of convergence of the Fourier-Bessel expansion near the

origin.

We

can now prove, by using Hardy's convergence theorem*, that, if ^ fit) total fluctuation in (0, b), while f(t) is also subject to the conditions 18-53, then

has limited of §

1 am x*Jv (jm x) JB=1 is

uniformly convergent in

— A)

(0, b

with

sum

x*f(x).

Let h {t) be an auxiliary function defined to be equal to f(t) in (0, b) and (6, 1); and let the Fourier-Bessel series associated with h(t) be

equal to zero in

eo

w=l co

Then, by

S am x*Jv (jm x) is uniformly summable (01) throughout

§ 18*54,

m=l (0, 6

— A)

with sum xlf(x), and, by Sheppard's theorem

0(l/m), while (jm x)lJv (jm x) Hardy's convergence theorem,

a bounded function of

is

a;

(§ 18*27),

a ml»Jjm

is

and m. Hence, by

m=l is

uniformly convergent throughout

(0, 6

— A),

with

sum

x*f(x).

Again

2

(am

-aJJJ. (jm x) = ** [ V(«) Tn ft a)
and this tends uniformly to zero in (0, b Riemann-Lebesgue lemma (§ 18*23).

— A)

as n-*-oo

by an analogue of the

n

Hence

2,

am x*Jv (jm x) tends uniformly

n~*~x; and this

18 56.

is

to the

sum

x*f(x) in

(0, 6

— A)

as

the theorem to be established.

Summability of Dini

series.

Except when x = 1, the summability (G 1) of the Dini series associated with 18*53. f(t) may be inferred by combining the results of § 18*33 and §§ 18*51

—

all

The summability (01) may, however, be established independently f for < x $ 1 by replacing A n and the functions Jv (w) and

points x such that

D

HJ

F„ (w), which occur in § 18*5, by y (w) and n and the functions wJJ (w) + to the analysis may be left the of details the respectively; HY„ (w) (w) + v reader, and he will find that when x = 1 the expression \ {f(x + 0) +f(x - 0>]

wY

'

must be replaced by/(l * Cf.

Modern Analysis,

— 0). § 8-5.

t Of course on the hypotheses concerning f(t) which were assumed in § 18-53.


616

The uniformity is

continuous in

of convergence

[CHAP. XVIII

+ A, 1) when f(x) same way as the uniformity

of the summability in the interval (a

(a, 1)

may be

dealt with in the

was dealt with in

§§ 18*33, 18*35.

The summability of Dini series (and of Fourier- Bessel series) by a modifimethod is of some physical importance. Thus, in Fourier's* problem of the Conduction of Heat in an infinite solid cylinder of radius unity, cation of Abel's

the temperature v at distance r from the axis satisfies the equation

dv __, (d 2v 1 dv dt~ \dr 2+ rdr

with the boundary condition

if

the initial distribution of heat

Normal

is

symmetrical.

solutions of the differential equation

satisfying

the boundary-

condition are J*

and so the temperature v

is

(*-m?\)

exp (.- k'kmH),

given by the series f

2 brnJoQ^r) exp (~k\mH),

m=\

where the

coefficients bm are to

be determined from the consideration that 00

is

the Dini series associated with the initial temperature f(r). It is expressible as

is

evident

that the initial temperature

00

2 6m J (Xm r)exp(-A-\TO

2

lim

and

this limit exists

18*6.

It has

when

the Dini series

The uniqueness of Fourier- Bessel

is

«);

summable

series

(R).

and Dini

series.

been shewn by YoungJ that the existence and the absolute con-

vergence of Jo are sufficient to ensure that if all the coefficients a m of the Dini series (or the Fourier- Bessel series) associated withf(t) are zero, then the function (t) must be a null-function.

f

* xii.

La

Theorie Analytique de la Chaleur (Paris, 1822), §§ 306—320. Cf Rayleigh, Phil. Mag. {6) 106—107 [Scientific Papers, v. (1912), pp. 338—339] ; and Kirchhoff, Berliner .

(1906), pp.

Sitzungsberichte, 1883, pp. 519

—524. H

> 0, and so there is no initial term to be inserted. t In this physical problem, t Proc. London Math. Soc. (2) xvni. (1920), pp. 174—175.


18-6]

To prove

this

617

= 0, 1, 2,

theorem we observe that, when p

...,

we may write

i-HrH- 5 am tiJAjm t), where the

coefficients

am are determined by the formula

Um =

7r4n P tv+ P+1J (i» *+i\Jm)Jo *

v

*) dt;

*'

and the series on the right converges uniformly in (0, 1 - A) and oscillates boundedly in (1 — A, 1). It is therefore permissible to multiply the expansion °y **/(*) and integrate term-by-term. It follows that

f\»+n>+*f(t)dt=

JO

5 am \\f{t)Jv {jm t)dt

m=0

JO

= 0. Since

all

the integrals

n t'

+w f(t)dt

(p« 1,2,3,...)

Jo

are zero, it follows that t"f(t) is a null-function, by Lerch's theorem*, and the theorem stated is proved for Fourier-Bessel series. The theorem for Dini

can be proved in precisely the same way, and it is theoretically simpler because the Dini series associated with t" +*> does not foil to converge uniformly in (1 — A, 1). series

It is possible to construct a theory of series of Bessel functions of the types 00

<3C

2 am Jv (jm x),

2

bm

J

v

(\ m x),

(where the coefficients am and bm are any constants) which resembles Riemann's theory of trigonometrical series f.

Such a theory

is,

however, more directly associated with Schlbmilch's

series of Bessel functions,

which will be discussed in Chapter xix seems convenient to defer the examination of the series ao

and

it

oo

2 am J¥ {jm x),

m-l

by Riemann's methods

;

X bm Jv (km x)

m-1

to § 19 7, when the discussion of the series forms a simple corollary to the discussion of Schlomilch series.

pp.

-

* Lerch,

Acta Mathematica, xxvn. (1903), pp. 345—347

37—43.

Cf. §12-22.

t Cf. Modern Analysis, §§ 9*6

—9632.

;

Yoang, Messenger, xl. (1910)

CHAPTER XIX SCHLOMILCH SERIES Schlomilch' s expansion of

19*1.

In Chapter real variable

x

xvm

we

a function of a

real variable.

dealt with the expansion of a function f(x) of the

in the form 00

f(x) =

where jm

That

is

2 am J¥ {jm x),

J¥ (z), so jm = (m + \v - i) 7T +

the with positive zero of

is


m,

(1/m).

argument of the Bessel function in a term of high rank approximately proportional to the rank of the term.

to say, the

the series

is

in

In this chapter we shall discuss the series in which the argument of the Bessel function in each term is exactly proportional to the rank of the term. By choosing a suitable variable, such a series may be taken to be 3D

2 OmJyimx). add an by making a

It will appear subsequently that it is convenient to

(§19-11;

cf.

§1833); and the analysis

fication in the

is

simplified

initial

term

slight modi-

form of the coefficients in the series (§19'2).

Series of this type were

first

by Schlomilch*. They are not

investigated

of such great importance to the Physicist as Fourier-Bessel series, though

= 0)

they present themselves naturally in the investigation of a periodic transverse vibration of a twodimensional membrane, if the vibration is composed of an unlimited number

Rayleighf has pointed

out that (when v

of equal one-dimensional transverse vibrations uniformly distributed in direction

through the two dimensions of the membrane.

Apart from applications the series present various features of purely mathematical interest; and, in particular, it is remarkable that a null-function can be represented by such a series in which the coefficients are not all zero (§19-41).

In some respects the series are more amenable to analysis than FourierBessel series, but the two types of series have many properties in common; and the reader will be right when he infers from a comparison of the arguments jm x and mx that the relevant range of values of x is (0, ir) for Schlomilch series, corresponding to the range (0, 1) for Fourier-Bessel series. * Zeitschrift

special cases

fUr Math, und Phya. n. (1857), pp. 155—158

;

Schlomilch considered only the

r=0 and r = l.

t Phil Mag.

(6) xxi. (1911),

pp.

567—571

{Scientific Papers, vi. (1920), pp.

22—25].

SCHLOMILCH SERIES

19-1, 19-11]

19*11.

Schlomilch's expansion in

a

series

619

of Bessel functions of order

zero.

We now state and prove the expansion theorem discovered by Schlbmilch. The theorem is concerned with the expansion of an arbitrary function f(x) of the real variable x, and, with modern terminology, it is to the following effect:

Letf{x) be an arbitrary function, with a derivate (x) which is continuous in the closed interval (0, v) and which has limited total fluctuation in this

f

interval.

Thenf{x) admits of the expansion CCr

/(#) = \a

(!)

+ 2 am J

(mx)

where o„

(2)

= 2/(0) + -

[' f 7TJ0 JO

uf(u sin

2 [ n [**

«m = and

this

)

d
-

expansion

interval (0,

is valid,

/

uf (u sin

/

and

)

{m >

cos raw ddu;

0)

the series is convergent, throughout the closed

tt).

Schlomilch's investigation

based on a discussion of the integral equation

is

-[% (* sin 0)d0,

/(*) =

(3)

of which he proved that a continuous solution is

g (x) =/(0) + x["f(x sin ) d$.

(4)

Jo

We

proceed to verify that the function g (x) defined by (4) actually is a solution of (3); we substitute the value given by (4) in the expression on the right of (3), and then we see that

-J o

g(xam6)de =

/(0) + x sin 6

7r j o

-/(°) +

Now

replace 6

by a new

2x

sin

x

f (x sin

sin

)

d

ft" ft*

—J

variable

j

]

f (x sin 6 sin

defined

)

sin0 ddd.

by the equation

x = sin 6 sin

\d6

,

[CHAP. XIX


620

We

and change the order of the integrations.

TT.'o

g(xsm0)dd-f(O)=— vr [*" [ 9 1,

2x

n*

= 2^

= 2x = #/

f»*

/ J^

sin

Jo

cos

)

sin

ddd

v d% d0

cos v d# c?y sin /(a;Smx) V(cos^-cos

/"*«

V.U

/' (# sin

I

J o

sin

x

deduce that

.

V0)

x ,

J

.

(x sin

/cos 0\~|**

.

X ) |- arc

sin

(—

,

cos

J J^

X dX

/'(a;sinx)cosxax

Jo

and

so,

when

Now

(x) is defined

by

(4),

g (x)

is

a solution of (3).

easy to verify from (4) that, when/'(«) is a continuous function with limited total fluctuation in the interval (0, ir), so also is g{x); and therefore, by Fourier's theorem, g (x) is expansible in the form it is

00

g(x) = %a

+ 2 am cosmx, m-l

where 2 [*

am = —

—_ and

this series for

\

g (u) cos mudu y(0) + ul

f

f (u sin

<£) d

cos

mwdw,

g (x) converges uniformly throughout the

Hence term-by-term

integrations are permissible, and so

interval (0,

tt).

we have

/<*)-!fV*™*)
*— •"Jo I

•U<*o

+ S a™ cos (wmj sin 0H d0 «»=i

I

J

= |a + 2 am J (mx), m=l

and

this is the expansion to

be established.

It is easy to verify that the

values obtained for the coefficients am are the

equation

same

as those

given by

(2).

When the restriction concerning the limited total fluctuation of /' (x) the Fourier series associated with g(x) is no longer necessarily removed, is convergent, though the continuity oif(x) ensures that the Fourier series

19 * 2 ]

SCHLOMILCH SERIES

uniformly summable (C 1) throughout

is

(0, tt);

621

and hence, by term-by-term

integration, the series 00

£a + X am J (mx) uniformly summable (Cl) throughout (0, tt), with sum f(x); an application of Hardy's convergence theorem* then shews that the additional condition is

am is sufficient

f(x) when x

=

(l/0n)

to ensure the convergence of the Schlomilch series to the lies in the half-open interval in which < x % ir.

For further theorems concerning the summability of Schlomilch the reader should consult a memoir by Chapman f.

sum

series,

[Note. The integral equation connecting f(x) and (x) is one which was solved in 1823 g by Abel, Journal fur Math. i. (1826), p. 153. It has subsequently been investigated* by Beltrami, 1st. Lombardo Rendiconti, (2) xm. (1880), pp. 327, 402 Volterra, Ann. di Mat. ;

(2)

xxv. (1897),

p.

104

;

C. E. Smith, Trans.

American Math.

Soc.

vm.

(1907), pp. 92

106.

The equation 2x fhw rj» /

rr

is

in

1

/ (x sin 6 sin

sin

6
most simply established by the method of changing axes of polar coordinates, explained §3-33 this method was used by Gwyther, Messenger, xxxm. (1904), pp. 97—107, but ;

in view of the arbitrary character of f(x) the analytical proof given in the text seems preferable. In connexion with the changes in the order of the integrations, ef. Modern Analysis, § 4-51.

19'2.

The

definition

of Schlomilch

series.

We have now investigated Schlbmilch's problem of expanding an arbitrary function into a series of Bessel functions of order zero, the argument of the function in the (m + l)th term being proportional to m; and the expansion is valid for the range of values (0, rr) of the variable.

Such series may be generalised by replacing the functions of order zero by functions of arbitrary order v\ and a further generalisation may be effected by taking the general term to contain not only the function ¥ (mx) but also a

J

function which bears to the Bessel function the same kind of relation as the sine does to the cosine. The latter generalisation is, of course, suggested by the theory of Fourier series, and we are thus led to expect the existence of

expansions valid for the range of values (-

The * Cf.

functions which naturally

tt, it)

of the variable.

come under consideration

for insertion are

Modern Analysis, § 8 5. t Quarterly Journal, xliii. (1911), p. 34. t Some interesting applications of Fourier's integral theorem to the integral equation have been made by Steam, Quarterly Journal, xvn. (1880), pp. 90—104.

.


622

[CHAP. XIX

Bessel functions of the second kind and Struve's functions; and the types of series to be considered may be written in the forms*:

£a

,

£ «m Jv (mx) + bm Y, (mx)

£ am J* (™x) + bm H„ (ma?)

£a«

I> + 1)

(\rnxy

«=!

Series of the former type (with i/= 0) have been considered by Coatesf; but his proof of the possibility of expanding an arbitrary function f(x) into

such a series seems to be invalid except in the trivial case in which f(x) defined to be periodic (with period 27r) and to tend to zero as x -*- oo Series of the latter type are of

much

greater interest, and they form a

direct generalisation of trigonometrical series.

Schlomilch

is

They will be

called generalised

series.

types of investigation suggest themselves in connexion with general-

Two

ised Schlomilch series.

first is the problem of expanding an arbitrary and the second is the problem of determining the

The

function into such a series;

properties of such a series with given coefficients and, in particular, the

construction of analysis (resembling Riemann's analysis of trigonometrical series) with the object of determining whether a generalised Schlomilch series, in

which the

coefficients are

not

all zero,

can represent a null-function.

Generalised SchlSmilch series have been discussed in a series of memoirs by Nielsen,

Math. Ann. Lil. (1899), pp. 582—587 Nyt Tidsskrift, x. B (1899), pp. 73—81 Oversigt 1901, pp. 1900, pp. 55—60 K. Danske VidensJkabernes Selskabs, 1899, pp. 661—665 127—146 Ann. di Mat. (3) vi. (1901), pp. 301—329. ;

;

;

;

:

Nielsen J has given the forms for the coefficients in the generalised Schlomilch expansion of an arbitrary function and he has investigated with great detail the actual construction of Schlomilch series which represent null-functions, but his researches are of a distinctly different character from those which will be given in this chapter.

The

investigation which

we

shall

now

give of the possibility of expanding

an arbitrary function into a generalised Schlomilch series is based on the in his memoir on applications investigation given by Filon§ for the case v = of the calculus of residues to the expansions of arbitrary functions in series of functions of given form. It seems to be of some importance to give such an

investigation *

||

because there

is

no obvious method of modifying the

The reason for inserting the factor x"

in the denominators is to

make the terms of

set of

the second

series one-valued (cf. § 1921).

t Quarterly Journal, xxi. (1886), pp. 189—190. + See e.g. his Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), p. 348.

London Math. Soc. (2) iv. (1906), pp. 396—430. has to be assumed that - £ < v < £. The results which will be proved in §§ 1941— 1962 of v. suggest that it is only to be expected that difficulties should arise for other values § Proc.

II

It

.

SCHLOMELCH SERIES

19*21] functions J, (mx),

H„ (mx)

set for the interval

(— ir,

623

so as to obtain a set

which is a normal orthogonal and consequently there is no method of obtaining the coefficients in a Schlomilch expansion in so simple a manner as that in which nr);

the coefficients in a Fourier-Bessel expansion are obtained

(§ 18-1).

The investigation, which forms the latter part of the chapter, concerning the representation of null-functions by generalised Schlomilch series, is of same character as the exposition of Riemann's researches on Modern Analysis, §§ 9*6 9632.

exactly the

—

trigonometrical series given in

19*21.

The application of

the calculus

of residues

to

the generalised

Schlomilch expansion.

We shall now explain the method* by which it is possible to discover the values of the coefficients in the generalised Schlomilch expansion which represents an arbitrary function f(x), when the order v of the Bessel functions lies

between

- \ and

\.

validity of the processes

When

this has

been done, we

shall not consider the

by which the discovery has been made, but we

shall

prove directly that the Schlomilch series in which the coefficients have the specified values actually does converge to the sum /"(#). This is analogous to the procedure which theorem in the expansion

is

adopted in* Dirichlet's proof of Fourier's

:

GO

/(#) — h°«+

2 m-l

(a m

cos»we+/3Bl sinfiM?)

the values of the coefficients are discovered by multiplying the expansion by cos mx and by sin mx, and integrating, so that the values of c^ and /3 m are taken to be given by the equations 1

<%=-

/"« |

/(0

If*

Pm =-l

cos mtdt,

f(t) sin mtdt.

We then take the series in which the coefficients have these values, 2^1' f(t)dt+~ and prove that

it

I

actually converges to the

namely

f (t) cos m(x-t)dt,

j

sum f(x).

It conduces to brevity to deal with the pair of functions

Jy (mx) ± i H„ (mx) (I

mx)'

'

instead of with the pair of functions

J, (mx)l(%mxy,

We

.

shall write

(1) *

H„ (mx)j(\ mx) v

Jv (z) + il3..(z) _

Apart from details of notation, the following analysis

memoir just cited, presents no difficulty in the

for the special case

is due to Filon; it was given by him, *=0, but the extension to values of v between ±£

—

)

.

[CHAP. XIX


624 so that*

v

variable z;

and uniform

(z) is analytic

complex

and evidently

J (mx)£ ± — —

t'H„ (mx)

v

We

for all finite values of the

now observe

m

that (—

i

i

¥

(mx)

'

s

.

,

(

.

,

+ mx).

the residue at z

is

w v

=m

of the function

(xz)

&\WKZ

m = 0,

where

±1, +2,

.

. .

and so we

;

shall consider the integral 7r<£„

(xz) ~dz,

S7TI J

in

SI

c

M

which the contour C is a circle, of radius + \, with its centre at the is an integer whioh will be made to tend to infinity. and

origin,

M

is assumed to be one- valued throughout the 2-plane, be analytic at infinity (cf. §19'24); its only singularity in the finite part of fche plane is an essential singularity at the origin.

The function F(z)

and

to

By

Jordan's lemma, the integral tends to zero as

provided that v

It is evident,

tends to infinity,

residues, that

by calculating

2 (-)m \F(m) is

M

> — |.

v

(mx)

+ F(-m)

„

(- mx)}

equal to the residue at the origin of

v

that

is

(2)

'

sin trz

to say

2 (-)» {F (m)

v

(mx)

+ F(-m)

v

=-

(- mx)}

— 2m 1

r

(0+)

J

.ir*(xz) , T dz. sin ttz

ET/ F(z)

of expanding an arbitrary function /(a?) into a generalised Schlomilch series is consequently reduced to the determination of the form of

The problem

F(z) in such a way as to make

-s— 2m

F(z)

.

differ

J

?" sin

y

'

dz

vrrz

by a constant from f(x).

* The insertion of the factor (J z) v in the denominator makes #„ (z) amenable to Cauchy's theorem when the contour of integration completely surrounds the origin.

SCHLOMILCH SERIES

19*22]

The construction of

19*22.

We now

625

the function F(z).

take the contour integral

2m J

smirz

'

and, in order to calculate

it in a simple manner, expansible in a series of Filon's type*

F(z) =

(1)

i w=l

we

shall

suppose that F(z)

is

?4#, *

where yfrn (z) denotes the sum of those terms in the expansion of ir- 1 sinirz whose degree does not exceed n, and the coefficients n will be defined later.

p

The reader

will observe that

•M*)=-M*)=*-*iry,

With

this definition of F{z), it is evident that, for small values of \z\,

^"sin^^/^sin^ X n+1 (s

- y „

i

»=i

*-<"> j

Tn+1 cos in7r + 0(^)1 (n+1)!

sin-w^

^

,

jt»\

.

/

It follows immediately that (2)

-'fV w *) & =6VfV(0) i »=i

*>» 7rWcos * W7r (/i

+ l)!

p»qtg)»

_ l

n^r^n + ^r^n + v+l)*

and consequently we proceed to identify n

Pn(%ix) -X .-ir(in+i)r(*n +

i-

+ l)

with /(a?) -/(0). For this purpose we have to assume temporarily that f(x) has differential coefficients of all orders at the origin, and then we define the coefficients

pn by the equation

/ (0)_ pn (£ i)n "n!'~ r(in+l)r(in + + l)'
(3\ K)

i-

We next transform of the series,

form

;

this equation defining

by which F(z)

is

pn in such a defined, is expressible in a

the transformation of the series for

= 1,2,3,...). way that the sum

compact symbolic F(z) can be effected by expressing

* This type of series is fundamental in Filon's theory, and is not peculiar to Schldmilch expansions; thus, in his work on Fourier-Bessel series, sin *z is replaced by z~"J {z) and v fn (z) denotes the sum of the terms whose degree does not exceed n in the expansion of that function.

[CHAP. XIX


626 the coefficients

pn

which involves n only as an exponent. For this

in a form

purpose we make use of Eulerian integrals of the that they

may be convergent, we

—£<

\.

v

<

We

then have

_

rgyrgn+v + i) r

= - *ru-l) -

)

first

kind, and, in order

shall find that it is necessary to

fm (0)

(1

suppose that

-**+-*»•-**

/!

^/W] dt ^-f-^r^ fa (1 ° d* U*'

r( *>

4

^r(i-,)Jo

and so we obtain the symbolic formula

* - - *m~) /> -

<">

where

D stands

Now,

if

^

it ['"

*•/«•>] „*•

for dfdu.

we arrange the

in descending powers of

series

z, it is

^ w=1

easy to verify that

tyn (z) D" n+1 in z

_

sinh irD it (iz

- D)

'

and therefore

Again, a consideration of (2) shews that

y pn and we are able to

we

effect

7T

we need

to

sum

the series

W cos £ ntr

our purpose by making use of formula

(4),

whence

find that

(6)

ii'"(n+T)r

We

have now obtained symbolic expressions for all the coefficients in the generalised Schlbmilch expansion of /(#), but it is necessary to transform these expressions into more useful forms, by finding the significance to be attached to the symbolic operator for the

value zero of z.

-jjyzT\ ^° fcn '

for

g6116'8,1 values

of *

and

SCHLOMILCH SERIES

19*23]

627

19*23. The transformation of the symbolic operators in the generalised Schlomilch expansion.

We

proceed to obtain an interpretation* of the symbolic expression

rê

/(
The usual

interpretation of

—

is

e-^fi^dv,

a constant of integration sinh irD

.._,

;

and therefore

sinh-wi) T

.

.

- f(tu) is

^

i:

where a

ô

f

.

u

7T

)

"1

,.

.

.

,

a

Now, by the symbolic form of Taylor's theorem, we have e±

where x (M )

is

"D

x(u ) = x( u ± 7r )>

an arbitrary function of u

= r sinh^(D-f^)

_™*Z£

/(ftf /V >] f L7T(i)-t^)-

and hence

;

'J w==0

[

that

is

= 0,

I

yV J«-0

I* 'e-^fitv) dv Ja J«-0

to sayf

The second term on the *

follows that

fJa

7T

- e-™

it

±1, ±2,

right has simple zeros at

all

the points at which

....

Therefore, so far as the calculation of residues of

F(z)

ir,

(xz)

sin irz *

The

numerous expressions involving symbolic operators of the types nnder l. (1839), pp. 22 32 and by Boole, Differential Equations (London, 1872), chapters xvi and xvn. t The expression for F(z) which is derived from this formula does not appear to have a singularity at the origin unless a is infinite or is a function of z ; but it seems unreasonable to be perturbed by this when we consider the nature of some of the analysis which has already been interpretations of

consideration have been discussed by Gregory, Cambridge Math. Journal,

used in the course of this investigation.

—

.


628

+

at 0,

and

+

1,

2, ... is

concerned,

we may omit

the second

[CHAP. XIX

term on the right in

(1),

calculate the residues of sin trz

where <

F

(z) is defined by the formula

*»-

2 >

Again, from

ww^km

§

1922

(6)

» j».g»coBinir (3)

(n

»ti

= r(i)/(0)

+ 1)!

The

(1 vx

r(i-»).l

_

py^dr "'" A

dt.

we have

eft

^W^^W^*

dt

v

thus obtain the expansion

f{x)

(4)

f

(1) of this section

»

/(tv) dv

term on the right in (3) is equal to T (v+ l)/(0), except when* = 0, the value of the term in question is zero.

first

= 0; when

We

^

- eyi"Jt [*' /". e

(1

and equation

2r(* v

F.

=

(0)

P(0)+ I (-)" [F (m)

= 0,

In the special case in which v additional term

When we

v

(mse)

+F(-m)

v

(- mx)\

the modified form of (3) shews that an

/(0) must be inserted on the right

in (4).

change the notation to the notation normally used

for Bessel

functions and Struve's functions, the expansion becomes

/W "^(y +

(5)

+

l)

"-

,

.t 1

(\rnxY

where f

am )

(6)

bm ~ This 19*24.

We

is

O-^-a

-T{\-v)T{\)),

r(|-,)r(D J

"

(1

<2)

f (tv) cos mv dv

I

J

— IT

./

—

*"*

dt,

dt.

j«

ir

the generalised form of Schldmilch's expansion

The boundedness of F(z), as

j

z

j

-*- oo

now prove that, when the function f{x) is restricted in a suitable manner, the function F(z) is bounded when [z|-»-oo, whatever be the value of arg«. The reader will remember that the assumption that F(z) is bounded was made in § 19"21 to secure shall

the convergence of the contour integral.

We take

the series of § 1922

(1),

by which F{z) was originally denned, namely

* When v is negative it is necessary to use a modified expression for the integrals ; cf. § 19*3. t When p = 0, the expression for a© flas *° he modified by tbe ins«rtion of the term 2/(0), in consequence of the discontinuity in value of the expression on the right of (3).

:

SCHLOMILCH SERIES

19-24, 19-3]

and divide it into two parts, namely the where is the integer such that

N terms

first

N

N^ When n ^ AT

the terms of

,

+n

^

(z)

n

it

j

z

< N+

|

do not exceed

ir

have

|

n

(?)

\

-

l

sinh

1

'

therefore,

whwn n

< JV,

|*|.(»-1)!

n-

1

2

, 1

and therefore l

'

sinh

\z\" +l

tends to zero as j

and

1

Since

I

!,

»—»

Sinh*r|3l " IP„+Ar+i S l^nk"- + ir|*|^t n ?o !*l»=i (n-l)» |*|*

IFMi^ W J_

as z

1.

\

\

yfr

and the remainder of the terms,

n ~ l \z n jn

n*»-i\sW(n l)

(z)

z\ n+1

When n ^ JV, we

629

-*- 00 is

\z

\

—

00

,

it is

evident that a sufficient condition for F(z) to be bounded

that the series

2 \Pn\ should be convergent

;

and

this is the case if f(x) is such that

2 »" + i|/(»)(0)| is

convergent.

The expansion of an arbitrary function

19*3.

into

a generalised Schlomilch

series.

Now that the forms of the coefficients in the generalised Schlomilch expansion have been ascertained by Filon's method, it is an easy matter to specify sufficient conditions for the validity of the expansion and then to establish

it.

The theorem which we

shall

Let v be a number such that in the interval (—ir, (I)

ir),

(II)

(III)

is

as follows

— \ < v < \;

The function h(x), defined by

and

is

and letf(x)

the equation

= 2vf(x) + xf'(x),

continuous in the closed interval (—

The function h(x) has limited

Ifv

be defined arbitrarily

subject^ to the following conditions:

h(x) exists

prove*

ir, ir).

total fluctuation in the interval

(— ir,

it).

is negative I the integral

if (*)-/(<>)}]** ri*£i\*r J O is absolutely convergent *

The expansion

when

A

is

a (small) number

either positive or negative.

by Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig, which he gives for the coefficients in the expansion seem to be quite inconsistent with those given by equation (2). t The effect of conditions (I) and (II) is merely to ensure the uniformity of the convergence of a certain Fourier series connected with h (x). X If v is positive, this Lipschitz condition is satisfied by reason of (II). is stated

1904), p. 348; but the formulae

[CHAP. XIX


630

Thenf(x) admits of the expansion £a £ am Jv (rruc) . (1) /(*) f(i7Tr)

+^

+ 6m H, (mx)

g^;

,

where

K=L wAen

m > 0;

r
j

w

the value

of a

[sin2v

^

{

f( U8in V-f(

M sin ™^*>

obtained by inserting an additional term

is

2I> + l)/(0) on the right in the first equation of the system

We

it will

(2).

on a discussion of the integral equation

shall base the investigation

be proved that a continuous solution

is


sr(*)=I> + l)/(0)

(4)

+ r^7>

f

sec2 " +1 '

~ sin *> /(0)1] d * o% [sin2" * [f{x

[Note. The (absolute) convergence of the integral contained in this formula

by condition equation (4)

be omitted when v

is

We

valid only

when

is

r

/:o

which

secured

It should be observed that the aggregate of terms containing /(0) in

(III).

may

is

-

positive in view of the formula

*

cUf>

r( ^

v is positive.]

proceed to verify that the function g (x) denned by (4) actually is a by taking g (x) to be defined by (4), substituting in the expres-

solution of (3),

sion on the right of (3),

The

and reducing the

result to /(#).

result of substitution is

2cosy7r IT

[* [

in

C082*

sec^+i

*

[gin*-

<*9

Jo Jo

{/(* sin

sin

{f(x sin

sin

)

-/(0)}]

dd0

+/(0).

Hence we have

^^ IT

** f

JO

to prove that

f ^cos

2"

$ sec 2" +1

^r

Jo

[sin"

d

)

-/(0)}] dd0

=/(*)-/(0). Replace

on the

left

by a new sin

variable

x — sni ^ s

i

%


n 9>

change the order of the integrations in the resulting absolutely convergent by a new variable ty defined by the equation integral, and then replace cos

— cos x sin ^.

^ SCHLOMILCH SERIES

19*3]

We

631

thus deduce that

/**/•

cos2 "

sin

d

r

.

,,

,

.

X) "WW*** * {/( "Jo Jo ( sin»fl-sin*xr> djc l n* rh* sin cos 2 " d r „ = [Sm" {/(a; Sm X) " /(0)1] 2 * (cos« io i x X - cos 0)"+* d£ .

*

^

.

,

rjir

'.Jit

= j'jjj tan2r *<**

•

^

[sin-

x {/(* sin x) -/(0)}]
= ir( v +|)r(i-v){/(^)-/(0)}, and hence the formula to be established is evident; and defined by (4), then equation (3) is satisfied.

Now, by

so,

when g(x)

is

Fourier's theorem, oo

g{x) = \a

(am cos ma; + bm sin ma;),

+ 2 JB-l

where \am

&m

and

= —I 7r

J

(5)

it is

easy to verify that

=—

is

an

is

a;

by

a;

(u) cos

mudu,

<7

(u) sin

mudu;

(— ir,

is 7r)

a continuous function with limited so also is

uniformly convergent when

arbitrarily small positive

Replace

I

g

when f(x)

total fluctuation in the interval

expansion for g (x)

'-'

sin

g (a:), and

therefore the

- ir + 8 ^ x ^ it - 8, where

in the expansion of

g (x), multiply by

cos2 *

has an absolutely convergent integral, and integrate term-by-term at once that

—

f( \ r{x)

and

^ a°

_i_

V a™J* ( ma! ) + ^t»H y (mx) ~~

-T( v + 1) +.*

W^f

this expansion converges uniformly

The

values of

m and

a.

those given by formula

8

number.

b m given

;

0,

which

we deduce

;

when — ir+ 8^x îr — 8.

by formula

(5) are easily reconciled with

(2).

It should be noticed that,

by the Riemann-Lebesgue lemma, a m and bm 0(l/m) when m is large. This seems to be connected with the fact that when we come to deal with any Schlomilch series (§ 1962) we are unable to make any progress without assuming that 26TO/m is convergent (or some equivalent hypothesis) this assumption will appear in § 1962 to be are both

;

necessary because the differential equation which Struve's function satisfies is not homogeneous, so that Struve's function is not of a type which occurs in solutions of Laplace's equation or the

wave equation

;

there would conse-

;


632

[CHAP. XIX

quently seem to be reasons of a physical character for the limitations which have been placed on f(x) in order to ensure the existence of the Schlomilch expansion. [Note.

Just. as in § 19*11,

if

condition (II) concerning the limited total fluctuation of

all statements made in this section up to this point about convergence of series have to be replaced by statements about sumtnability (Cl).]

±vf{x)-\-xf'(x)

is

not

satisfied,

then

There is one important consequence which follows from the fact that a m (1/m) when 2vf{x) + xf (x) has limited total fluctuation and bm are both in (— it, 7r), namely, that in the neighbourhoods of — it and it, the general (l(m v+ %), and so the expansion repreterm of the Schlomilch expansion is sents a continuous function

hence the expansion converges (uniformly) to

;

the sum/(a;) throughout the interval (—

it, it).

Special functions represented by Schlomilch series.

19*4.

There are a few problems of Mathematical Physics (other than the problem mentioned in § 19*1) in which Schlomilch series occur in a natural manner, and we shall now give an account of various researches in which Schlomilch be found.

series are to

A

very simple series

is

1+1 this series is convergent

e~ mz J (mp)

when p and

cylindrical-polar coordinates,

of space above the plane z

z are positive, and, if p and z denote a solution of Laplace's equation at all points

it is

= 0.

Various transformations of the series have been given by Whittaker*; thus, by changing to Cartesian coordinates (x, y, z) and using § 2*21, we have 00

(1) v/

e- mz

1+ %

J

n= i

,

If™' : (mp) v r/ = z2ir J _» 1

du

— exp

>

.

When a? + y* + z < 1, the integrand may be of z + ix cos u + iy sin k. 2

If this

is

done,

1+2

fir

1

e~ mz Jo (mp) r/

= 5J

2-7T

7W=l

_.„.

—+

z

—

+7T— 2 -—7s

(r,

—

:

-.

expanded in ascending powers

1

fill .

ix cos u

-r-^

(2m)l

27T OT=!

where

:

+ %x cos u + xy sin u)\

we getf

CO

(2) v

.

{— (z

I

(z v

+ ~r~. + %y sin u 52

+

ix cos

VI

u + iy sin

J_,r

m) 2**" 1

du

6) are the polar coordinates corresponding to the cylindrical-polar

coordinates

(p, z),

and * I

B

1}

B.2

,

B

,

...

are Bernoulli's numbers.

i.vn. (1903), pp. 341—342. and Modern Analysis, §§ 7-2, 18-31.

Math. Ann. Cf. § 4-8

s

SCHLOMILCH SERIES

19'4]

633

Another transformation of the series, also given by Whittaker, is obtained from the expansion for 1/(1 — e~ l ) in partial fractions; this expansion is

^L_ = I + l4. 2

\-e-

1

t

l

1

i

f

w=1

\t - 2miri

}

I

t

+ 2miri)

'

whence we deduce that (3)

1+ I= 1M

1

e-™Jo(mp) = i + s & T

+ 2

?! \jj{(%nviri

+ zr + a? + tf) "
'

due to a set of unit charges (some positive and some negative) at the origin and at a It follows that the series represents the electrostatic potential

set of imaginary points.

The reader may find it interesting to discuss the Lipschitz-Hankel integral of § 13'2 as a limiting form of a series of Whittaker's type.

Some other series have been examined by Nagaoka* in connexion with a problem of Diffraction. One such series is derived from the Fourier series for the function which is equal to 1/V(1 — a?) in the interval (— 1, 1). The Fourier

series in question is

1

(4)

-r-z

°°

^ = £•" +

•*"

2

J„ (wwr) cos withe,

and it converges uniformly throughout the is any positive number. Multiply by to

e""*

I e%

Jj

goaii

(- 1

+ A,

1

- A), where A

and integrate, and we then obtain the formula

Nagaoka) <3 >

interval

fa

w=*>

=

~b igoxi

n

+

„ « 2 »?.

Jr

'

(also

^^

.a cos mirx — lairi sin mmsc\

,

(mv)

•

J

The series on the right in (5) converges uniformly throughout the (-1,1) and so we may take - 1 and 1 as limits of integration. Hence,

for all values (real

Jr

/r\ (6)

A

more general

\

t (a)

v

'

sina Ti

- -j-

result, valid

[1

_l.9„ 8 3 + **£

when

(-)"J,(m7r) l a?

_m

^

•

J

R (v + §) > 0, is i

^/'^

lm" (a- - nt'ir*)]

* Journal of the Coll. of ScL, Imp. Univ. of Japan, iv. (1891), pp. 301—322. formulae are quoted by Cinelli, Nuovo Cimento, (4) i. (1895), p. 152.

W.

B. P.

interval

and complex) of a,

+*,(!)• i Ma)m ™\<&r a \T(v+l) WJ m=

(7)' v

due

Some

of Nagaoka's

21


634 This expansion

is also

by expressing

obtainable

.

[CHAP. XIX as a sura of partial

fractions*.

Various representations of the integral on the left of (5) were obtained by Nagaoka the formula quoted seems to be the most interesting of them.

Finally

This

is

we

formula f

shall give the

deducible from the Fourier series

" cos(2wi-l)a?

by replacing

a;

by

a;

sin

then, if/(#) denotes the

sum

=

9

^ T O

of the

(

fare

+

(JO

<

a?

<

2-7T,

19*41.

We

—

^

To

rts

now prove

/"arc sin (nix)

— ^(Tr-2x8m6)d0 o

- 3tt) d0,

we have !

... /tt = v (^ — t ) — *a; — 7r arc cos — + 2

8"

series.

ir

;

the series oscillates

when x =

and diverges

to

when x = ir.

This theorem has no analogue in the theory of Fourier it is

if

(-rJ.(m»)-0, \+2 * m-l

provided that oo

\f(xsmd)d0

the remarkable theorem that

(1)

+

\tr.

J arc sin {nix))

Null-functions expressed as Schlomilch

shall

to

")

(2# sin " 'w arc sin (ir/x) £

S J".[(2m-1)«} (9)

fi*

-/"""

+ -n-

from

x

7r, ir),

sin (»/«)

TTJ o

when

.

sum of a Schlbmilch series when we shall take ir < x < 2tt, and Fourier series, we see that

'

so that,

<—<«<«•)

calculation of the

outside the interval (—

lies

,,

6 and integrating with respect to

As an example of the the variable

ir.

ol -g(*-»MX

(2m-iy

„?,

definitely

knownj that a Fourier

function throughout the interval (0, * Cf.

Modern Analysis,

series, and, in fact,

cosine-series cannot represent a null-

ir).

§ 7*4.

t This was set as a problem in the Mathematical Tripos, 1895.

+ Cf. Modern Analysis, §§ 9-6— 9632.

SCHLOMILCH SERIES

19*41] It is easy to prove (1)

integer,

635

M

by using Parseval's integral; when

a large

is

we have

M

1

+ 2

s

(-)m J (mx)

2

M

ft" il

k+

= -\

)

2

(-)«• cos

_ ("^ /*** cos {(J/ + £) 7r

a;

cos (£# sin

Jo

_ (~) M [ x cos (^ + h) u cos

J

ir

\u

'

(mx sin t)\dt sin

t]

,

£)

du ^(x2 —

v?)

= 0(1), as

itf -*- oo

,

by the Riemann-Lebesgue lemma*, which

is

applicable because

the integral

du

f Jo cos \u '

o

exists

and

< x < ir and ;

\\+ 2

this is the

It is easy to prove in

(i)

when — £ <

By

v

a it )

< x<

ir.

that

lim

when

—

\Z(ar*

when


is

Hence we have proved

.

(-)» J, («»*)]

theorem

=0

stated.

a similar manner that

< a; < 7r,

$ J and

(ii)

when

>

v

using Poisson's integral we have (since

£ and

< x ^ tt.

v> — \)

"

\

(-)*Jp(imB)

g

,

r
"•",.-!

{\mxy

-

" ro+Lr^ r {^ + .?i r C08 (

r(* + *)r<*)Wo

cos**

(rw * 8in

•<*

M

°(

cos" tdt

'

= o(l), as Af -* co

,

provided that the integral

*(a?-u*y .'o

exists

and

is

cos£u

dw

absolutely convergent; and this

is

the case

when x and

v satisfy

the conditions stated.

The truth If

n

is

of (2)

is

now

evident.

a positive integer, and

if v is

so large that v

d [ d xdx\ dx * Cf.

{

2

Modern Analysis,

)

j §

941.

— In > — \

t

the operator


636

may

be applied n times to equation

once to the function therefore,

when

Jv {mx)j{\mx)

The

(2).

v

effect of

[CHAP* XIX

applying the operator

to multiply the function

is

by

—w

s ;

and

< x < ir,

(-jm )" (-)*"«/,, (ms) = 2

J

that

is

{)

(Jwa?)"

m=i to say,

y (-)"'/>

l<\\

(3)

H

_ -

or

(ii)

(i^r-

J5i

'

provided that either (i)

— \ < v — In ^ \ and

The formulae given

—

^x<

ir,

section are

in this

- In > \ and

v

^ x ^ 7r.

due to Nielsen*, Math. Ann.

lii.

(1899),

pp. 582 587 two other papers by Nielsen on this subject were published at about the same time, Nyt Tidsskrift, x. B (1899), pp. 73 81 ; Ooersigt K. Danske Yidenskabernes ;

Selskabs, 1899, pp. 661

—

—665.

In the

first

two of these three papers integral values of

only were considered, the extension to general values of v being Shortly afterwards t Nielsen gave a formula for the this formula is easily obtained

sum of the

v

in the third paper.

series in (2)

when

x>

ir

;

from the integral of Dirichlet's type

(M+$)u

f* cos

M

cosît

;

^_ ur -j d

by considering the behaviour of the integrand at u = ir, It is

made

thus found that, when x

is

positive

and q

is

3»r,

5w,

the integer such that

(2q-l)ir
W

2 w^TTT+ r(i/ + l) m=1

(~)m Jv (mx)

(%mxy

«

2r(J)

(2tt-l) 2

J

7ry-*

xT{*

The importance of Nielsen's formulae lies in the fact that they make it when a function f{x) is defined for the interval (— ir, ir), if the

evident that,

function can be represented by a Schldmilch series throughout the interval

number of points) the representation is not unique and there are an unlimited number of Schlomilch series which are equal to the function f(x) throughout the interval, except at a finite number of points, namely the points already specified together with the origin and (when (except possibly at a finite

— £ < v K. i)

the end-points ±

The converse theorem, coefficients

which

— ir
that the only Schldmilch series with non-vanishing

represent

(whenl

—\ <

ir.

null-functions

h

I> + 1) *

Formula

(1)

at

points

all

|

interval

(-) m J,(mx)

^-x

($mxy

'

was rediscovered by Gwyther, Messenger, xxxm.

f Oversigt K. Danske Yidenskabernes Selskabs, 1900, pp. 55 Kielsen, Ann. di Mat. (3)

of the

v ^.\) except the origin are constant multiples of

vi. (1901),

pp.

301—329

for

see also a later paper

more complicated

by

194(9). It would be interesting to know whether

J The theorem is untrue when y>f cf. formula (3). any Schlomilch series other than the one given can represent a ;

(1904), p. 101.

—60;

results.

null- function

Cf. §

when J<.r<|.

SCHLOMILCH SERIES

19*5]

of course, of a

is,

much deeper

We

yet been published.

up

theorem

to this

character,

now

shall

and

it

637

seems that no proof of it has

discuss a series of propositions which lead

the analysis which will be used resembles, in

;

features, the analysis*,

due to Riemann, which

is

its

main

applicable to trigonometrical

series.

19*5.

Theorems concerning

the convergence

of Schlomilch

series.

We shall now discuss the special type of Schlomilch series in which v = 0, and in which Struve's functions do not appear the object of taking this particular case is to avoid the loss of clearness due to the greater complication in the appearance of the formulae in the more general case. With a few ;

exceptions, the complications in the general case are complications in detail only; those which are not matters. of detail will be dealt with fully in §§

19-6— 1962.

The series now

to

be considered

is

ao

£a + 2 am J

(1)

(ma;),

m=\

in which the coefficients a m are arbitrarily given functions of m.

We

shall first

condition that

of X,

lemma f, namely that the -*Oosm-»>oo,aiaW points of any interval of values

prove the analogue of Cantor's

am J

is sufficient to

(mac)

ensure that

am [Note. .

is

,

.

If the origin is

.

= o ( *Jm).

a point of the interval in question, then the theorem that

«m=o(l)

n

obviously true.]

Take any portion^ of the let this portion

be called It

Throughout Ix we have am J (mx) and,

— am I

)

(

.

.

which does not contain the Let the length of J, be L

interval

(cf. §

t

origin,

and

.

7'3)

[P (mx, 0) cos (mx — \ir)—Q (mx, 0) sin (mx — |tt)]

;

asm-*-»,

P (ma?, 0)^1, Hence,

Q(mco,

for all sufficiently large values- of

P(mx,0)>$,

0)^0.

m, (say

all

values exceeding

ra,,)

\Q(mx,0)\^$

at all points of/,.

Now suppose

that am

is

not o (^/m);

we have

to

shew that

this hypothesis

leads to a contradiction. * Cf. Modern Analysis, §§ 9'6—9*632. t Sinee J (mx) is an even function of

the origin without loss of generality.

x, the portion

t Ibid. % 9 '61. be supposed to be on the right of

may


638 If

am

is

number

not o (\/m), a positive

am >

e

|

I

€

must

[CHAP. XIX

exist such that

slm

whenever m is given any value belonging to a certain unending sequence * Let the smallest member of this sequence which exceeds both wh, raa m* m and 2ir/Li be called ?«/. ,

(m/ x — \tr) goes through be a portion f of I say J2 such that

Then

cos

x ,

at all points of

Next and

let

2tt/Z« 2

Then

J2

.

If

cos

at all points of

|

member

— \ir) < |

sin (ra/a?

the length of i2 , then of the sequence

j

Z2 = \irjm-l.

mr which

exceeds both

m/

.

If

is

i3

— \ir) ^ £ \/3,

is

|

cos

|

|

sin

(tfij'a?

— ^7r) < £ |

the length of /„ then L,

contained in

lies inside all

,

,

cos (rn^'x

7S

phases in 72 and so there must

all its

I3 such that

continuing this process,

such that each

m

V3,

(w8'a; — ^7r) goes through

j

when

L% is

the smallest

,

which

|

be called mj.

be a portion of 7a say

By

— far) > £

cos (w/a?

|

phases in Ilf and so there must

all its

,

= $w/m2

we obtain a sequence of intervals Iu I9 its

predecessor ; there

these intervals, and at this point

(mX — lir) ^ £ \/3» j

has any of the values

'.

sin I

,

1,,...

therefore a point

is

X

we have

(mX — \tt) < \, j

to/, to/, to,',....

For such values of to we consequently have \an J

(mX)\>\am \^/[^~ I) x [P(mX,0)

.

|

cos

(mX - ±tt) - \Q(mX,0)

>6 V and all

this is inconsistent

points of

The

/

x

|

.

j

|

sin

(toX -

J*-)

|

]

3-1 //2\ 4

vUi'

with the hypothesis that a^ J (mx) tends to zero at

.

contradiction which has

now been obtained shews

that

Om must be

o {\Jm).

The next theorem which we

shall prove is that, if the Schlomilch series

converges throughout any interval, then the necessary

and

* It is supposed that mx
sufficient condition

may

be uniquely deter-

SCHLOMILCH SERIES

19*51]

that the series should converge

639

for any positive value of x (whether a point of

the interval or not) is that the series

^ a *»*/(

] .

cob (wmj

- iw) +

—

s

sin

(mx -

\ir)\

should be convergent for that value of x.

This theorem

evident from the fact that the general term of the

is

trigonometrical series differs from a m (am m~S)

= o (ra~ 2)

and So (m -2)

;

is

J (mx)

by a function of

a convergent

m

which

is

series.

The associated function.

19*51.

Let the sum of the

series

%a + 2 am J

(mx),

m-l

at any point at which the series

is

convergent, be called f(x).

Let

F(*)- ¥ o,*»- 2

(1)

Then series

P (x)

will

whose sum

a » J«< mc> .

be called the function associated with the Schlomilch

is f(x).

It is easy to see that, if the series defining

of any

of

interval, then the series defining

P (x)

f(x) converges at

all points

converges for all real values

x.

For am J

(ntx)

—

as m-*-ao at all points of the interval,

and therefore

(§195)

a« = o ( Vw). Again, by § 2*5

(5), for all real

values of x

l«J.(«*)ki; and consequently am J (mx)

^ o / 1\

Since

*•£) is

convergent,

it is

obvious that the series on the right in (1) must be

convergent. It is evident, moreover, not only that the convergence is absolute,

that

it is

but also uniform throughout any domain of values of the real variable x.

[CHAP. XIX


640

Lemma

19*52.

We

I.

now prove that, if Y(x) sum is f(x\ and

shall

Schlomilch series whose -*/ G(x,a)x

,i\ (1)

function associated with the

is the

if

= (x + a)T(x-r2a) + (x-a)T(a;-2a)-2xT(a;)

^

,

then

lim G(x,a) = xf(x)

(2)

a-fcO

at

any point x at which

the series defining f(x) is convergent, provided that*

am = deduce from (1) that

It is easy to

-

00

G- (x, a)

= \a x — 2 ,. l

x

[(x

4o2

m

3

+ a) J (mx + 2ma) + (x — a) J (mx — 2ma) — 2xJ

and, from l'Hospital's theorem,

lim

o (Jm).

it

(mx)]

;

follows that

—— \(x + a) J (mx + 2ma) + (x — a) J (mx — 2ma) — 2xJ (mx)~\

.

2

a +o4>m cP

= lim pr—— [Jn (mx + 2ma) — J

(mx — 2ma)

+ 2m(x + a) J (mx + 2ma) — 2m (x — a) Jo' (mx — 2ma)] = x Jo" (mx) + J^ (mx)fm = — xJ (mx). '

Consequently the limits of the individual terms of the series defining Cr(#,a> are the individual terms of the series defining xf(x). It is therefore sufficient to prove that the series for Ot(x, a) converges when x uniformly with respect to a in an interval including the point o — has any value such that the series forf(x) is convergent. It shall

may be assumed, without

loss of generality, that

then take a so small that |

j

it

does not exceed \x

that the series for Cr (x, a) converges uniformly

By

x ;

is positive f,

we

when - \x ^

shall a.

^

and we

now prove

\x.

observing that

«±«-V{«(«±2«)}

% tg + V

fc

(g±2g)}

3

< |a /^ and that the

series

S a™ m »i w2

Jo [x

±a

(mx ± 2ma)

+ *J{x(x ±

2a)}]

* Since we are not assuming more than the convergence of / (x) at a single point, it is not permissible to infer from § 19-5 that am must be o (»Jm). (0, o) =0, the special t The functions under consideration are even functions of x ; and since

O

case in which

x=0

needs no further consideration.

SCHLOMILOH SERIES

641

uniformly convergent (upper or lower signs throughout being taken),

we

19*523 is

see that Or (x, a) differs from

*"—.*,

W?

[V(« + 2a) Jo (mx + 2wa) .

by the sum of two

series,

+ */(x — 2a) J (mx - 2ma) -2*Jx.J .

each of which

is

(mx)]


It is therefore sufficient to establish the uniformity of the convergence of

the last series which has been written down.

Now -

take the general term of this

^^ V(# + [

and write

it

2a)

.

J

{mx + 2wa) +

»ftx

namely

- 2a) J (mx - 2ma) - 2 sjx J .

.

(mx)]

in the form

//2x\[

.

,

If

m

_ aw + am where 4> (y)

series,

is

V

2x\

\«wr/

'

(mx — ±?r)"l

.

sin

cos

(mx — ^7r) sin2ma

8m (a? - 4a2)

//2a?\ sin (to# —

V

a

/(2x\

f 2 (imp)

2wa

m*

'

V UJ-

ma\ 2

ma

'

|7r) cos

V"™7 *»« («" - 4a2)

/sin

'

— $> (ma! + 2ma) — $> (« — 2ma)"

4m^

L

'

J

denned by the formula

* (y) = {-P <* 0) - 1} cos (y - i») - |i + Q (y, 0)j sin (y - Jtt). The general term

is

thus expressed as the

proceed to prove that each of the four general terms,

is

series, of

sum

of four terms, and we which these terms are the


The first two series are proved to be uniformly convergent, in connexion with the theory of trigonometrical series*; and the third is obviously uniformly convergent from the test of Weierstrass.

To

deal with the fourth series,

theorem, numbers f

and

1

we observe

that,

by the

first

mean-value

exist such that

-1<0<1, -1<0,<1, * Cf. Modern Analysis, §§9'62, 9-621. It has been the general (bat not invariable) custom to obtain various properties of the series without establishing the uniformity of their convergence. The convergence of the series for f(x) is required to deal with the first serits the second series can be dealt with in consequence of the less stringent hypothesis that a m =o\ >/m). ;

t

The number

is

a function of a variable

t

which

will be introduced immediately.

for

[CHAP. XIX


642 which

-

24> (mx)

= 2ma

(mx + 2ma) —

— 2ma)

(mx - 2mat) - ' (mx +

{<£'

I

(nix

2mat)} dt

Jo

= — 2ma

4>mat

I

(mx —

2moi6t) dt

Jo

= — 4m

2

= 0(l/y ) when

Since 4>"(^)

(« - 2ma6

a2 4>"

2

3/

is

1

).

large, it is evident that

m=l is

uniformly convergent with respect to

Hence

G (x, o)

is

a.

sum of six series each of which when — \x < a < \x\ and therefore

expressed as the

uniformly with respect to a

lim

converges

G (x, a)

a-»-0

is

equal to the

sum

of the limits of the terms of the series for

equal to xf(x), provided that the series the lemma to be proved.

is

19*53.

We that a m

Lemma (\/m)

is sufficient to

As

G (x, a),

convergent

;

i.e. it

and this

is

of%

19*51, 19*52, the condition

ensure that

+ a) F (x + 2a) + (x-a)F(x- 2a) - 2xT (x) _ Q a

a— all values

is

II.

shall next prove that, with the notation

—

(x

for

for/O)

of

x.

in § 1952,

the series for a

we need

consider positive values of x only

;

and we express

G (x, a) as the sum

of six series each of which is easily seen to ±x < a < \x, by applying the theorems con-

be uniformly convergent when cerning trigonometrical series which were used in

§ 19*52.

Hence lim [a

G (x, a)]

a—0

= lim(W)- £ \im-^~[(x + a)J (mx + 2ma) + (x - a) Jo (mx - 2ma) - 2xJ

{mx)\

= 0, and

this is the

19-54.

We

lemma

to

be proved.

The analogue of Riemanns theorem* on trigonometrical

series.

can now prove that,

if

=

0,

two Schlomilch series of the type now under and with Struve's function absent) converge

Cf.

Modrrv Analysis, §963.

consideration

(i.e.

with v

**

SCHLOMILCH SERIES

19-53, 19*54]

643

and have the same sum-function throughout the interval (0, ir), then corresponding coefficients in the two series are equal. The formal statement of the theorem is as follows

Two Schlomilch

series, of the special type, which converge and are equal at of the closed interval (0, tr), with the possible exception of a finite number of points, must have corresponding coefficients equal, unless the end-

all points

and

points

If these

7T

are both exceptional points.

points are exceptional points, the two series

may differ by a

constant

multiple of the series

£+2

(-T

(nix).

Let the difference of the two series be

$a + 2 am J

(mx),

m-l

and

let

the

sum

of this series be f(x), so that f(x) converges to zero for and nr, except the exceptional values.

all

values of x between

Let £x

,

&

be any points (except the origin) of the interval

(0, ir),

such

that there are no exceptional points inside* the interval (£, £2).

We

proceed to prove that,

if

V(x)

is

the function associated with the

Schlomilch series for fix), then P (x) is a linear function of log x in the This is the analogue of Schwarz' lemmaf.

interval (£„ £2 ).

= 1, orif0 = -l,andif

If

*(«)-^[F(»)-F(ft)-^^{F(6)-F(ft)j]

then

(x) is

continuous when Zi^x^g,, and

*(fc)-*(&)-o. If the first term of will first

be some term of

(x) is not zero J throughout the interval (&, £2), there point c at which it is not zero. Choose the sign of so that the

(c) is positive at

c,

and then choose h

so small that

(c) is

still positive.

Since

continuous in (&,£»),

(x) is

positive since

(c) is

positive.

Let

it

it

bound which is bound at clt so that

attains its upper

attain its upper

ft
Now by Lemma j.

(c,

I (§

+«)

1952)

(c,

+ 2a)+(d - a) 4o

«ô *

The

points £ 2

,

|2 themselves

may

(ct

- 2a) - 2c,

(cQ

=^

be exceptional points.

f Cf. Modern Analysis, § 9631. X If it is zero throughout (£ x |.2 ), then ,

2

P {x)

is

obviously a linear function of log x.

[CHAP. XIX


644

But

(Cj 4-

2a)

^

(cj),

— 2a) ^

(Cj

tf>

(d), so the limit

on the

left

must be

term of £ (cc) must be zero throughout (f1} f2), that is to say that F(#) must be a linear function of log x and this is the theorem to be proved. This contradiction shews that the

negative or zero.

first

;

Hence the curve whose equation

y

is

— F (a?)

consists of a set of

segments

of logarithmic curves with equations of the type

y= A Now, by

§

1951, F(#)

is

log x

continuous in

+ B.

(0, ir),

and so these logarithmic curves y = 'F(x) cannot have

are connected at the exceptional points; and the curve

an abrupt change of direction at an exceptional point, because, by

a-*-o

la

{_

even when £

is

2a

an exceptional point

that

;

is

Lemma

II,

J

to say

fF'(£+0) = £F'(f-0).

A and B cannot be discontinuous at the exceptional and so they have the same values for all values of x in the

Hence the constants points,

interval (0,

-n-).

when

Consequently,

H*

2

Make

x•-*

;

ir,

- 2 -*-^

Consequently,

when

^x %

on the

left

Therefore

A \ogx + B

has a limit

when

ir,

m=\ series

= A\ogx + B.

the series on the left has a limit, namely

because it is uniformly convergent. x -* 0, and so A is zero.

and the

'-

"'*

r»=l

if*

converges uniformly throughout

(0, ir), so

integrations

term- by- term are permissible.

Replace x by x sin

by §12

0,

multiply by sin

11,

&a x*-B=

2.

~

I

0,

tt

nd integrate from

"j (mx sin sin 0d0 0)

s am sin ma? Hence, when

^ x ^ tt, "

2

am

smmx = -

D fâ x3 — Bx.

to \tt.

Then,

.

SCHLOMILCH SERIES

19'6]

Multiply by sin

mx and

integrate from

to

ir

645 it is

;

then evident that

7ra-

2mi*8 Since dm

m

[2m3

'

J

given to be o (*Jm), this equation shews that

is

B = J9 n*a

a m = (-)w a

,

.

Hence we must have f(x) = a w=l

From the

J

results contained in § 19*41 concerning the behaviour of the

and at #= ir, it is evident that f(x) cannot be a series on the right at a? = convergent Schlomilch series at either point unless a is zero; and this proves the theorem stated at the beginning of this section. 19*6.

Theorems concerning

We shall now

study briefly the series " am Jv (mx)

fro,

{)

We

i> + l).ti

shall first prove that,

-*- oo

+ w H, (mx) fr

QmwY

when v<\,

the series tends to zero as ra is sufficient to

of generalised Schlomilch series.

the convergence

(m +

the condition that the

at all points

of any

\)th

term of

of x

interval of values

ensure that

am = [Note. If the origin

o(ra" + *),

6w

= o(i»r+ *).

a point of the interval in question, then the theorem that

is

«m=0(l) is

obviously true.]

Since the series under consideration of x

if

the signs of

by considering an

lost

We call

.\ 5 ($mxy

H

r

(mx) *

unaffected

by a change

in the sign is

interval on the right of the origin.

this interval /,

am Jv (mx) + bm

is

the coefficients bm are also changed, no generality

all

;

we

and, at all points of /,,

= 71

cm

yii / +i */'jr (%mxy

rD/ [P (w*,

have,

,

.

v) cos

by

§

1041

(4), v

,

, (mx - \vir -\ir- n m )

— Q (mx, v) sin (mx — \vir — \ir — vm )] + b, n (»n _1 where am = cm cos m b m = cm sin M We now suppose that a m and bm are not both o(w + *); we have to shew .

r\

,

i/

),

,

that this hypothesis leads to a contradiction. If

am and b m are not both

o (m" + *), a positive

number

e

must

exist such

that cm

whenever

m

x

,

wio,

m

m

is

3 ,....

>em v+i

given any value belonging to a certain unending sequence

[CHAP. XIX


646

We now

prove, exactly as in § 19*5, that, at

some point

X

of

Ilf the

inequalities

|

P(mX,v)>l \Q(mX,v)\ \ */3, cos (mX

w has

are satisfied whenever is

j

any value belonging to a sequence (mr ) which

a sub-sequence of the sequence

m

For values of

(mr

).

which belong to this sub-sequence we have

(ml) + bm H„ (mX)

a m Jv

qmxy —\

and, since v

This

small.

is

must both be

r)

|

|

>m&-°***>

negative, the expression on the right cannot be arbitrarily

is

the contradiction which

is sufficient

(m + l)th term

o(ra"+*) if the

to prove that

am and

bm

of the Schlomilch series tends to

zero at all points of J,.

The reader may now prove

(as in §

19 5) that, when v <\,tf the generalised

Schlomilch series converges throughout any interval, the necessary and condition that

it

may

sufficient

x {whether a point of

converge for any positive value of

the interval or not) is that the series

2

•—

a

Hcos(ra#— kwr — \ir — vm )

\

4i/

2

-l

Smx

Bin

(mx -

W

l-rr

- Vm )\ +

-

.

r(l + i)r(i) ] ,

should be convergent for that value of x.

19*61.

The associated function.

Let us take

—£< v<

\

,

and

any point at which the

the

sum

of the series

| am Jv (mx) + bm ($mxy

£a r(i/+l) at

let

H

,

(mx)

«=i

series is

convergent be called f, (x).

Let

m {) Then

— v_ *"

2

W ~8I>-|-2) /

floff

5 a m J, (mx) + bm H„ (mx) mti

m\(tmxy

F„(a?) will be called the function associated with the Schlomilch series

whose sum isf¥ (x).

f

It is easy to prove that, if the series defining r (x) converges at all points interval, then the series defining F„ (x) converges for all real values of x.

of any

The only

respect in which the proof differs from the analysis used in

§ 19*51 is that the additional

theorem that

the real variable x has to be used.

H

r

(x^x"

is

a bounded function of

SCHLOMILCH SERIES

19-61, 19-62]

Again,

a

(2)

¥

let

= l[(x + 2va + a)F,(x + 2a) + (x - 2va - a) F„ (x - 2a) - 2x F„ (x)]/**.

(x,a)

Then, just as in

we may prove that*

§ 19*52,

»

Hm».(^«)-./.(.)at

647

any point x

at

rfr+ )r(t)

which the series defining

fy (x)

j^ i

is convergent,

provided that

ao

am and bm are both Further we

o

may

(m -"'*) and

2

that the series

bm/m is convergent.

prove that lim [a Gr„ {x, a)] =

0,

a-»-0 ao

provided only that am and bm are both o

(to" + *),

whether the series

X m=l

bmfm is

convergent or not.

The analogue of Riemann's theorem.

19*62.

We

can now prove that, if two generalised Schl&milch series of the same — \ < v < ^) converge and have the same sum-function at all

order v {where

ir, ir) with the possible exception of a finite supposed that the origin and the points ± tt are not all exceptional points), and if the coefficients of the terms containing Struve's functions in the two series are sufficiently nearly equal f each to each, then all

points of the closed interval (—

number of points

(it is

corresponding coefficients in the two series are equal.

Let the difference of the two

_Joo__

I> + 1) and

let

the

sum

series

am Jv (mx) + bm H.,(mx)

|

The convergence 7r)

($mxy

»ti

of this series be

f

¥ (x),

to zero at all points of the interval

(— 7r,

(— tt,

*

two

that 6m

We

tt)

with a

finite

number

of exceptions.

of the series for /„ (x) nearly everywhere in the interval

The statement that the tions in the

now

for /„ (x).

so that the series for /„ (x) converges

necessitates the equations

am = o (m"+*),

mean

be

bm

= o (m"+i).

coefficients of the

terms containing Struve's func-

be sufficiently nearly equal » -»-Oasm-*oo in such a way that i. series are to

to be interpreted to

is

L. —

is

convergent.

discuss the function F„(a") associated with the Schlomilch series

It can

The presence

be proved J that

of the series

if

on the right

the interval is

(£,,

f2 )

is

such that the origin

due to the lack of homogeneity in the

equation satisfied by Struve's function. f This statement will be made definite immediately. t It seems unnecessary to repeat the arguments already used in § 19*54.

differential


648

[CHAP. XIX

and the exceptional points (if any) are not internal points of the interval, then F,(#) is a linear function* of a? - 2" in the interval. It may then be shewn "

that the exceptional points do Dot cause any discontinuity in the form of

F

r (x),

and hence we deduce that

iT v (x) = Ax-*" + B,

(0
\Fv {x) = A'\x\-" + B where A, B,

Now

A\

,

(0>x>-tt)

>

B' are constants.

take the equation

p

m = 8I>a^+

" K

>

1 aw Jv (mx) + bm H„ (mx) m\{\mxy mZx

2)

x by #sin0, multiply by sin2"* convergent integral) and integrate from

1

replace

0/cos2v

(which has an absolutely

0,

The

to \ir.

converges uniformly in this interval of values of

0,

series for

Fr (#sin0)

so term-by-term integra-

tions are permissible. It is thus found that +1 ^ sin "—gT/ 0,0 = 2

f**-,

,

F„ (X Sin 0) •

,

cos "0

Jo fj *

00

- 2

"a

2

m

_ g^T(\ — v)

Y(\

12r(£)

T(i)

^

^

;

and a similar

Since am and 6 OT are both o

we

If

Zx

— cos mx)

m*x

we deduce a*a?

4a?

that 1

- 2"

T

(§) „ „ -^r(, = 12-- r + i), ( f-,) equation may be obtained when O^x^ — tt.

(m v+*),

it is

it

,

permissible to differentiate (1) twice only be differentiated once if

may

once as the case may be, the resulting series tends to a limit as x -* 0, but the resulting expression on the right

left

fails to

do so unless infer that

F„ (x) at the

(2)

cos2"0

differentiate, twice or

on the

We

JQ

—(Iff

'

g m sin mx +b m (l

*

>*/>-£; but

term-by-term when

\

— v)

d0

0) sin" +1

.(£m#)"

substitute for P„ (x sin 0)

when 0^#^7r

It

2

v aTO sinm#+&m (l-cosma;)

/ 1X (1)

of

^

m J, (mxsm0) + bm H„ (mx sin

m=l JO

When we

fi'8m iv+3 cue* 5^=-^ 8r(i/ (i/ + 2)J cos2 "0

now

A

is zero.

A = 0,

and

in like

manner A' must be zero the continuity B and must be equal. ;

B

origin then shews that

follows from (1) that

% am Sm mX + km (1 - COS mx)

Jr

„ _ -^-BxTiv+l) (^X3

.

when — 7r $ x ^ ir. *

When

v is zero

x~iv has

to be replaced

by logx.

SCHIidMELCH SERIES

19*7]

— ir

Multiply (2) by cos mx and integrate from 6TO

(3)

649 to it

;

and then

= 0.

Again, multiply by suitor and integrate; and then

(-) m dm This equation

is

=
inconsistent with the fact that

and then

am

Hence the

series for /„ (#)

a

Now

= (-)m a

must reduce

+

lr(r+i)

the series for/r (a?)

is

o(mv+l)

unless

.

to

a^y

mt 1

at least one of the points 0,

am

}.

- 7r

J-

not an exceptional point and cannot converge at that point unless a* is zero, so that am ir,

is

;

is also zero.

We

have therefore proved that, if the series 2 bm/m is convergent all the am and bm must vanish that is to say, the two Schlomilch series with which we started must have corresponding coefficients equal. And this is the theorem to be proved. coefficients

;

We have therefore established for Schlomilch series in which — \ < v < £ theorems analogous to the usual theorems concerning the representation of null-functions

19*7.

by trigonometrical

series.

Theorems of Riemanris type concerning

series

of Bessel functions and

Dini's series of Bessel functions.

We

shall

now

give a very brief sketch of the method by which the series

discussed in Chapter xviii,

namely

2 am Jv {jm a;),

t»=i

(in

which v

2

bm

Jv (\m

a;),

«t=i

> — £) may be investigated

after the

manner

of Riemann's investi-

gation of trigonometrical series.

The method

is identical with the method of investigation of Schlomilch given in §§19 6 1962, though there are various points of detail*, which do not arise in the case of Schlomilch series, due to the fact that m j and Xm are not linear functions of m.

series just

—

These points of detail are very numerous and there is no special difficulty in discussing any but it is a tedious and lengthy process to set them out in full, and they do not seem to be of sufficient importance to justify the use of the space which they would require. The reader who desires to appreciate the details necessary in such investigations may consult the papers by C. N. Moore, Trans. American Math. Soc. x. (1909), pp. 391—435; xn. (1911), pp. 181—206; xxi. (1920), pp. 107—156. *

of

them

;

In the if

[CHAP. XIX


650

first place, it is

easy to prove by the method used in §

195

that

the series 00

00

2

2 bmJyQ^x)

OmJ.ijmti),

converge throughout any interval of values of x, then

Next we consider the

associated function

we

;

write

OO

f(x)

= 2 am Jv (jm x\ m=l

and then the function associated with /(a?)

2 5s/» (*»*>.

F(x) =

m—ljm It

may be proved

defining /(a?)

is

that,

«"

positive value for which the series

when x has any

convergent, and


is

if

the expression

^-s l(x + 2va + a) F(x + 2a) - 2x F(x) + (x~ 2v* -«)F(x- 2a)] arranged as a series in which the rath term has am for a factor, then the with respect to a in an interval containing the point o = 0, and that its limit when a -* is — ^""/(a?).

is

latter series is uniformly convergent

It

may

also

be proved

that,

the condition that am =*o (*/m)

whether the

is sufficient

t- [(# + 2vo + o) F(x + 2d) tends to zero with

The

f(x) converges or

not,

- 2x F(x) + (x~ 2i>« - a) F(x - 2a)]

a.

proofs of these theorems depend on a ?2*

number

2 a bounded function of a

;

proofs of the

of

lemmas such as the lemma* that

ain2/

- rt

«8

m=l! Jm+1"' is

series for

to ensure that

„2

fm<**

!

lemmas can be constructed on the v=$.

lines of the

proofs in the special (trigonometrical) case in which

It

now

follows in the usual

manner

function throughout the interval (0,

(cf.

1),

equation

^=

*-rf^ + (2,, + 1) F(x) = A + Bx-*>,

and sof where

A

§ 19'54) that,

then F(x)

and

B are

constants. * Cf.

t

This equation

Modern Analysit,

is

§ 962.

is

when

a null-

the differential

>

valid

When v=0, F(x)=A+B'logx.

when/(«)

satisfies

< x ^ 1.

SCHLOMILCH SERIES

19*7]

Now

v>-±,

since

be the value of

m

and

;

converges uniformly

Hence F{x) zero

is

when

J,(jm^)/(jf>^y so,

when

when*

^x<

<

v

Oâ^l

bounded when

is

whatever

£, the series

by the

1,

651

test of Weierstrass.

a continuous function of x in the closed interval and so positive and is zero in the case v = 0.

is

v is

B

;

B

For any assigned value of n multiplying the series for F(x) by x¥+1 J,(j x) n does not destroy the uniformity of its convergence; and, when we integrate,

we

find that

o«

J? 0») = j«

•

2

{Ax* J

+ Bar 9 ) x Jv (jn x) dx

J

B

=jn [

Now, when n

'^rW

j''«->i-V(£),

and so the formula just obtained

= o (00 when

*

]

is large,

i

«n

~ (A + B) J: ° n)

v

>—

for

£ unless both

an

is

inconsistent with the equation

A +B and B are zerof;

and then an

is

zero.

Hence a series of Bessel functions (in which v > — |) cannot converge to sum zero at all points of the interval (0, 1), with the possible exception of a finite number of points (the origin not being an exceptional point} when the

v

>

£),

unless

all

the coefficients in the series are zero.

We infer that two series of Bessel functions, in which v > — \, cannot converge and be equal at all points of the interval (0, 1), with the possible exception of a finite number of points, unless corresponding coefficients in the two

series are equal.

Dini's series § 00

f(x)= 2 may be

dealt with in the

6m «7F (\JB a;)

same manner. The

associated function

is

defined

by the equation

F(x) = i

^J^zl,

* When *>}, the convergence of the series for /(a)/*" at.r=0 is sufficient to ensure the uniformity of the convergence. t An exception might occur when v - 1= - £ ; but this is the trigonometrical case.

% The

series divided

by x" then has to converge at the

§ It is supposed for the present that

H+v>0,

origin.

so that no initial term

is

inserted.

and

it is

inferred that

A

constants

and

when f(x)

B exist such

bn

(l

a null-function throughout

is

- £) «/„* (\n) + JJ 2 (X.)] = v/^ (A*P + Bar*) xJv (X»«) efo

5=

when v ^ 0. Now, when n is large

<*•)}]

5

5^0, °n

~ 2T( V) bn

> — £. J? is zero,

is

= o (\/n),

and therefore

-

6n [(l

This equation

'

with the equation

this is inconsistent

Hence

f^p - ^.

«/-iOO = 0(\n-»),

J, +1 (X„)=0(X«-*), so that, if

since v

then

A + Bar*»,

- \„ r^^+1 (\„) +• b

and

(0, 1),

that

F(x) =

and

[CHAP. XIX


652

£) ^

2

2

+ J* (Xn)] = 4\„ J,+1 (X»).

(X.)

inconsistent with the equation bn

—

o (\/n)

A is zero, since «/„+, (\„) is not zero and then bn is zero. We next consider what happens when H + v is zero or negative

unless

;

cases Dini's series

;

in these

assume the forms oo

b

+ 2

Xp

bmJyCKnX),

w=l oo

6 /„ (\ x)

+ 2 bm J, (\m x),

respectively.

In the second of the two cases the previous arguments are unaffected by the first of the two cases needs more careful consideration because the initial term to be inserted in the associated the insertion of an initial term function

;

is

b x*

4(i/+l)' and hence, when n

>

1, l

bn JS (\„)

= X„

[Ax*

2

J

+

^f^ + Bar*} xJ„

{\nx) dx

SCHLOMILCH SERIES

19-7]

Since bn (£^n)">

= o (0i) we infer first that b = and so bn =

and then that

;

2?

= 0,

653

by considering the term in

for all values of n.

We

infer also that, as in the limiting case of series of Bessel functions, Dini's series of Bessel functions cannot represent a null-function throughout the interval (0, 1), and that if two of Dini's series (with the same v and H)

converge and are equal at of a finite

are equal.

number

all

points of the interval (0,

1),

with the exception

of points, then corresponding coefficients in the two series

CHAPTER XX THE TABULATION OF BESSEL FUNCTIONS Tables of Bessel Functions

20' 1.

It is evident

VIII

and

associated functions.

from a consideration of the analysis contained in Chapters

vii,

and XV that a large part of the theory of Bessel Functions has been con-

structed expressly for the purpose of facilitating numerical computations connected with the functions. To the Mathematician such computations are of less interest and importance* than the construction of the theories which make them possible; but to the Physicist numerical results have a significancef

which formulae may

fail

to convey.

As an application of various portions of the Theory of Bessel Functions, has been considered desirable to insert this Chapter, which contains an historical account of Tables of Bessel Functions which have been previously published, together with a collection of those tables which seem to be of the

it

greatest value for the present requirements of the Physicist.

The reader will not be concerned with the monotony and technical irrelevance of this Chapter when he realises that it can be read without the efforts required to master the previous chapters and to amplify arguments so ruthlessly condensed.

The first Tables of J (*) and J (x) were published by Bessel himself in memoir on Planetary Perturbations, Berliner Abhandlunyen, 1824 [1826], x

his

pp. 46

— 52.

decimals

fo*r

These tables give the values of J (x) and J, (x) to ten places of a range of values of x from x = to x = 3'20 with interval O'Ol.

A short Table of J»(x) and J Hx) to four places of decimals was constructed by Airy,

Phil.

with interval the same

Mag. 0*2.

(3)

xvm.

(1841),

p.

7; its

range

is

from

a?

=

Airy J had previously constructed a Table of

to

x — 100

2«/j (x)/x,

of

scope.

The function Ji{x)jx was subsequently tabulated to six places of decimals by Lommel, und Phy». xv. (1870), pp. 164—167, with a range from x-^0 to #=20*0 Table, with a Table of JJ (a?)/^2 was republished by, Lommel, 0*1 this interval with Munckener Abhandlungm, xv. (1886), pp. 312—315.

ZeiUchrift fiir Math,

,

;

* Cf. Love, Proc.

London Math.

Soe. (2) xrv. (1915), p. 184.

" in formulas unless I feel their arithmetical f Cf. Lord Kelvin's statement I have no satisfaction magnitude at all events when formulas are intended for definite dynamical or physical problems."

—

Baltimore Lectures (Cambridge, 1904),

p. 76.

% Tram. Camb. Phil. Soc. v. (1835), p 291. A Table of 2JX (x)/x and its square, to four or five 1125° (with interval 15°) places of decimals, in which the range is from to the circular measure of was given by Sohwerd, Die Beugung$er$cheitmngen (Mannheim, 1835), p. 146. :

TABULATION OF BESSEL FUNCTIONS

20 *1]

655

In consequence of the need of Tables of Jn (x) with fairly large values of for Astronomical purposes, Hansen constructed a Table of J (x) and (x) to six places of decimals with a range from x = «/"i to x = 10-0 with

n and x

01

was published in his Ermittelung der absoluten StOrungen und Neigung (Gotha, 1843). Hansen's Table was reprinted by Schlomilch* and also by Lommelf who extended it interval

;

this

in Ellipsen von beliebiger Excentricitdt

toa: = 20.

These

J (x)

and

x

tables,

however, are superseded by Meissel's great Table of

to twelve places of decimals]:, published in the Berliner

lungen, 1888; its range

is

from x =

to

x

— 15*50

J (x)

Abhand-

with interval 001.

by Gray and Mathews, A Treatise on Bessel Functions (London, 1895), pp. 247 266, and an abridgement of it is given in Table I infra, pp. 666—697. Meissel's Table

was reprinted in

full

—

A

Table of

x = 60 with

to

lxvi. (1900),

J

(x)

and

«/,

twenty-one places of decimals, from x = been constructed by Aldis, Proc. Royal Soc.

(x) to

interval 01, has

p. 40.

A Table of J (ntr) to six places of decimals for n = Nagaoka, Journal of the

The value Mag.

of

J

(40)

Coll.

l, 2, 3, ...,

of Sci. Imp. Univ. Japan,

50 has been computed by

iv. (1891), p. 313.

was computed by W. R. Hamilton from the ascending

series, Phil.

(4) xiv. (1857), p. 375.

A

Table of Ji(x) to six places of decimals from ^=201 to # = 41 with interval 0*1 or by Steiner, Math, und Naturwiss. Berichte aus Ungarn, xi. (1894),

0-2 has been published

372—373.

pp.

The

earliest table of functions of the second

kind was constructed by is a Table to four

B. A. Smith, Messenger, xxvi. (1896), pp. 98—101; this 0) places of decimals of Neumann's functions (x) and

Y

from x =

to

x =100 with

interval

<

01 and from

Y (x). Its range is x= 10 to x = 102 with <"

interval 01.

A

more extensive table of these functions

ciation Report, 1914, pp.

76

—82;

is

given in the British Asso-

this is a Table to six places of decimals

whose range is from x = to x= 1550 with interval 002; a year later a table was published, ibid. 1915, p. 33, in which the values of F (x) and F<» (x) were given to ten places of decimals for a range from x = to x = 6'0 with interval 0*2 and from x = 6*0 to x= 160 with interval 05. (0»

Shortly after the appearance' of Smith's Table, an elaborate table was constructed by Aldis, Proc. Royal Soc. lxvi. (1900), p. 41, of Heine's functions § O (x) and Gx (x) to twenty-one places of decimals; the reader should be * Zeitschrift

far Math, und Phijs. n. (1857), pp. 158—165. t Studien iiber die Bessel'schen Functionen (Leipzig, 1868), pp. 127 135. X Meissel's Table contains a misprint, the correct value of J (0-62) being +0-90618..., not + 0-90518.... An additional misprint was made in the reprint of the Table by Gray and Mathews.

—

§

123

;

These functions were also tabulated by B. A. Smith, Phil. Mag. the scope of this table

is

the

same

as that of his Table of I'M (x)

(5),

and

xlv. (1808), pp. 122 I'*

1

)

(,c).


656

[CHAP.

XX

reminded that these functions are equal to — \irY (x) and — |w Y t (x) reto x = 6'0 with interval 0*1. spectively. The range of Aldis' Table is from x =

Another table of these functions with a smaller interval was published in

—

the British Association Report, 1913, pp. 116 130; this table gives the functions to seven places of decimals for a range from x = to x = 16*00 with interval 001.

decimals from

The

The Report for 1915, p. 33, contains a to x — 15*5 with interval 0"5.

table to ten places of

x— 65

functions

in Table I infra

;

Y

(x)

Y

and

(x) are tabulated to seven places of decimals

t

this table has

an appreciable advantage over the British

Association Tables*, in that the auxiliary tables

make

matter; in the British Association Tables interpolation

By means of the

interpolation a trivial is

impracticable.

recurrence formulae combined with the use of the tables

which have now been described, it is an easy matter to construct tables of functions whose order is any integer. Such tables of «/",» (x) were constructed by Hansen and reprinted by Schlomilch and Lommel after their Tables of J (x) and J (x). Subsequently Lommel, Miinchener Abhandlungen, XV. (1886), 316, published a Table of Jn (x) to six places of decimals, in which pp. 315 n = 0, 1, 2, ..., 20, and x - 0, 1, 2,..., 12; this Table is reprinted in Table IV 731. A Table of Jn (x) of practically the same scope was also infra, pp. 730 published by Meissel, Astr. Nach. cxxviii. (1891), col. 154 155.

—

x

—

A

much more

—

Jn (x) was computed by Meissel, but it He communicated it to Gray and Mathews 279. This table gives Jn (x) to pp. 267

extensive Table of

seems that he never published

it.

for publication in their Treatise,

eighteen places of decimals

—

when n = 0,

1, 2, ...

Some graphs of Jn (x) were constructed, with table,

by Hague, Proc. Phys.

,

60,

Soc. xxix. (1917), pp.

The corresponding Tables

and x =

0, 1, 2, .... 24.

the help of the last-mentioned

211

— 214.

of functions of the second kind are not so ex-

tensive.

The

British Association Report, 1914, pp.83

to five places of decimals forf

with interval 0*1 and x Similar Tables f of

= 60

Y

{n)

to

= 0,

1, 2,

160 with

—86 contains Tables of G

...,13 for the range

appeared in the Report

values of Hankel's function

Table of

Yn (x)

Yn (x)

for 1914, pp.

—

36.

— 114.

to seven (or more) significant figures

is

G

contained in (x)

and

* In the coarse of (x)

34

had been given previously by

Table IV infra. This has been computed from Aldis' Table of

and 6,

n (x) to 6*0

=

interval 05.

Nicholson, Proc. London Math. Soc. (2) XI. (1913), pp. 113

A

x

(x) to six places of decimal- iwith the intervals

in the earlier part equal to 0*2)

Some

w

G

x

(x).

computing Table I, a small part of the British Association Table of G (x\ was checked, and the last digits in it were found to be unreliable in about 5 °/ of the

entries checked.

t For the larger values of

n the functions are not tabulated

for small values of x.

20,1 J


H

657

Tables of log10 [s/iîrx) ^> (x) v ] to eight significant figures are given in the British Association Report, 1907, pp. 94—97. The values assigned to v are 0, \, 1, 1§, ..., 6£, and the range of values of x is from 10 to 100 .

j

j

x=

(interval 10)

and 100

1000 (interval

For this range of values of x, the asymptotic expansion (§ 7*51) gives so rapid an approximation that the Table is of less value than a table in which the values of x and the intervals to

100).

are considerably smaller.

Functions of the first kind with imaginary argument have been tabulated in the British Association Reports. The Report for 1896, pp. 99—149, contained a Table of I (x) to nine places of decimals, its range being from to x = 5-100 with interval 0001. Table of /, (x) of the same scope had been

x=0

A

—

published previously in the Report for 1893, pp. 229 279; an abridgement of this (with interval 001) was given by Gray and Mathews in their Treatise, pp.

282—284.

I (x) and 1^ (x) to twenty-one places of decimals have been conby Aldis, Proc. Royal Soc. lxiv. (1899), p. 218. The range of these Tables is x = to x = 60 with interval 01; Aldis also gave (ibid. p. 221) the values of I (x) and I (x) for x = 7, 8, 9, 10, 11. Tables of

structed

x

Extensive tables connected with I (x) and I\ (x) have been published by Andmg,Sechsstellige Tafeln der Bessel 'schen Funktionen imagindren Arguments

These tables give log ]0 1 (x) and log 10 /, (x)/x] from #=0 with interval 001. They also give the values of the functions

(Leipzig, 1911). to

x= 1000

vX27nr)
I

{

(x),

^(2irx) e~*Ix

(x),

.

x from x = 100 x = 200 to x = 1000

log 10

= 500

for values of

to

(interval 1),

(interval 10),

x

[Jx IQ (x)) and log10 .

(interval 01),

and

# = 50

{y/x /, (*)} .

to

#=200

for various larger values of x.

—

Table II infra, pp. 698 713, gives the values of e~ x IQ (x) and er* J, (a?);, these have been computed, for the most part, by interpolation in Aldis' Table.

The

earliest tables of functions of the type

Kn (x) were

constructed bv These give (x) and x (x) to twenty-one places of decimals for values of x from ^= to x = 60 with interval 01, and also to between seven and thirteen significant figures from x = 5*0 to x = 12*0 with interval 01. Aldis, Proc.

Royal

K

The

values of

219—221.

Soc. lxiv. (1899), pp.

e*Ka (x) and e*K

x

(x) in

K

Table II infra were computed with

the help of Aldis' Table, like the values of

and eÎ^x).

By means of recurrence formulae, In (x) has been tabulated to twelve signiw = 0, 1, 2, ..., 11 over the range of values of x from x = x = 60 with interval 02. These Tables of In (x) were published in the

ficant figures for

to

—

British Association Report, 1889, pp. 29 32, and reprinted by Gray and Mathews in their Treatise, pp. 285 288. An abridgement (to five significant figures) of these Tables has been given by Isherwood, who added to them

—


658

Kn (x) to five significant figures for n =

Tables of

of values of x from

published in the

x=

to

x = 6'0 with

0, 1, 2,

interval 0*2.

Mem. and Proc, Manchester

Lit.

. .

[CHAP. .

,

XX

10 over the range

Isherwood's Tables were

and Phil. Soc, 1903

— 1904,

no. 19.

K

Tables of e~ x In (x) and n (x) to seven places of decimals are given in Table IV infra, pp. 736—739.

The

earliest

who

Meissel, 11,

. .

.

,

Tables of Bessel functions of large order were constructed by

has calculated

Jm (n)

to seven significant figures for col.

*° twelve significant figures for

21, Astr. Nach, cxxix. (1892), col. 284; Meissel also calculated ?i

= 1000,

999,

154—155. The values of Jn (n), Jn-An),

n=10,

Jn (1000)

981, ibid, cxxviii. (1891), n (n), Y< ~v(n), G n (n), G n ^(n)

...,

Y™

n from n = l to 7? = 50 (interval 1), = w 50 to w=100 (interval 5), n = 100 to n = 200 (interval 10), n = 200 to n = 400 (interval 20), n = 400 to n = 1000 (interval 50), n = 1000 to n = 2000 (interval 100) and for various larger values of n, are given in the British to six places of decimals for values of

Association Report, 1916, pp. 93

Tables of

VI

in Table

The

— 96.

Jn (n), Jn '(n), Yn (n), Yn '(n)

infra, pp.

functions ber

to seven places of decimals are given

746—747. (x), bei (x),

ker (x) and kei (x) have been extensively

tabulated on account of their importance in the theory of alternating currents.

A brief Table of ber (x) and bei (x), computed by Maclean, was published by Kelvin, Math, and Phys. Papers, ill. (1890), p. 493. Tables of J (xy/i) and y'2 J (x \fi) to twenty-one places of decimals have been constructed by Aldis, Proc. Royal Soc. lxvi. (1900), pp. 42 43; their range is from .

x

—

x=

x = 60 with

These are extensions of the Table of J (x *Ji) to nine places of decimals for the range from x = to x = 60 with interval 02 published in the British Association Report, 1893, p. 228, and reprinted by Gray and Mathews in their Treatise, p. 281. to

interval O'l.

Tables of ber(#), bei

x= 1, 2, 3,

...,30,

(a;), ker(#) and kei (x) to four significant figures for have been published by Savidge, Phil. Mag. (6) xix. (1910),

p. 53.

The

functions ber

(x),

places of decimals, from

bei (x), ber' (x)

x

=

to x

=

and

bei' (x) are

10*0 with interval

tabulated to nine

01

in the British

—

Association Report, 1912, pp. 57 68; and a Table of ker (x), kei (x), ker' (x) and kei' (x) of the same scope (except that only six or seven significant figures

—

were given) appeared in the Report for 1915, pp. 36 38. Tables of squares and products of the functions to six significant figures from x = to x= 100 with interval 02 were given in the Report for 1916, pp. 118 121.

—

The functions
—

,

.

.

.

,

.

. .

,


20-1]

with x =

659

34 with smaller ranges of values of x see and n = 15, Table V infra, pp. 740 743. A Table of the same functions to four places of decimals with n =0, 1, 2 and from a; = to x = 80 with interval 02 is given by Dinnik, Archiv der Math, und Phys. (3) xx. (1913), pp. 238—240. 1, 2,

. .

.

,

20,

. . . ,

;

—

Functions related to J±(n +h)( x) have recently been tabulated in the British The notation used is

Association Reports.

s/Qirx)

.

Jn+i (x) = Sn (x), i-Yîrx)

.

J_ n . h (x) = Cn (x),

En (x) = \Cn (x) + iSn (x)\, tabulated are Sn (x), Cn (x), En (x),

and the functions and their logarithms. In the Report

2

tabulated to seven significant figures for n

and in the Report

88

for 1914, pp.

Sn'(x), Cn'(x),

En'*(x),

— 102, the functions are

= 0,1, 2, ...,17 and x = 1,2,3, ...,10, for n = 0, 1, 2, ..., 10 and x =11,

97—107,

for 1916, pp.

12,..., 19.

Functions of order + J, ± §, have been tabulated by Dinnik, Archiv der Math, und Phys. (3) xvni. (1911), pp. 337 338, to four places of decimals; the functions tabulated are T(l + J) J±i (x), T(l ±%)J±$(x) from « = to x = 8*0 with interval 0*2; and Dinnik has also tabulated I±±(x), I±$(x), ibid.

—

(3) xxii. (1914), pp.

226—227 and J±i (x), J±i (x),

ibid. (3)

xxi. (1913),

—326. All these have the range x = x = 8 with 02. The Tables of T(l + $)J±i(x) are extensive than Table III 714 — 729 with the exception of Dinnik's there 324

pp.

tables

to

less

pp.

;

but,

tables,

infra,

exist

no

and f

tables of functions of orders f, J

In connexion with functions of order + cos

I

interval

^rn-

(w3

—

£, Airy's miff)

Table of his integral

dw

Jo

must be mentioned

;

Airy calculated by quadratures and by ascending series from — 5'6 to + 5'6 with interval

the values of this integral for values of

m

m

= — 4tom = 4is given in the Trans. Camb. a seven-figure Table from Phil. Soc. vi. (1838), p. 402, and a five-figure Table from ra = - 56 to m = 56,

0'2

;

ibid.

vin. (1849), p. 599.

Apart from the work of Euler described in § 15 5 the earliest computation of the zeros of J (x) and */, (x) is due to Stokes, Trans. Camb. Phil. Soc. ix. (1856), p. 186 [Math, and Phys. Papers, n. (1883), p. 355]. Stokes gave the values of the first twelve zeros (divided by if) of J (x) and J, (x) to four places of decimals. In the same memoir he gave the first fifty zeros of Airy's integral, and the first ten stationary points of this integral. -

The Ann.

sci.

first

nine zeros of

J (x), J (x), ...,J (x) t

de Vficole norm. sup.

ill.

given to three places of decimals

;

S

were computed by Bourget,

—

87. Bourget's results are (1866), pp. 82 some corrections in his Tables have recently

been made by Airey*. * Phil.

Mag.

(6) xxxii. (1916),

pp.

7—14.


660

[CHAP.

XX

Bourget's Tables have been reprinted so frequently that their authorship has been overlooked by the writers of the articles on Bessel Functions in the Encyclopiidie der Math. Wiss. and the Ewrgclopedie de.i Set. Math.

The first five zeros of J (x) and J2 (x) were given to six places of decimals by Lommel, Zeitschrift fur Math. und. Phys. xv. (1870), p. 167 and Munchener Abhandlungen, XV. (1886), p. 315. x

The

first

J

ten zeros of

were computed to ten places of decimals bv

(x)

Meissel, Berliner Abhandlungen, 1888.

The

first fifty

zeros (and their logarithms) of

J

(x)

were given to ten places

of decimals by Willson and Peirce, Bulletin pp. 153

— 155;

zeros to eight

The

they also

American Math. Soc. Hi. (1897), gave the values of J (x) and log \Ji{x)\ at these x

and seven places of decimals

J

respectively.

and the corresponding values of J (x) were to sixteen places of decimals by Meissel*, Kiel Programm, 1890; Table is reprinted by Gray and Mathews in their Treatise, p. 280. first fifty

zeros of

x

(x)

computed this

Tables of roots of the equation

Jn (x) Yn (kx) — Jn (kx) Yn (x) = have been constructed by Kalahne, Zeitschrift fur Math, und Phys. liv. (1907), 86 the values taken for k are 1*2, 15 and 2 0, while n is given the pp. 55

—

values

;

0, £, 1, f, 2, f.

Dinnik in his Tables of functions of fractional order mentions the values of a few of the zeros of each function, while Airey, Phil. Mag. (6) xli. (1921), 205, has computed the value of the smallest zero of pp. 200 v {x) 'for small fractional values of v by Euler's method.

—

i.

J

Rayleigh, Proc. London Math. Soc. x. (1878), pp. 6 363—364], has calculated that

—

7 [Scientific Papers,

(1899), pp.

has a

maximum when

a?

(1-tfW, (#)//„ (x) = 0*4858.

Airey, Archiv der Math,

the

first

ten zeros of

3xJ

und Phys.

(x)

—

(3) xx. (1913), p. 291, has

2J\ (x) and of

2xJ

(x)

—J

x

computed

(x) to four places

of decimals.

In his memoirs on Diffraction, Munchener Abhandlungen, xv. (1886), has published tables connected with his functions of two variables,

Lommel

but these tables are so numerous that a detailed account of them will not be given here. His Table of Fresnel's integrals (p. 648) to six places of decimals from x = to x — 50'0 with interval 0*5 (with auxiliary tables for purposes of interpolation) must, however, be mentioned, and with it his Table of the first sixteen maxima and minima of these integrals. * Jahrbitch iiber die Fortschritte der

Math. 1890,

p. 521.

In consequence of the inaccessibility

of Meissel's table, the zeros of J\ (x) were recomputed (to ten places of decimals) for insertion in

Table VII,

p. 748.


20*2]

Lommel's form

for Fresnel's integrals

661

was

J±i(t)dt; J o

a different form was tabulated earlier by Lindstedt, Ann. der Physik und Ch&mie, (3) xvn. (1882), p. 725.

M (x) and N (x) by the equations cos dt = M {x) cos x — JV (x) sin x

Defining the functions

2

%

t

/

2

,

J X

sin

and writing x =

{(y

of decimals from

The

y

Pdt =

M (x) sin x* + N (x) cos x

+ £) tt}^ Lindstedt tabulated M(x) = to y = 9 with interval O'l.

2 ,

and

N (x) to six places

-

function I(x) defined as 00

if Hi 2(2«) rf* t

has been tabulated to four places of decimals by Struve, Ann. der Physik und Chemie, (3) xvn. (1882), pp. 1008—1016, from x = to 4'0 (interval 01), from

x

= 40

A

to

70

(interval

02) and from x =

7*0 to 11-0 (interval 0'4).

table of values of the integral

J

#

in which the limits are consecutive zeros (up to the forty-ninth) of

J

x

(x),

has

been published by Steiner, Math, und Naturwiss. Berichte aus Ungarn xi. (1894), pp. 366 367 this integral occurs in the problem of Diffraction by

—

;

a Circular Aperture:

No

Tables of Struve's functions- seem to have been constructed before the

Table of

20*2.

to

Ho (x) and Hi (x) which

is

given on pp. 666

— 697.

Description of the Tables contained in this book.

Preliminary considerations on the magnitude and character of the tables be included in this book led to the following decisions (I)

That space did not usually admit of the inclusion of more than seven

places of decimals in the tables.

of

(II) That the tables should be so constructed as to minimise the difficulty making interpolations. In particular, it was decided that a table with a

moderately large interval (such as 0*02), together with an auxiliary table to facilitate interpolation, would be more useful than a table with a smaller interval (such as 0*01), occupying the auxiliary, in

same space

which interpolation was impracticable.

as the first table

and

its


662

That

(III)

in

computing

tables, calculations

places of decimals in order to ensure that the

[CHAP.

XX

should be carried to ten

number

of cases of inaccuracy

in the last figure of the published results should be trivial*.

This does not

apply to the auxiliary tables of angles which are entered in Tables I and III.

In order to obtain seven-figure accuracy,

it is

not sufficient to tabulate to

tenths of a second of arc, because the differences per minute of arc in a sevenfigure table of natural sines

may be

as large as

00002909 on the other hand, ;

an error of a hundredth of a second does not affect the value of the sine by more than 0*00000005. Hence, for seven-figure accuracy, it was considered adequate to compute to nine places of decimals the sines (or cosines) of the angles tabulated and then to compute the angles from Gifford's Natural Sines (Manchester, 1914)

;

these are eight-figure tables with an intervalf of 1".

The angles tabulated may consequently frequently be last

digit,

in error as to the

but, in all probability, the error never exceeds a unit

hundredth of a second of

"We now proceed

(i.e.

a

arc).

to describe the tables in detail.

J

Y

J

and F, (x) from and Jîx) up to 1550 are taken from Meissel's Table J, while the values of Y (x) and Y {x) were computed partly by interpolation in Aldis' Table of G (x) and Gx (x) and partly from the asymptotic expansions of J 2 (x) + F 2 (x) and J^ (x) + Y? (x) Table I consists primarily of Tables of

x=

1600 with

to

interval of 0*02.

The

(x),

(x),

values of

J

1

(x)

{x)

x

given in

§ 7*51.

H

{1) auxiliary tables § give the values of \H n w(x)\ and arg n (x) for and n=l. In these tables the first differences are sufficiently steady

The n

=

(except for quite small values of x) to enable interpolations to be effected

with but

trouble on the part of the reader ; thus,

little

the second differences of

Hn

|

H

W

when x

is

about 10

(x) do not exceed 0*0000009. \

and arg H„ (x) can consequently be computed by the reader for any value of x less than 16, with the exception of quite small values. The corresponding values of Jn (x) and Yn (x) can then be calculated immediately by the use of seven-figure logarithm tables.

The values

j|

of

|

ll)

(x)

{1)

j

and it was found that the by more than a unit in the tenth place so it is hoped that the number of errors in the last figure retained does not exceed about one in every thousand entries. tNo tables with a smaller interval have been published; the use of any tables with a larger interval and a greater number of decimal places would have very greatly increased the labour of constructing the auxiliary tables of angles, and the increased accuracy so obtained would be of no advantage to anyone using the auxiliary tables for purposes of interpolation. X I must here express my cordial thanks to the Preussische Akademie der Wissenschaften zu *

The

tables were di. T erenced before removing the last three figures,

ten-figure results were rarely in error

Berlin for permitting §

The

me

to

make use

;

of this Table.

idea of constructing the auxiliary tables grew out of a conversation with Professor Love,

which he remarked that it was frequently not realised how closely Bessel functions any given order resemble circular functions multiplied by a damping factor in which the rate

in the course of

of

of decay is slow. I|

The remarks immediately following

of course presuppose that n

is

or

1.


20*2]

The most

663

between the various functions tabulated may be expressed w (x) and arg n (a?) as the polar coordinates by regarding n

relation

briefly

|

of a point in a plane

and 7„ (x).

;

H

H

|

{1)

Jn (x)

then the Cartesian coordinates of this point are

Thus, from the entry for x

= 800,

+ 01716508 = 0-2818259 cos 412° 28' 40"60, + 02235215 = 02818259 sin 412° 28' 40"60.

H

H

Table I also contains the values of Struve's functions (x) and These functions are included in Table I, instead of being contained separate

Table,

Hn (x) — Yn (x)

to facilitate is

interpolation;

The Tables

§ 10'41 (4),

a positive monotonic function, and

steadily for interpolation to be easy

and Ho'(ar)

by

when x

is

it

x

(x).

in a

the difference

varies sufficiently

not small.

H

computed by calculating the values of (x) when x=\, 2, 3,..., and then calculating these values o( x from the differential equation § 10-4(10) and the

of Struve's functions were

directly from the ascending series

Ho"(#), Ho"'

(#),... for

equations obtained by differentiating

it.

A few differential coefficients are adequate to calculate Ho (x) and Ho' (x) by Taylor's theorem for the values 0-5, 0'6, 07, ... of x. Interpolation to fiftieths of the unit is then effected by using Taylor's theorem in the same manner. This process, though it seems at first sight to be complicated and lengthy, is, in reality, an extremely rapid one when a machine* is used. It is very much more effective than the use of asymptotic expansions or the process suggested in the British Association Report, 1913, p. 116. As an example of the rapidity of the process, it may be stated that thewalues ofe~ x I (x) and e~ x Ix (x) in Table II took less than a fortnight to compute of course the time taken over this tabulation was appreciably shortened by the use of Aldis' Table as a framework for interpolation. ;

K

Table II consists of Tables of e-* I (x), e~x Ij (x), e? (x), and e x A", (.r), and a Table of e* is inserted, in case the reader should require the values of the functions'^ I (x), Ix (x), (x) and (x); the functions are tabulated

K

K

from

to

1600 with

x

interval 002.

Interpolation by differencing

is

easy in the case of the

first

four functions

throughout the greater part of the range.

The Table of e* was constructed with the help of Newman's Table of e~ x Trans. Camb. Phil. Soc. XIII. (1883), pp. 145—241. Unlike the other Tables in this book, the Table of e* is given to eight significant figures J, and care ,

has been taken that the lation in this Table

is,

last digit

given

is

accurate in every entry.

Interpo-

of course, effected by multiplying or dividing entries

by exponentials of numbers not exceeding O'Ol

;

such exponentials can be

calculated without difficulty. * The machine on which the calculations were carried out is a Marchant Calculating Machine, 10 x 9 recording to 18 figures. t These functions were not tabulated because tables of them are uusuited for interpolation.

X Nine figures are given in parts of the Table to avoid spoiling

its

appearance.


664

[CHAP.

XX

Newman's Table gives e~ x to a large number of places of decimals, but the actual number of significant figures in the latter, part of the Table is small and less than half of the Table of e* was constructed by the process of calculating reciprocals the rest was con;

;

structed from

Newman's Table by using the values

the value of e -15,6 given by

Newman

of e13

and

in a short table of e~ x with interval 0-1.

ponentials were employed because the tenth significant figures in

and third are

significant figures in the first

Table III consists of Tables of of the

same scope as Table

A

already explained.

I,

e*

all

These ex-

three and the eleventh

zero.

J

7

(x),

h

4

(*),

|

H^ (x)

and arg

|

\

interpolations are effected in the

and

Table of

given by Glaisher* and

e 16

K$ (x)

is also

H^ (x)

\

manner

These Tables are

included.

of importance in dealing with approximations to Bessel functions of large

order

and

(§ 8'43),

also in the theory of Airy's integral.

The reader can

compute values of J_ 4 (x) from

easily

this table

by means

of the formula

J_j (x)

Table

IV

=

|

H^ (x)

|

cos {60°

+ arg H^ (x)}.

Jn (x), Yn (x), e~x In (x) and Kn (x) for various of Jn (pc) are taken from Lommel's Table t, with

gives the values of

values of x and

n.

The

values

but the remainder of Table IV, with the exception of some values of n (x) taken from Isherwood's Table f, is new they have been constructed in part by means of Aldis' Tables of functions of orders zero and unity.

some

corrections,

K

V

;

Lommel's Tablet of J ±(w+i )(a;) and Fresnel's integrals with some modifications and corrections. Table VI gives the values of Jn (n), Yn (n), Jn '(n), Yn (n) and n*Jn (n), Table

is

'

n Y„ (n), n*Jn (n), n» n ' (n) for n = 1, of the last four of the eight functions '

!>

Y

Table VII gives the

first

of n; part of this Table

is

2, 3,

. . .

50.

'

Interpolation in the tables

is easy.

forty zeros of

Jn (x) and Yn (x)

for various values

taken from the Tables of Willson and Peircef.

Forty zeros of various cylinder functions of order one-third are also given. Table VIII gives the values of X

X

li

J

(t)dt,

I J

Y

(t)dt,

= to 50 with interval 1, together with the first sixteen maxima and minima of the integrals. The former table of maxima and minima can be used to compute the coefficients (cf. § 18'12) in certain Fourier-Bessel series from x

for

which v * Trans. t I

= 0.

Camb. Phil. Soc. xni.

must here express

my

(1883), p. 245. cordial thanks to the Bayerische

Akademie der Wissenschaften zu

Miinchen, to the Manchester Literary and Philosophical Society, and to the American Mathematical Society for permitting me to make use of these Tables. The non-existence of adequate trigonometrical tables of angles in radian measure has made it impracticable to check the last digital in

the entries in the greater part of Table V.

TABLES OF

BESSEL FUNCTIONS

1


666

Table

Y

/.(*)

I.

Functions of order zero

(x)

!«?(*)!

+ i-ooooooo

HoW

arg#„(*) 90

0-02 0-04 O'OO o-o8 O-IO

+ + + + +

o o o o o

9999000 9996000 9991002 9984006 9975016

-

2-5639554 2-1219006 1-8626264 1-6780254 I-5342387

2-7520297 2-3455622 2-1136647 •952581 •8299993

0-12 o«I4 0-16 0-18 0-20

+ + + + +

o o o o o

9964032 9951060 9936102 9919164 9900250

-

1-4161969 1-3158701 1-2284710 1-1509166 1-0811053

0'22 0-24

o o o o o

9879366 9856518

030

+ + + + +

9804958 9776262

-

0-32 o-34 0-36 0-38 0-40

+ + + + +

o o o o o

9745634 9713081 9678615 9642245 9603982

0-42 °-44 0-46 0-48 0-50

+ + + + +

o o o o o

0-52 °*54 0-56 0-58 o-6o

+ + + + +

0-62 0-64 o-66 o-68 0-70 0-72 0-74 0-76

68° 64° 61° 5Q ° 56°

o-ooooooo 41 42*16 4 6 3J*73 47 28*29 52*70 58 10*79

+ + + + +

0-0127318 0-0254603 0-0381819 0-0508934 0-0635913

0*02 o«04 0-06 0-08 O-IO

•7315984 •6497727 •5800007 •5193772 •4659257

54° 52 I 4 '<42 5 2 54 5I ° 2 0*32 49° 14 37'o5 3i 4^89 47

+ + + + +

0-0762722 0-0889328 0-1015697 0-1141796 0-1267590

0*12 0-14 O-IO

1-0175412 0-9591221 0-9050133 0-8545676 0-8072736

•4182414 •3752907 •3362914 •3006375 •2678500

45° 44 42° 41° 39

6*41 37 46*91 4 27"57 32 53*32

+ + + + +

0-1393046 0-1518131 0-1642813 0-1707056 0-1890829

0-22 0'2A 0-20 0-28 0-30

-

0-7627204 0-7205732 0-6805558 0-6424376 0-6060246

•2375444 •2094070 •1831788 •1586436 •1356190

38° 2 51*54 36° 34 12 "42 35° 6 47*25 33° 40 28*14 32° 15 8*97

+ + + + +

0-2014099 0-2136834 0-2380565 0-2501497

038

9563838 9521825 9477955 9432242 9384698

-

0-5711520 o-5376789 0-5054836 0-4744608 -4445i87

•1

30° 50 29 27 28 4 26 42 25 20

19*71 11*96 42*72

+ + + + +

0-2621765 0-2741336 O-2OOOI0O 0-2978265

0-42 o-44 0*46

o o o o o

9335339 9284179 9231233 9176518 9120049

-

0-4155768 0-3875642 0-3604182 0-3340833 0-3085099

•0218560 -0060645 0-9909884 0-9765738 0-9627727

23° 59 49*07 22 39 28*42 21° 19 36*41 20° O 16*94 i8° 4 i 22*10

+ + + + +

0-32I2033 0-3327655 0-3442396 0-3556226 0-36691 14

+ + + + +

o o o o o

9061843 9001918 8940292 8876982 8812009

- 0-2836537 - 0-2594751 - 0-2359383 - 0-2130113 - 0-1906649

0-94954I7 0-9368418 0-9246378 0-9128976 0-9015920

17° 22 52*16

+ + + + +

0-3781032 0-3891950 0-4001841 0-4110675 0-4218424

o-8o

+ + + + +

p o o o o

8745391 8677147 8607300 8535868 8462874

-

0-1688729 0-1476114 0-1268587 0-1065950 0-0868023

0-8906945 0-8801807 0-8700283 0-8602168 0-8507273

10° 55 45*35 39 16*05 to 8° 23 3*05 7° 7 5*4» 5° 5i 22*51

0-82 0-84 o-86 o-88 0-90

+ + + + +

o o o o o

8388338 8312284 8234734 8155711 8075238

-

0-0674640 0-0485651

0-0120311 + 0-0056283

0-8415424 0-8326455 0-8240231 0-8156598 0-8075434

0-92 0-94 0-96 0-98 I-OO

+ + + + +

o o o o o

7993339 7910039 7825361 7739332 7651977

+ + + + +

0-0228974 0-0397860 0-0563032 0-0724576 0-0882570

0*7996618 0-7920038 0-7845590 0-7773I77 0-7702706

026 0-28

078

0-03009*17

139500

•0935036 •0741648 •0558337 •0384231

H

rh

50 44-63 13

44*27 9*25

16

4 45*55 47 0*82 13° 29 36^67 12° 12 31*88

K

02258999

O3O95559

+ 0-4325061 + 0-4430558

018 0-20

O32 o-34 0*36 0-40

048 050 0*52 o-54 0-56

058 060 0-62

064 o-66 o-68 0-70 0-72 oo-7

+ 0-4534888 + 0-4638026 + 0-4739944

7% o>7

4° 35 3° 20 53*4J 37*48 2 ° 5 34*o6 o° 50 42*55 o°2 3 57*6i

+ + + + +

0-4840616 0-4940018 0-5038124 0-5134909 0-5230350

0-82

i° 3 8 26*96 2° 52 45*99

+ + + + +

0-5324422 0-5417103 0-5508368 0-5598197 0-5686566

0-92

55*i6 54*92 6° 34 45*67

6 < 5° 20

084 o-86 o-88 0-90

094 096 098 I-OO

TABLES OP BESSEL FUNCTIONS Table

AM

X

-

IH^WI

00

0-02 0-04 O-OO 0-08 O-IO

+ + + + +

00099995

00499375

- 31-8598128 - 15-9643089 - 10-6757892 - 8-0376696 - 6-4589511

0-12 0-14 o-io T8 0-20

+ + + + +

0-0598921 0-0698286 0-0797443 0-0896360 0-0995008

-

5-4094402 4-6619853 4-1030547 3-6696037

0-22 0-24 O-20 0-28

+ + + + +

01093358 0-1191381 0-1289046 0-1386325 0-1483188

+ + + + +

0-0199960 0*0299865 0-0399680

Functions of order unity

Y^x)

O-OOOOOOO

o

I.

667

00

31-8598144

159643214 10-6758314 8-0377690 6-4591441

arg rfi(*)

Hit*)

-90 -

X

o-ooooooo

§9° 58' 55-26 89° 55 4I-64 89 50' 20*64 89° 42; 54*34 89 33 25*29

+ + + + +

0-0000849 0-0003395 0-0007638 0001 3575 0-0021207

0-02 0-04 O-OO 0-08 O-IO

+ 0-0030528 00041539 + 0-0054232 + 0-0060607 + 0-0084657

0-12 0-14 0-16 0-18 0-20 0-22 0-24 0-20 0-28 0-30

3323825b

5-4097717 4-6625082 4-1038295 3-6706983 3-3253I40

- 89 21' 56*38 -89 8' 30*73 -88° 53' 11*68 -88° 36' 2*65 - 88° 17' 7*17

-

3-0416730 2-8071277 2-6091059 2-4396971 2-2931051

3-0436375 2-8096547 2-6122883 2-4436328 2-2978968

- 87° 56' 28*81 - 87° 34' 11*11 - 87° 10' 17*63 -86° 44' 51*87 - 86° 17' 57*28

+ + + + +

0-0102377 0-0121762 0-0142806 0-0165502 0-0189843

0-1579607 0-1675553 0-1770997 0-1865911 0-1960266

-

2-1649866 2-0520233 1-9516372 1-8617949 1-7808720

2-1707415 2-0588527 i-959656i 1-8711218 1-7916282

" -

43'

16 36*72 6*78

+ + + + +

0-0215820 0-0243427 0-0272652 0-0303489 0-0335925

0-46 0-48 0-50

+ + + + +

0-2054034 0-2147188 0-2239699

-

I-7075549 1-6407704 I-579633I

1-7198647 1-6547603 1-5954320 1-5411449

- 83° 8' 26*86 - 82° 32' 39*76 - |i° 55 48*18 X " 54;69 -8o° 3 9' 1*79

+ + + + +

0-0369952 0-0405559 0-0442733 0-0481463 0-0521737

0-42 0-44 0-46 0-48 0-50

0-52 o-54 0-56 0-58 o-6o

+ + + + +

0-2513105 0-2602774 0-2691665 0-2779752 0-2867010

-79°

+ + + + +

0-0563542 0-0606865 0-0651691 0-0698006 0-0745797

0-52 o-54 0-56

0-62 0-64 o-66 o-68 0*70

+ + + + +

0295341

0-0795046 0-0845739 0-0897860 0-0951392 0-1006317

0-62 0-64 o-66 o-68 0-70

0-72 o-74 0-76

0-72 0-74 0-76 0-78 o-8o

030 0-32 o-34

036 038 0-40 0-42

044

078 080 0-82"

0-84

086 o-88 0-90 0-92

094 0-96 0-98 i-oo

02331540 0-2422685

x

5234063

§5! 85 84° 84°

-83°

K

49' 37-22 19' 55'oo

48' 53*80

K

+

I-47I4724

1

I-4453258 1 -4028308

-

1-4233094 1 '3784737 1-3365858 1-2973191 1-2603913

0-3038932 0-3123547 0-3207230 0-3289957

-

1-2255572 1-1926026 1-1613400 1-1316043 1-1032499

1-2606415 1-2307120 I-2020I22 1-1761767

II5I2595

-

23' 42*38

+ + + + +

+ + + + +

0-337I705 0345244 8 0-3532164 0-3610829 0-3688420

-

1-0761476 1-0501828 1-0252532 1-0012677 0-9781442

I-I2773I2 1-1054763 1-0843920 1-0643861 I-0453758

- 72° 36' 12*87 - 71° 48' 6*68 - 70° 59' 25*22 - 70° io' 9*80 - 69° 20' 21*73

+ + + + +

0-1062619 0-1120279 0-1179279 0-1239601 0-1301225

+ + + + +

0-3764916 0-3840292

-

03914529

-

09558093 09341970 0-9132475 0-8929069 0-8731266

1-0272864 I-OI00507 0-9936077 0-9779022 0-9628837

- 68° 30' 2*20 - 67° 39' 12*39 - 66° 47 53*42 -65° 56 6*35 - 65° 3' 52*21

+ + + + +

0-1364131 0-1428299 0-1493716 0-1560343 0-1628175

+ + + +

0-4130184 0-4199649 0-4267871

-

08538622

0-9485066

+ 0-1697186

09347290

-

ii' 11*97

0-8350735 0-8167241 0-7987806 0-7812128

1 8'

+ + + +

0-3987603 0-4059495

04334829 + 0-4400506

-

-

1

4912830

3634193

1-3267657 1 -2925880

0-9215126 0-9088223 0-8966259

-

79

59' 11*85 18 27*16

78° 36 49*90 77° 54 22*18 77° ii' 5*99

- 76° 75° 74° 74° 73

64° 63 62° 6l° 6o°

27' 3*26 42' 15*81 56' 45*4i

10 33*73

6*57 24' 36*89 30' 43*79 36' 28*10

0-1767354 0-1838656 0-1911070 0-1984573

0-32 o-34

036 038 0-40

058 o-6o

0-82

084 o-86 o-88 0-90 0-92 o-94 0-96

098 I-OO

TABLES OP BESSEL FUNCTIONS

668

Table

I.

Y9 i*)

*

7oW

1-02

+ 07563321 + 07473390 + 0-7382212 + 07289813 + 07196220

+ + » + +

0-1480406 0-1621632

+ + + + +

07005564

0-1037085 0-1188188


|ffJW

H

arg H^(*)

X

(*)

074386I4 07376671

7 48' 27*80 9° 2' 1*68 io° 15' 27*64 11° 20' 46*01 12° 41' 57*10

+ + + + +

0-5773455 0-5858842 0-5942706 0-6025028 0-6105787

+ 0-1759670 + 0-1894567

0-7316228 0-7257225

0-6908557 0-6810469 0-6711327

+ 0-2026367 + 0-2155111 + 0-2200835

O7I99606

13° 55' J*20 15° 7 58-58

I7,° 33' 34*20 1 8° 46' 12*92

+ + + + +

0-6184965 0-6262544 0-6338504 0-6412828 0-6485500

+ + + + +

0-66^1163 0-6510004 0-6407880 0-6304822 0-6200860

+ + + + +

0-2865354

19° 21° 22° 23° 24

+ + + + +

0-6556502 0-6625819 0-6693434 0-6759334 0-6823503

1-22

0-6981944 0-6930499 0-6880157 0-6830879

0-6096023 0-5990343 0-5883850 0-5776576

+ + + + +

0-2973645 0-3079127 0-3181824 0-3281758 0-3378951

0-6782630 0-6735372 0-6689073 0-6643701 0-6599226

26° 27° 28° 29° 30°

132

24' 11*88 36' 5*57 47' 55*o8

+ + + + +

0-6885928 0-6946595

07005492 07062606

136

40

+ + + + +

1-42 1-44 1.46 1 -48 1-50

+ + + + +

31° 59' 40*54 33 11' 22 "08 34° 22' 59*82 35° 34' 33*9o 36° 46' 4*43

:a i-o8 I-IO

112 i-i4 i-i6 i-i8 1-20 1-22 1-24 1-26 1-28

130 1-32

m

1-38 I

1

52

*% 1-58 i-6o 1-62

!ts 1-68

170 1-72 1-74

176 178 i-8o

0-7101461

0566855

OI335943

0-2403577 0-2523369 0-2640243 O-27'54220

0-7634092

07507255 0-7502I20

o7 I 433i7 07088309 07034533

16

20 49*49

58' 45-88 II' 13-30 23' 35^37

35 52-28 48 4 ? 22 0' 11*36 12' 13*86

I'02

53 1

08

r-io I-I2

114 116 118 1-20

XXI 1-28 1

30

i-34 1-38

0-7117925

140

+ + + + +

0-7171439 0-7223136

1-42 1-44 1-46 1

-48

0-7367235

1

50

+ + + + +

0-7411573 0-7454051 0-7494662 0-7533398 0-7570255

1-52 i-54 1-56 1-58 i-6o

07605226 07638306 07698781 07726168

1-68

1-72

0-5559807 0-5450376 0-5340289 0-5229579 0-5118277

+ + + + +

0-3473424 0-3565195 0-3654285 0-3740710

O3824489

0-6555618 0-6512850 0-6470895 0-6429728 0-6389325

+ 0-5006415 + 0-4894026

0-3905639 0-3984176 0-4060116 0-4I33476 0-4204269

0-6349662 0-6310717 0-6272469 0-6234898 0-6197983

37° 39° 40° 4i° 42°

57' 31*50

+ 0-4667797 + 0-4554022

+ + + + +

+ + + + +

+ + + + +

0-4272512 0-4338219 0-4401404 0-4462083 0-4520270

0-6161706 6-6126049 0-6090994 0-6056526 0-6022627

43° 45° 46° 42° 48°

53' 5**77

38' 16*30

+ + + + +

+ o-4575979 + 0-4629223 + 0-4680019

+ 0-4728378 + o-47743i7

0-5989282 o-5956477 0-5924198 0-5892429 05861 159

49° 49' 14*10 51° 0' 9*41 52° il' 2*31 53° 21' 52*85 54 32' 4i*n

+ + + + +

07751652 07775230 07796902 07816666 0-7850474 0-7864518

1-82

07876658 07886897 07895236

lit

26 10*14

+ + + + +

45*88 I9 ? 73 5I-7I 21*89 50*32

+ + + + +

0-7901680 0-7906233 0-7908898

192

-f-

+ + + + +

0-4781143

0-4439850

04325313 0-4210446 0-4095280 0-3979849 0-3864185 0-3748321 0-3632292 0-3516128 0-3399864

55*24 20' 15*74 3i' 33-io 42 47*42 8'

5, 7*26 16' 12*95

27/,

J 5*94

1-82 1-84 1-86 1-88 1 -90

4 0-3283532

+ + + +

0-3167166 0-3050797 0-2934460 0-2818186

+ + + + +

0-4817849 0-4858989 0-4897751 0-4934149 0-4968200

0-5830374 0-5800061 0-5770210 0-5740809 0-5711845

55° 56° 58° 59° 6o°

43' 27*14 54' 11*00

1-92 1-94 1-96 1-98 2-00

+ + + + +

0-2702008 0-2585959 0-2470071 0-2354376 0-2238908

+ + + + +

0-4999917 0-5029315 0-5056411

05683310

6i° 62° 6 3° 65° 66°

36' 47' 57' 8' 18'

0-'j08l220

0-5103757

0-5655192 0-5627481 0-5600168 o-5573243

4' 52*75 15' 32*45

07273008 07321043

0-7669493

0-7834523

07909681 07908588

162

sa 170

m it i-88 1

90

i-94

196 198 2-00

8

TABLES OF BESSEL FUNCTIONS Table

Y

ZiW

X 1-02 I-04 1-06 1-08 I-IO I-I2

x

I.


{*)

K'wi

arg

H

IT* 1 '/ ,

H

\

(x)

x

(*)

X

+ + + + +

0-2059142 0-2134753 0-2211382 0-2289005 0-2367597

1-02 1-04 I -oo 1-08 I-IO

+ + + + +

0-2447133 0-2527589 0-2608939 0-2691157 0-2774218

I-I2 1-14

50 18' 6*48 49 20' 12*33 48 22' 3*58 47° 23' 40*70 -46° 25' 4*12

+ 0-2858095 + 0-2942761 + 0-3114357 + 0-3201231

1-22 1-24 1-26 1-28 i-3°

0-3288788 0-3376999 0-3465837 0-3555273 0-3645280

1-32 i-34 1-36 1-38 1-40

0-4464882 0-4527939 0-4589660 0-4650027 0-4709024

-

0-7639930 0-7470959 0-7304984 0-7141794 0-6981196

0-8848938 0-8735987 0-8627154 0-8522205 0-8420926

-

59° 58 57° 56° 55

4 1 ' 50*60 46' 52*03 5i' 33'i2 55 54-55

+ 0-4766634 + 0-4822840

-

0-6823011 0-6667078 0-6513248 0-6361385 0-6211364

0-8323117 0-8228591 0-8137178 0-8048715 0-7963055

- 55°

3' 41 '07

-

0-7880056

-

+ + + + +

669

59 56*99

-54°

7-40 7 - 53 10' 16*56 - 52° 13' 9*IQ - 5i° 15' 45*57

116

I'l6 1-18 1-20

+ 0-4877629 + 0-4930984 + 0-4982891

1-22 1-24 1-26 1-28

+ 0-5033336 + 0-5082305

I

30

+ 0-5220232

0-6063070 0-5916398 0-5771253 0-5627546 - 0-5485197

I

32

+ + + + +

0-5263174 0-5304580 o-5344439 0-5382741 0-5419477

- o-5344i33 - 0-5204287 - 0-5065597 - 0-4928008 - 0-4791470

0-7500717 0-7431229 0-7363647 0-7297888 0-7233873

_ -

-41

28 29*03 28' 49*98

+ + + + +

+ + + + +

0-5454638 0-5488215 0-5520200 0-5550586 o-5579365

- 0-4655936 - 0-4521367 - 0-4387723 - 0-4254973 - 0-4123086

0-7171528 0-7110785 0-7051576 0-6993840 0-69375 1

" -

28' 28' 28' 28' 27'

59*54 58*03 45*78 23*11 50*30

+ + + + +

0-3735830 0-3826894 0-3918443 0-4010450 0-4102085

1-42 1-44 1-46

03992036

0-6882554 0-6828896 0-6776493 0-6725298 0-6675267

-35°

27' 7'65 34° 26' 15*44 33° 25, 13*93 32 24' 3*39 31 ° 22' 44*05

+ + + + +

0-4195719 0-4288924 0-4382471 0-4476330 0-4570472

1-52

0-6626355 0-6578524 0-6531734 0-6485948 0-6441131

- 3O 2l' 16*17 - 29 19' 39*97 - 28 17' 55*68 - 27 16' 3*52 - 26 14' 3*70

+ + + +

0-4664869 0-4759490 0-4854306 0-4949288 + 0-5044407

1-62 1-64 1-66 1-68 1-70

- 25

56*41 41 '85 20*21 51*68 16*43

+ + + + +

05139633 0-5234937 0-5330289 0-5425661 0-5521021

1-72 1-74 1-76 1-78 i-8o

" 19° 59' 34*63 - 17° 53' 52*04 - 16° 50 51-57 - 15 47' 45*18

+ + + + +

0-5616342 0-5711594 0-5806748 0-5901775 0-5996645

1-82 1-84 1-86 1-88 1-90

- M° 44' 33*oi - 13° 4i' 15*22 - 12° 37' 51*93 - n° 34' 23*27 - io° 30' 49*38

+ + + + +

0-6091329 0-6185800 0-6280027 0-6373982 0-6467637

1*92 1-94 1-96 1-98

1-34 1-36 1-38 1-40 1

42

1-44 I -46 1-48

150

+ 0-5129786 + 0-5175766

07799589 0-7721533 0-7645772 0-7572200

+ + + + +

0-5606532

1-54 1-56 1-58 i-6o

0-5656003 0-5678298 0-5698959

-

1-62 1-64 1-66 1-68 1*70

+ + + + +

0-5717984 o-5735368 0-5751108 0-5765204 0-5777652

-

03348619

1-72 1-74 1-76 1-78 i-8o

+ + + + +

0-5788453 0-5797604 0-5805107 0-5810962 0-5815170

-

0-2723716 0-2600881 0-2478757 0-2357345 0-2236649

0-6397250 0-6354274 0-6312171 0-6270914

1-82 1-84 i-86 i-88 1-90

+ + + + +

0-5817731 0-5818649 0-5817926 0-5815566 0-5811571

-

0-2116671 0-1997416 0-1878891 0-1761102 0-1644058

0-6190823 0-6151939 0-6113796 0-6076371 0-6039642

1-92

+ + + + +

0-5805946 0-5798695 0-5789825 0-577934J 0-5767248

-

0-1527766 0-1412236 0-1297478 01 183504 0-1070324

0-6003588 0-5968189 0-5933424 0-5899276 0-5865726

152

194 1 -96

1

98

2-00

05632079

0-3861800 0-3732356 0-3603088 o-347578o 0-3222194 0-3096498 0-2971522 0-2847262

06230473

-

-

45° 44° 43° 42°

4°° 39 38° 37 36°

26' 14*26

27

27' 56*34

ii' 9' - 23° 7 - 22 4' - 21° l'

-24

n*54

- 18 56' 46*45

4-

0-3028191

1-18 I-20

1-48

150

1-56 1-58 i-6o

2-00


670

Table

X

;.w

2-02 2-04 2-06 2-08 2-IO

+ 0-2123697 + 0-2008776 + 0-1894177

212

I.

Y9 (x)

Functions of order zero (

arg

|tf ?(*)l (

{1

+ 0-1779931 + 0-1666070

+ + + + +

0-5124038 0-5142080 0-5157900 0-5171513 0-5182937

0-5546698 0-5520523 0-54947 10 0-5469250 o-5444i37

67 29' 17*02 68° 39' 42*05 69° 50 5*44 71° 27*23 72° io' 47*47

2-16 2-l8 2-20

+ + + + +

0-1552625 0-1439626 0-1327106 0-1215095 0-1103623

+ + + + +

0-5192190 0-5199289 0-5204252 0-5207097 0-5207843

0-5419362 o-53949i7 0-5370796 o-534699i o-5323496

73° 74° 75° 76° 78°

2-22 2-24 2-26 2-28 2-30

+ + + + +

0-0992720 0-0882416 0-0772742 0-0663726

+ 0-5206508 + 0-5203112 + 0-5197675 + 0-5190215

00555398

+ 0-5180754

0-5300304 0-5277408 0-5254803 0-5232482 0-5210439

79° 8o° 81° 82° 83

2.32 2-34 2-36 2-38 2-40

+ + + + +

0-0447786 0-0340921 0-0234828 0-0129538 0-0025077

+ + + + +

0-5169311 0-5155908 0-5140565 0-5123304 0-5104147

0-5188670 0-5167167 0-5145926 0-5124942 0-5104209

§5° 86° 87° 88° 89°

2-42 2-44 2-46 2-48

-

0-0078527 0-0181247 0-0283057

0-5083116 0-5060233 0-5035522 0-5009004 0-4980704

0-5083723 0-5063478 0-504347 1 0-5023696 0-5004149

9 °°53'

0-0483838

+ + + + +

2-54 2-56 2-58 2-60

-

0-0582758 0-0680664 0-0777531 0-0873*34 0-0968050

+ + + + +

0-4950645 0-4918851 0-4885347 0-4850157 0-4813306

0-4984826 0-4965722 0-4946834 0-4928157 0-4909608

96° 97° 99° IOO° IOI°

42' 48*94 52' 42*52

2-62 2-64 2-66 2-68 2-70

-

0-1061654 0-1154123 0-1245434 0-1335565 0-1424494

+ + + + +

0-4774820 o-4734724

0-4891422 o-4873357 0-4855488 0-4837812 0-4820325

102° 103° 104° io6° 107°

2-72 2-74 2*76

0-1512198 0-1598658 0-1683852 0-1767759

0-4558761 0-4511009 0-4461806 0-4411181

01850360

+ + + + +

04803025

278 280

-

2-82 2-84 2-86 2-88 2-90

- 0-1931636 - 0-2011568 - 0-2090137 - 02167325 - 022431 15

2 -92 2-94 2*96 2-98 3.00

-

250 252

00383929

0-2317491 0-2390434 0-2461931

02531964

- 0-2600520

HW

H J{x) + + + + +

X

0-7905626 0-7900800 0-7894119 0-7885590 0-7875222

2-02 2-04 2-06 2-08 2-IO

+ 0-7863025 + 0-7849006 + 0-7833178 + 0-7815550 + 0-7796135

2-12 2-14 2-16 2-l8 2-20

+ + + + +

0-7751986 0-7727279 0-7700834 0-7672665

2-22 2-2A 2-20 2-28 2-30

5*90 6*62

+ + + + +

0-7642787 0-7611214 0-7577962 0-7543047 0-7506485

2-32 2-34 2-36 2-38 2-40

6*25 92° 3 4-83 93° 13' 2*36 94° 22' 58-88 95° 32' 54*40

+ + + + +

0-7468293 0-7428488 0-7387088 0-7344112 0-7299577

2-42 2'44 2-46 2-48

+ + + + +

07253504

252

0-7205912 0-7156821 0-7106251

2-54

07054223

2-58 2*60

32'

7*69 4 I' 56*77 51' 45*OI 1 32*41 11' 19*00

+ + + + +

0-7000759 0-6945880 0-6889609 0-6831967 0-6772977

2-62 2-64 2-66 2-68 2-70

108° 2i' 4*79 109° 30' 49*80 no° 40' 34*04 111° 50; 17*53 0' 0*27 1 13°

+ + + +

0-6712664 0-6651050 0-6508160 0-6524017 + 0-6458646

2-72 2-74 2-76

04359160

0-4785907 0-4768970 0-4752209 o-473562i

+ + + + +

0-4305772 0-4251045 0-4195008 0-4137689 0-4079118

0-4719204 0-4702955 0-4686871 0-4670950 0-4655187

114° 115° Il6° 117° 118°

+ + + + +

0-6392073 0-6324323 0-6235420 0-6185392 0-6114264

+ + + + +

04019323 03832893

0-4639582 0-4624131 0-4608831 o-459368t

0-3768500

04578678

19° 121° 122° 1 23° 124°

+ + + + +

0-6042062 0-5908814 0-5894546 0-5819206 0-5743061

04693043 0-4649805 0-4605035

0-3958334 0-3896181

1

21' 6*18 3i' 23*41 41 39;i9 51' 53*55 2'

6*53

12' 18*15

22' 28*45 32' 37*46

42 45*20 52' 51*71 2 ' 57*01 13' 1*12 23' 4*08

33 43'

35*17 12' 26*90 22' 17*74 2'

9' 42*29 19' 23*59 29' 4*20

38' 44*12 48' 23*36 58' 7'

1*94 39*86

17' 17*15 26' 53*82

36' 29*87

07774943

250

256

278 2-80 2-82 2-84 2-86 2-88 2-90 2 92

2-94 2-96

298 300


2-02 2-04 2-06 2-08 2-IO

Y

7iW

X

+ + + + +

1

I.

l#7(*)l

0-5753554 0-5738267 0-5721393 0-5702942 0-5682921

-

0-0957951 0-0846398 0-0735677 0-0625801 0-0516786

o-5832757 0-5800353 0-5768497 o-5737i74 0-5706370

-

0-0408645 0-0301393

0-5676071 0-5646262 0-5616930 0-5588064 0-5559650

+ + + + +

0-5661342 0-5638212 0-5613543 0-5587345 0-5559630

+ 0-0014878

2-22 2*24 2-26 2-28

+ + + + +

0-5530410 0-5499696 0-5467502 0-5433841 0-5398725

+ + + + +

230 232


(x)

2-12 2-14 2-l6 2-l8 2-20

671

00195045 0-0089616 0-0118422 0-0220999 0-0322594 0-0423191 0-0522773

0-5531678 0-5504135

0-54770H 0-5450295 o-5423977

arg

H

,

H

(x)

x

(*)

X

I9 37 " ll ;^ - 6° 15 43*88 - 5 45*59

+ + + + +

0-6560964 0-6653933 0-6746517 0-6838688 0-6930418

2-02 2-04 2-06 2-08 2-IO

4° 7' 42 ? 76 3° 3 35*49 i° 59' 23*88 0° 55' 8*03 o° 9' 11*98

+ + + + +

0-7021680 0-71 12445 0-7202680 0-7292381 0-7381496

2-12 2-14 2-16 2-18 2-20

J 3' 3 6 ;°4 2 i8' 4*07 3° 22' 35*97 4 27' 11*68 5 31' 51*10

+ + + +

0-7470008 0-7557890 0-7645117 0-7731661 + 0-78I7498

2-22 2-24 2-26 2-28

+ 0-7902603 + 0-7986950

2-32 2-34 2-36 2-38 2-40

- 9° 27' 10*37

-8° 23 26*38

"

-

K

230

0-0910627 + 0-1004889

0-5398047 o-5372496 o-53473i5 0-5322494 0-5298025

6° 7° 8° 9° IO°

+ + + + +

0-5273901 0-5250111 0-5226650 0-5203509 0-5180682

12° I' 1*31 6' 4*62 13 I4 II' II*o6 16' 15° 20*57 16 21 33*IO

+ + + + +

0-83I6270 0-8396463

+ + + + +

0-8707993 0-8783453 0-8857900 0-893I3I4 0-9003674

2-52 2-54 2-56 2-58 2-60

+ 0-0621324 + 0-0718828

2-34 2-36 2-38 2-40

+ + + + +

0-5362170 0-5324190 0-5284801 0-5244016 0-5201853

+ 0-0815267

2-42 2-44 2-46 2-48 2-50

+ + + + +

0-5158327 0-5113456 0-5067256 0-5019745 0-4970941

4-

0-1098039 0-1190059 0-1280934 0-1370647 0-1459181

36' 34-16 41' 20*77 46' 10*86 5i' 4"37 1*21 56

+ 0-80705I4 + 0-8I53272 + 0-8235I98

^8475755 0-8554I22 0-863I542

2-42 2-44 2-46 2-48

250

2-54 2-56 2-58 2-60

+ + + + +

0-4920863 0-4869528 0-4816957 0-4763168 0-4708183

+ + + + +

0-1546522 0-1632654 0-1717560 0-1801226 0-1883635

0-5158160 o-5i3593« 0-5114009 0-5092366 0-507 roo3

17° 18° 19° 20° 21°

2-62 2-64 2-66 2-68 2-70

+ + + + +

0*4652020 0-4594700 0-4536245 0-4476676 0-4416014

+ 0-1964774 + 0-2044627 + 0-2123179

22° 53' 48*11 23 59 20*07 25° 4'54'6o 26 10 31*65 27° 16' 11*16

+ 0-9074958 + 0-9I45I48 + 0-92I4224

2-62 2-64

+ 0-2200416 + 0-2276324

0-5049913 0-5029092 0-5008534 0-4988232 0-4968182

+ 0-9282I67 + 0-9348957

2-68 2-70

2-72 2-74 2«76 2-78 2-80

+ + + + +

0-4354281 0-4291500 0-4227693 0-4162882 0-4097092

+ + + + +

0-4948378 0-4928816 0-4909490 0-4890395 0-4871528

28° 29° 30° 31° 32°

+ 0-94I4577 + 0-9479008 + 0-9542233 +0-9604235 + 0-9664998

2-72 2-74

2-82 2-84 2-86 2-88 2-90

+ + + + +

0-4030346 0-3962667 0-3894079 0-3824607 0-3754275

+ + + + +

0-2703106 0-2769339 0-2834140 0-2897497 0-2959401

0-4852883 0-4834456 0-4816243 0-4798240 0-4780443

33° 50' 57 4 8 34° 56 53-oi 36° 2' 50*66 37° 8' 50*39 38° 14' 52*17

+ + + + +

0-9724504 0-9782739 0-9839687 0-9895333 0-9949663

2-82 2-84 2-86 2-88 2-90

2-92 2-94 2-96 2-98 3

+ + + + +

0-3683108 0-3611130 0-3538368 0-3464846

+ + + + +

0-3019839 0-3078802 0-3136281 0-3192264 0-3246744

0-4762847 o-4745449 0-4728245 0-4711232 0-4694406

39° 40° 4i° 42° 43°

20' 55*96 27' 1*72 33' 9 ? 42 39' 19*02 45' 30-50

+ + + +

I-0002663 I-00543I8 I-OI046I7 1-0153547 1-0201096

2-92 2-94 2-96 2-98 3-oo

252

03390590

0-2350890 0-2424099 0-2495937 0-2566393 0-2635454

20' 48*57

6*93 32 37 28*12 42 52*08 48' 18*76

21' 27' 33' 39' 45'

53*09 37*39 24*03 12*95 4"i

1

?

+

2-66

276 2-78 2-80

1


672

Table

X

302 3-04 3 06 3-o8 3-10

Y

/„(*)

-

I.

(

(x)

+ 03703033 + 0-3636522

02667583 0-2733140 0-2797178 0-2859683 0-2920643


\H l\x)\

axgH

+ 0-3568997 + 0-3500489 + 03431029

0-4563820 0-4549103 o-4534528 0-4520090 0-4505787

125° 46' 5*31 126° 55 40*16 128° 5' 14*42 129° i 4 ' 48*11 130° 24' 21*23 J 3i°

3-12

- 0-2980048

3M 3-16

-

0-3037884 0-3094142 0-3148811 0-3201882

+ + + + +

0-3360648 0-3289376 0-3217245 0-3144287 0-3070533

0-4491618 o-447758i 0-4463673 0-4449893 0-4436239

132° 43' 25*82 133° 52' 57*3i 135° 2' 28*27 136 11' 58*71

-

0-3253345

4-

03303193

+ + + +

0-2996013 0-2920760 0-2844806 0-2768182 0-2690920

0-4422708 0-4409300 0-439601 0-4382841 0-4369787

137° 1 38° 139° 140° 141

0-2613052 0-2534609 0-2455624 0-2376128 0-2296153

0-4356849 0-4344023 o-433i3io 0-4318706 0-4306210

J 43°

3'i8 3-20

322 3*24 3-26 3-28 3 30

332 3-34 3-36 3'38 3'4°

3H2 3-44 3-46 3-48

350 352 354 3-5f>

3-58

360 3-62 3-64 3-66 3-68

370 372 374 376 378 3'8o 3-82

3-86 3-88

390 3 92

394 3 96 ;

398 4-00

0-3351416 0-3398009 0-3442963

-

0-3567934 0-3606277 0-3642956

+ + + + +

-

0-3677967 0-3711306 0-3742972 0-3772963 0-3801277

+ + + + +

0-2215732 0-2134896 0-2053678 0-1972108 0-1890219

0-4293822 0-4281539 0-4269360 0-4257283

-

0-3827914 0-3852873 0-3876155 0-3897760 0-3917690

+ + + + +

0-3486272

03527931

33' 53'-8o

21' 30' 40' 49'

28-65 58*08 27-02 55*47 59 23*45

§'50*95

144° 18' 18*00 14 fo 2 7' 44-59 146° 37 10*74

147

H

(x)

46 36*44

+ + + + +

(*)

0-5665900 0-5587829 0-5508877 0-5429073 0-5348444

+ 0-5267021

+ + + +

0-5184831 0-5101905 0-5018270 0-4933957

+ + + + +

0-4848996 0-4763415 0-4677245 0-4590516 0-4503257

+ 0-4415499 + 0-4327272 + 0-4238607 4-

0-4149532

+ 0-4060080

X

3-02 3-04 3-06

308 310 312 3-M 3-16

318 3'20 3. 22 3-24 3*26 3-28

330 332 3*34 3-36 3-38 3-4°

04245308

148° 56' 1*71 150° 5' 26*56 I 5 I ° 14' 5o"98 152° 24' 15*00 153° 33' 38*6i

+ + + + +

0-3970279 0-3880161 0-3789757 0-3699095 0-3608208

0-1808043 0-1725612 0-1642956 0-1560109 0-1477100

o-4233432 0-4221655 0-4209974 0-4198389 0-4186898

154° 155° 157 158° 159°

1*82

52 24*63 1 47*05 11' 9-10 20' 30*77

+ + + + +

0-3517124 0-3425876

- o-3935947 - o-3952533 - 0-3967452 - 0-3980707 - 0-3992302

+ 0-1393962 + + + +

0-1310727 0-1227424 0-1144086 0-1060743

0-4175501 0-4164195 0-4152980 0-4141855 0-4130817

160° 161° 162° 163° 165°

29' 52*06 39' 12*99 48' 33*56

+ 0-3059833 + 0-2968211 + 0-2876605

57 53-78 7' 13*65

+ 0-2785044 + 0-2693559

3'62 3-64 3-66 3-68 3-70

-

0-4002242 0-4010532 0-4017178 0-4022187 0-4025564

+ + + + +

0-0977426 0-0894167 0-0810994 0-0727939 0-0645032

0-4119867 0-4109003 0-4098223

166° 167° 168° 169° 170°

16' 25' 35' 44'

33*17 52*36

+ 0-2602179

3-72

+ 025IO933

374 376 378

-

0-4027318 0-4027456

0-0562303 0-0479782 0-0397498 0-0315481 0-0233759

04066383 0-4055933 0-4045561 0-4035269 0-4025054

172° 173° 174° 17 fa 176°

3'

0-4022918 0-4018260

+ + + + +

5*81 23*37 40*63 57*57 14*22

+ + + + +

0-2147883 0-2057749 0-1967923 0-1878435 0-1789312

0-4012023 0-4004218

+ 0-0152362 + 0-0071319

0-4014915 0-4004853

I7 49' 30*56 Zo 178° 58' 46*60

03994854 03983943 03971498

- 0-0009343

03994865 03984951 o-3975"o

180

+ + + + +

0-1700582 0-I6I2273 0-1524412 0-1437027 0-1350146

-

04025986

k

- 0-0089594 - 0-0169407

04087528 0-4076915

43'

11*21

29*74 53 47*93

12 21

3° 40

V

2*36

181 17' 17*86 182° 26' 33*05

O3334492 0-3243003 0-3151440

+ 0-2419852 + 0-2328964 + 0-2238298

3H2 344 3H6 3-48

350 352 3-54

3-5f 3-58 3-6o

3-8o 3-82 3'8o 3-88 3-90 3-92

394 396 3-98 4-00

1


X

302

J lit)

I.


Yi(*)

Ifl'I'WI

+ + + + +

0-3299712 0-3351158 0-3401076 o-3449457 0-3496295

0-4677763 0-4661301 0-4645016 0-4628904 0-4612963

44° 45° 47° 48° 49°

0-2931100 0-2852440 0-2773257 0-2693579 + 0-2613432

+ + + + +

0-3541583 0-3585314 0-3627483 0-3668084 0-3707113

o-4597!90 0-4581582 0-4566135 0-4550847 o-45357i6

50° 23' 16*83

+ + + + +

0-2532845 0-2451844 0-2370457 0-2288711 0-2206635

+ + + + +

o-3744565 0-3780436 0-3814723 0-3847421 0-3878529

0-4520738 0-4505911 0-4491233 0-4476701 0-4462312

+ + + + +

0-2124255 0-2041599 0-1958696 0-1875574 0-1792259

+ 0-3908045 + 0-3935966 + 0-3962292 + 0-3987021 + 0-4010153

0-4448064 o-4433955

+ + + + +

0-1708779 0-1625163 0-1541439 0-1457634 0-I373775

+ + + + +

0-4031689 0-4051628 0-4069973 0-4086724 0-4101884

0-4378863 0-4365415 0-4352093

+ + + + +

0-41 15455

358 360

0-1289892 6-1206010 0-1122159 0-1038365 + 0-0954655

0-4127440 0-4137843 0-4146667 0-4153918

362 364 3 1§ 3-68 370

+ + + + +

0-0871059 0-0787602 0-0704312 0-0621215 0-0538340

+ + + + +

0-4159599 0-4163716 0-4166275 0-4167282 0-4166744

0-4249824 o-4237552 0-4225387

372 374 376 378

+ 0-0455712

+ + + + +

0-4164668 0-4161062 0-4155934 0-4149293 0-4141147

0-4189527 0-4177779 0-4166131 0-4154582 0-4143131

308 310 312 314 3-16 3-i8

320 3.22 3*24

326 3-28

330 332 334 3-3o 3'38

340 342 3'44 3-46

3-48 3'50

352 354 3-56

3-80 3-82

3 1i 3-88

390 3.92

394 390 3-98 4-00

+ + + + + + + + +

+ + + +

+ 0-0373359 + 0-0291307* + 0-0209582 + 0-0128210

+ + + + +

1-0456765 1-0494347

312

0530469

316

1-0565124 1-0598303

3-i8 3-20

55° 55' 31*29 2' 2*86 57 58° 8' 35*92 59 15 10*45 6o° 21' 46*43

+ 1-0630001 + 1 -06602 1 + 1-0688928

3-24

+ 1-0716147

3-28

+ 1-0741863

330

6i° 28' 23*84 62° 35' 2*65 63° 41' 42^85

1-0766072 1-0788770 1-0809955 1-0829624 1-0847774

332

64 48 24-40 65° 55' 7*28

+ + + + + + + + + +

1-0864406 1-0879516 1 0893 1 06 1-0905175 1-0915723

3*42 3-44

3H6

0-4325819

1' 51*48 67 68° 8' 36*98 69° 15' 23-75 70 22 11*77 71 29' 1-04

0-4312864 0-4300020 0-4287305 0-4274699 0-4262206

72°35'5i r 5i 73° 42 43-18 74° 49' 36 "04 75° 56' 30-05 3' 25*21 77

+ + + + +

1-0924752 1-0932264 1-0938260 1-0942743 1-0945716

3-52 3\54 3-56 3-58 3-6o

21*50 18*89 17*38 16*94 17*57

+ + + + +

1-0947183 1-0947147 1-0945614 1-0942589 1-0938077

3-62 3-64 3-66 3-68

83° 45' 19*23

+ + + + +

1-0932084 1-0924617 1-0915683 1-0905289 1-0893444

3-72 3-74

1-0880156 1-0865434 1-0849288 1-0831727

+ + + + + 1-08-12762

3-82

+ + + + +

1-0792403 1-0770662 I-074755I 1-0723082 1-0697267

392

04419983 0-4406145

04392439

04338895

04213329 0-4201376

51°. 29' 40*52

52° 36' 53° 42'

54 49

78 79° 80° 81° 82°

io' 17' 24' 31' 38'

5*84.

3276 1*25

84*52 21*94 59 25*65 f5° 87° 6' 30*37 88° 13' 36*07

04131776 0-4120516

- O-OI 13524 - 00193223 - 0-0272440

0-4131506 0-4120381 0-4107780 0-4093717 0-4078200

04109349

90 27' 50*38 91° 34' 58-96

0-4098274 0-4087290

92° 42' 8*47 93° 49' 18*8^9

-

+ + + + +

0-4061243 0-4042858 0-4023056 0-4001851 0-3979257

0-4076396 0-4065590 0-4054871 0-4044238 0-4033691

94° 96° 97° 98° 99°

00506953 0-0583995 0-0660433

X

1-0247251 1-0292003

4 15*82 10 34*45 16' 54*80

89° 20'

0-0351151 0-0429330

X

+ + + + +

51' 43 ? 8i 57 58*93

+ + + + +

+ 0-0047218 - 00033369

HW

argtf^*)

0-3315626 0-3239979 0-3163677 0-3086746 0-3009211

3-°4 3.06

673

4275

56' 30*22 3' 42*44 10' 55*54 18' 9-50 25' 24*33

1

0335340

1-0377252 1-0417730

1

302 304 306 308 310 3-14

322

326

3-34 3-36 3-38 3-40

3-48 3-5°

370

376 378 3-80

3 'h

3*2 3-88 3 90

3-94 3 96 3-98 4-00


674

Table

X

/oW

4-02 4-04 4-06 4-08 4-10

- 0-3957530 - 0-3942053 - 03925079 - 0-3906622

Y

I.

Functions of order zero (

{*)

l# >)l

0-0248755 0-0327610

- 0-3886697

-

4-12 4-14 4-16 4-18 4-20

- 0-3865318 - 0-3842500 - 0-3818259 - 0-3792610 - o-376557i

-

0-0637561 0-0713550

4'22 4'2 4 4*26 4-28 4-30

-

0-3737I57 0-3707386 0-3676276 0-3643845 0-3610111

-

4-32 4'34 4'30 4'38 4-40

- 0-3575093 - 0-3538810 - 0-3501281 - 0-3462527 - 0-3422568

-

4-42 4*44 4-46

- 0-1698081 - 0-1761827

.0-3775411

4-50

- 0-3381424 - 033391 16 - 0-3295666 - 0-3251095 - 03205425

- 0-1824583 - 0-1886330 - o- 1947050

4-52 4-54 4-5o 4-58 4-60

-

0-3158678 0-3110877 0-3062045 0-3012204 0-2961378

-

4-62

-

0-2909591 0-2856866 0-2803228 0-2748700 0-2693308

-

-

0-2637076 0-2580029 0-2522193

4H8

$

4-68

470 472 474 476 4-78 4-80 4-82

4-86 4-88

490 4-92 4-94 4-96

498 5-00

arg

H

H„(*)

(x)

X

0-396534° 0-3955643 0-3946015 o-3936457 0-3926967

35 47*96 ô 184° 45' 2^62 185° 54' 16^99 187° 3'31-n 1 88° 12' 44*96

+ + + + +

0-I263794 0-II77998 0-1092784 0-I008I79 0-0924208

0-0863551 0-0937512

o-39i7546 0-3908191 0-3898903 0-3889680 0-3880522

189 21' 58*54 190 31' 11*87 191 ° 40' 24*95 Ig2 49' 37*78 l 193 58 50*36

+ + + + +

0-0840896 0-0758269 0-0676351 0-0595166 0-0514740

4-12 4-14 4-i6 4-18 4-20

0-1010748 0-1083234 0-1154947 0-1225863 0-1295959

0:3871428 0-3862397 0-3853428 0-3844522 0-3835676

195° 196 197 198° 1 99°

+ 0-0435095 + 0-0356255

4-22 4-24 4-26 4-28 4-30

0-1365213 0-1433602 0-1501104 0-1567699 - 0-1633365

0-3826891 0-3818166 0-3809499 0-3800891

200 54' o-88 3' 11*83 202 203 12' 22*56 204 21 ' 33*08 205 30' 43*37

+ 00049399 -

0-0025077 0-0098616 0-0171197 0-0242798

39' 53*45 49' 3*33

0-3767030 0-3758705 0-3750434

206° 207° 208 2IO° 211

-

0-0313400 0-0382984 0-0451530 0-0519019 0-0585433

4-42 4*44 4-46 4-48

0-3742217 0-3734053 o-3725943 0-3717885 0-3709878

212° 2I3 214°o 215° 217

25' 40-76 34; 49;62 43' 58*28 53' 6*75

-

00650755

4-52 4-54 4-5§ 4-58 4-60

0-2288780 0-2341813 0-2393683 o-2444376 0-2493876

0-3701923 0-3694018 0-3686164 0-3678359

218 11' 219 20' 220° 29' 221° 38' 222° 47'

0-2542172 0-2589248

0-3662896 0-3655237

02635093

03647625

02463592 O2404253

-

0-2679693 0-2723038

0-3640061

-

0-2344201 0-2283462 0-2222062 0-2160027 0-2097383

-

0-2765116

03625071

02805915

0-3617645 0-3610265 0-3602929 o-3595637

-

0-2034158 0.1970377 0*1906067 0-1841255 0-1775968

-

00405944 0-0483732 0-0560946

0-0.788889

0-2006723 0-2065332 0-2122859 0-2179287 0-2234600

0-2845427 0-2883640 0-2920546

02956136 0-2990401

0302333s 03054928 0-3085176

03792341 0-3783848

03670603

03632543

0-3588389 0-3581185

03574023 03566904 03559828

',

8' 271 17' 14*81 26' 26*67

35' 38-31 44' 49*71

58 12*99 7' 22*45 16' 31*70

2' 15*03

+ 0-0278243 + 6-020IOOI + 0-0124793

0-0714966 0-0778050 0-0839990 0-0900771

4-02 4-04 4-06

408 4-10

4-32 4-34 4-30 4-38 4-4°

450

- 0-0960376 - 0-1018790 - 0-1075998 - 0-1131987 - 0-1186742

4-62

-

0-1240251 0-1292500 o-i343477 0-1393170 0-1441567

4-72 4-74 4-76 4.78 4-80

229° 42' 34*17 230° 51 40-33 232° O' 46*33 233° 9' 52*17 234° 18' 57-85

- 0-1488659 - o-i534435 - 0-1578884 - 0-1621997 - 0-1663766

4-82

4-f§ 4-88 4-90

235° 28' 2*37 236° 37' 8*74 237° 46' I3 ?95 23»° '55 19*01 4' 23*93 240

- 0-1704182 - 0-1743238 - 0-1780925 - 0-1817237 - 0-1852168

4-92 4-94 4-96 4-98 5-00

223° 225 226° 227° 228

57'

23*12 31*03 38*75 46*29 53*65 0*83

6' 7*84 15' 14*68

24' 21*34 33' 27*84

1% 4-68 4-70

1


/»W

X 4-02 4-04 4-06 4-08 4-10

-

4-12

I.


Yx(»)

!<(*)! 0-4023226 0-4012845 0-4002545

01032733

0-3955287 0-3929956 0-3903277 0-3875267 0-3845940

- 01 105054 - 0-1176609 - 0-1247378 - 0-I3I7339 - 0-1386469

+ + + + +

0-3815313 0-3783401 0-3750222 o-37i5792 0-3680128

03972123

0-3932638

no°

-

0-1454750 0-1522160 0-1588679 0-1654287 0-1718966

+ + + + +

0-3643248 0-3605171 0-3565914 o-3525497 0-3483938

03922953 03913340

in

-

0-1782695 01 845457 0-1907233 0-1968005 0-2027755

+ 0-3441256 + o-3397472

+ 0-3352606

4-44 4'42 4-48 4-5°

-

414 416 4-18 4'20 4'22 4-24 4-26 4-28

03992325 0-3982185

0-3962138

03952229 03942396

HxW

arg rf\\x)

+ + + + +

0-0736243 0-0811401 0-0885884 0-0959669

675

ioo° 101 102° 103° 105

32' 39' 47' 54'

40-00 56*49

+ 1-0641653 + 1 -0611801 + 1-0580818 + 1-0548479

4-02 4-04 4-06 4-08 4-10

10*59 31*08 52*34 14*36 38' 37*12

+ + + + +

1-0514880 1-0480034 1-0443959 1-0400671 1-0368186

4-12 4-14 4-16 4-18 4-20

46' 0*62 53' 24*85

+ + + + +

1-0328522 1-0287695 1-0245724 1-0202627 1-0158422

4-22 4-24 4-26 4-28 4-30

+ 1-0113128

4*32 4'34 4-36 4-38 4-40

i'

13*81

31-94 50*87

9' 106 107 16' 23' 108 109 31'

112

1-0670119

X

4-

0-3903799

"4°

03894328

115°

0-3884928

116

+ 0-3306677 + 0-3259707

0-3875596 0-3866333 0-3857136 0-3848007 0-3838942

117° 23' 8*84 118 30' 36*56 1 19 38' 4*96 120 45' 34*02 121 53' 3*74

0-2086467 0-2144125 0-2200710 0-2256209 0-2310604

+ + + + +

0-3829943 0-3821008 0-3812136 0-3803327 o-3794579

0' 34*10 123 8' 5*10 124 125° 15' 36*73 126 23' 8*99 127 30' 41*86

+ + + + +

0-9871006 0-9819591 0-9767229 0-9713939 0-9659744

4-42 4-44 4-46 4.48

452

- 0-2363882 -.

450

- 0-2467026 - 0-251 686a - 0-2565528

+ + + + +

0-3785893 0-3777267 0-3768700 0-3760193 0-375I744

+ + + + +

0-9604664 0-9548724 0-9491944 o-9434347 o-9375956

452

4*54

128 38' 15*34 1 29 45' 49*42 130° 53' 24*09

o-3743352 0-37350I8 0-3726740 0-3718517 0-3710350

134° 16' 2 *35l 3 136 31 I3 Zo 3 i/ 1 38° 46'

+ + + + +

0-9316793 0-9256883 0-9196249 0-9134914 0-9072901

4-62

0-3702237 0-3694177 0-3686171 0-3678218 0-3670317

139° 54' 21*96

0-9010236 0-8946941 0-8883042 0-8818563 0-8753528

472

141° 2' 1*65 142° 9' 41 -8 7 143° 17 22*60 i44°25' 3*85

+ + + + +

M5°°

+ + + + +

0-8687963 0-8621891 0-8555338 0-8488330 0-8420890

4-82 4-84 4-86 4-88 4-90

11' 21*79

+ + + + +

0-8353045 0-8284820 0-8216241 0-8147332 0-8078119

4-92 4-94 4-96 4-98 5-00

430 4-32 4'34

430 4-38 4-40

4H2

4-58 4-60 4*62

x\ 4-68 470

-

0-2416027

0-2613006 0-2659284

02704352 0-2748196 0-2790807

0-3211716 0-3162725 0-3112757 0-3061832 0-3009973 0-2957202

02903542 0-2849015

02793644 0-2737452

+ 0-2680464 + + + +

0-2622702 0-2564190 0-2504952 0-2445013

132° 133°

°'

49*79

8 15*45 15 41*80

©'59*33 8' 35*18

11*59

4f;56 26*09 4 ;o 7 42*80

0-2832174 0-^872286 0291 1 133 0-2948707 0-2984999

+ + + + +

02384397

4-78 4-80

-

4-82 4-|4 4-86 4-88 4-90

-

0-3019999 0-3053702 0-3086098

0-2072020 0-2007860 0-1943198 0-1878058 0-1812467

0-3662467 0-3654068 0-3646919

0-3146947

+ + + + +

0363 1 57

32' 45*6i *4 6 40 27*87 147° 48' 10*62 148° 55' 53*86 150 3 37"59

492 494

- 031 75386 - 0-3202495 - 0-3228269 - 0-3252702 - 0-3275791

+ + + + +

0-1746449 0-1680031 0-1613238 0-1546097 0-1478631

0-3623971 0-3616418 0-3608913 0-3601456 0-3594045

151 *5 2 ° 153 154° 155

472 4-74

476

4-96 4-98 5-00

03117182

0-2323128 0-2261230 0-2198730 0-2135652

03639221

*?',

6 "A7

26 5 J *6i 34; 37*22 42 23-28

1-0060764 + 1-0019350 + 0-9970906 + 0-9921451

4-

450 4-54 4-56 4-58 4-60

#* 4-68 4-70

4-74 4-76 4.78 4-80


676

Table

X

Y

7o(*)

I.

Functions of order zero {

(x)

\H l\x)\

arg //„(*)

-

0-1710232 0-1644075 0-1577524 0-1510606 0-1443347

-

0-3114072 0-3141609 0-3167784 0-3192590 0-3216024

o-3552793 o-3545799 0-3538847 o*353!934 0-3525062

241 242 243° 244° 245

-

0-1375776 0-1307919 0-1239803 0-1171456 0-1102904

-

0-3238083 0-3258764 0-3278063 0-3295978 0-3312509

0-3518230 o-35ii437 0-3504684 o-3497968 0-3491291

246 58' 50*32 248° 7' 54*22 249 1 6' 57*98 250 26' 1 "61

-

0-1034176 0-0965297 0-0896295 0-0827198 0-0758031

-

0-3327654 0-3341413 0-3353785 0-3364772 0-3374373

0-3484652 0-3478051 0-3471487 o-3464959 0-3458469

252° 44' 8?47 253° 53 n*70 2' 14*80 255 256 n' 17*76 257 2o' 20*60

-

0-0688822 0-0619598 0-0550386 0-0481211 0-0412101

-

0-3382591 0-3389428 0-3394886 0-3398969 0-3401679

0-3452014 o-3445595 0-3439212 0-3432863

-

0-0343082 0-0274180 0-0205422 0-0136833 0-0068439

-

+ + + +

- 0-0000266 0-0067661 0-0135315 0-0202673 0-0269709

+ + + + +

5-8o

+ + + + +

5-82 5 'h 5 *fi 5-88 5-90 5*92 5-94 5'96 5-98 6-oo

5-02 5-04 5-06 5-08

510 5-12 5-14

516 5-18 5-20 5-22 5-24 5-26 5-28

530 5'32 5'34 5-36 5-38 5'4<>

5H2 5H4 5-4^ 5-48 5 50 5-52 5.54 5-56 5-58 5 60

5-62 5-64 5-66 5-68

570 572 574 576 578

H„(*)

X

-

0-1885712 0-1917864 0-1948610 0-1977971 0-2005919

5-02 5-04 5*o6 5-o8 5-10

-

0-2032458 0-2057586 0-2081301 0-2103600 0-2124483

5-12 5-14 5-16 5'i8 5-20

- 0-2143949 - 0-2161998

- 0-2178630 - 0-2193846 - 0-2207647

5*22 5-24 5-26 5-28 5-30

03426550

258 29' 23*32 259 38' 25*91 260 47' 28*37 261 56' 30*72 5' 32*95 263

-

0-2220035 0-2231013 0-2240583 0-2248748 0-2255513

5-32 5-34 5-36 5-38 5-40

0-3403021 0-3402999 0-3401619 0-3398886 0-3394806

0-3420271 0-3414027 0-3407816 0-3401639 0-3395496

264° 14' 35*05 265 23 37*04 266 32' 38*91 267 41' 40*66 268 50' 42*30

- 0-2260882 ^ 0-2264860 - 0-2267451 - 0-2268062 - 0-2268499

5-42 5-44 5-46 5-48 5-50

-

0-3389385 0-3382631 0-3374550 0-3365151 o-3354442

0-3389385 0-33^3307 0-3377262 0-3371249 0-3365267

269 271° 272° 273° 274

-

0-2266969 0-2264079 0-2259836 0-2254249 0-2247327

5-52 5-54 5-56 5-58 5-6o

0-0336398 0-0402716 0-0468638 °-°534 I 4 I 0-0599200

-

0-3342432 0-3329130 0-3314545 0-3298689

0-33593I7 o-3353399

275° 44' 276 53 2' 278 279 ii'

- 0-2239078 - 0-2229512 - 0-2218639 - 0-2206469 - 0-2193014

5-62 5-64 5-66 5-68 5*70

0-0663792 0-0727894 0-0791482 0-0854533 0-0917026

-

0-3263203 0-3243597 0-3222763 0-3200715 0-3177464

53*07 53*42 53*67 53*82 5' 53*87

-

0-2178284 0-2162291 0-2145048 0-2126567 0-2106861

572 574

+ + + + +

0-0978937 0-1040245 0-1100928 0-1160964 0-1220334

- 03153025 - 0-3127411 - 0-3100636 - 0-3072714 - 0-3043659

<

- 0-2085942

5-82

+ + + + +

0-1279015 0-1336987 0-1394230 0-1450725 o- 1506453

-

0-328157,1

0-3013488 0-2982215 0-2949856 0-2916428 0-2881947

0-33475" 0-33+1655 o-3335828

03330033 0-3324267 °-33 I 853° 0-3312824 0-3307146

03301498 0-3295878 0-3290286 0-3284723 0-3279188 0-3273681 0-3268201 0-3262749

03257324 03251925

13' 28*68 22' 33*30 31' 37*76

40 42*09 49' 46*28

251° 35'

5*"

59' 43*83 8' 45*24

17 46*54 26' 47*73

35 48*81

49*79 50*65 51*41 52*07 280 20' 52*62 28l° 282° 283° 284

286 2

29' 38' 47' 56'

l 53*83 f ll 288° 23 53*68 289° 32' 53 ? 44 290 41' 53*11 291 ° 50' 52*68

292° 294° 295° 296 297°

59' 52*16 8' i* 5 54 17' 50-84

26 50-05 35' 49 ? i6

- 0-2063827 - 0-2040527 - 0-2016058 - 01 990435

-

0-1963672 0-1935787 0-1906794 0-1876711 01 845553

5-76

578 5-80

lit 5-88 5-9o 5 92 5-94 5-96 5-98 6-oo


X 5-02

504 5-06 5-08 5-io

Y

/iW -

03297533 03317925 03336963 03354646 0-3370972

l

I.

(x)

+ 0-1410869 + 0-1342835 + 0-1274556 0-1206057 + 0-1137364 4-

520

- 03385940 - 03399550 - 0-3411802 - 03422695 - 03432230

+ + + + +

0-1068504 0-0999502 0-0930384 0-0861176 0-0791903

5-22 5-24 5-26 5-28 5-30

-

+ + + + +

0-0722592 0-0653269 0-0583958 0-0514685 0-0445476

532 534

0-3457337 o-3453448

+ + + + +

00376356

5HO

-

5'42 5*44 5*4 6 5'48 5'50

-

0-3448242 0-3441725

+ 0-0032975 - 00035083

5-52 5'54 5'5o

-

5-12

516 5-i8

5'36 5-38

0-3440409

03447234 03452707 03456831 03459608 03461043 0-3461140

03459903

03433905

0-0307351 0-0238485 0-0169784 0-0101273

- 0-0102879


argH^x)

i^J'wi

13' 31*99 l6l° 2l' 20*26

0-8008629 0-7938886 0-7868916 0-7798745 0-7728398

29' 8*96 163 36' 58*08 164° 44' 47*61 165 52 37 r 56 27*92 167

+ + + + +

0-7657902 0-7587281 0-7516562 o-744577o 0-7374930

8' 18*68 168 169 16' 9*84 170 24' 1*40 i7i°3i'53 ? 34 1 72° 39' 45-63

+ 0-7304068

522

+ + + +

0-7233211 0-7162382 0-7091607 0-7020912

5-24

0-6950321 0-6879061 0-6809555 0-6739428 0-6669506

532 534

ii' 18*80 19' 13*00

+ + + + + + + + + +

0-6599812 0-6530372 0-6461209 0-6392347 0-6323810

542

§5: 6 ; 45-72 14' 42*38 187° 22' 39*38 188 30' 36*72 189 38' 34*39

+ + + + +

0-6255623 0-6187809 0-6120390 0-6053391 0-5986835

190 46' 32*39 I9i° 54' 30 ? 7i 2' 29*35 193 194 io' 28*30 18' 195° 27*57

+ + + + +

0-5920743 0-5855138 0-5725481 0-5661472

5-68

196 26' 197° 34' 198 42' 199 50' 200° 58'

27*15 27;04 27*23 27*72 28*52

+ + + + +

0-5598038

572

05535201

5-74 5'7| 5-78

+ + + + +

0-5290231 0-5230685 0-5171858 0-5113768 0-5056434

+ + + + +

0-4999876 0-4944111 0-4889157 0-4835031 0-4781753

i56°5o'

03579362

157° 57' 56-75 159° 5' 44**5

0-3572088 0-3564859 0-3557675

03550534 0-3543437

03536383 0-352937 1

03522402 o-35i5474 0-3508587

03501742 o-3494936 0-3488171

1 62

0-3481446 o-3474759 0-3468112 0-3461503 0-3454933

173° 47' 38;39 174° 55 31-49

0-3448400

179

03441904

I

0-3435445 0-3429024 0-3422638

27' 7*56 2 4 ; 2 l°o 35 42 57-78 182° 50 53 r 4i 183 58' 49 T 39

*K

176 177 178

0-3402696

-

00304443

0-3416288

J

0-0370944 0-0437062 0-0502774 0-0568050

03409975

186

00632886

03385069

- 03264245 - 03241477

-

0-0697241 0-0761099 0-0824437 0-0887233

0-3378928 0-3372821 0-3366748 0-3360708

-

0-3217534 0-3192429 0-3166176 0-3138787 0-3110277

-

0-3080661

5-86 5-88 5'90

-

-

5*92 5-94 5'9o 5-98 6-oo

-

0-2916501 0-2880563 0-2843629 0-2805715

5-60 5-62

S3 568 5-7© 5'72 5-74 5-7°

578 5'8o 5-82

o-33755i8

0336004s 03343328

- 0-3325379 - 0-3306208 - 0-3285826

03049952 0-3018166 0-2985318

02951424

0'2 766839

0-0949466 0-1011115 0-1072157 0-1x32573 0-1192341

9 ? 8o

160

- 0-0170386 - 0-0237582

558

0-3403696 o-3397452

03391243

0-3354700

03348725 0-3342782

03336871 0-3330991

3'

24*96

0-125 1 442

0-3325I43

o- 1309855

03319326

0-1367560 0-1480772

0-33I3540 0-3307784 0-3302059

202° 29*60 203 14' 30*98 22' 32*65 204 205° 30' 34*61 206 38' 36*85

- 0-1536240 - 01590925 - 0-1644809 - 0-1697874 - 0-1750103

0-3296363 0-3290697 0-3285061 o-3279453 0-3273875

207° 208 2IO° 211° 212°

01424539

X

Ht(*)

+ + + + +

0-3586680

0-3424788 0-3414382

03389739

677

6'

46' 39*37 54' 42*17 2' 45*25

IO' 48*60 l8' 52*23

05790044

0-5472981 0541 1399 0-5350476

5-02

504 5-06 5-08

510 512 5-14

516 518 520

526 528 53°

5'3° 5-38

54° 5-44 5-46 5'48

550 552 554 5-56 5-58 5-60

562 5

s&

570

580 582 5§4

5-86 5'88

590 5 92

594 5-96

598 6-oo

2


678

Table X

Y

/«(*)

I.


(x)

K'(*)l 0-3246553 0-3241208 0-3235889

6-02 6-04 6-o6 6-o8 6-io

+ + + + +

0-1561393 0-1615527 0-1668837 0-1721306 0-1772914

-

0-2846430 0-2809893 0-2772356 0-2733835 0-2694349

6-12 6-14 6-i6 6-i8 6-20

+ + + + +

0-1823646 0-1873484 0-1922411 0-1970413 0-2017472

-

0-2653917 0-2612556 0-2570287 0-2527128 0-2483100

0-3220087

6-22 6-24 6-26 6-28 6-30

+ + + + +

0-2063574 0-2108705 0-2152848 0-2195991 0-2238120

-

0-2438221 0-2392513 0-2345996 0-2298691 0-2250617

6-32 6-34 2' 3 f 6-38 6-40

+ + + + +

0-2279222 0-2319283 0-2358292 0-2396237 0-2433106

-

0-2201798 0-2152253 0-2102005 0-2051075 0-1999486

6-42

+ + + + +

0-2468888 0-2503573 0-2537151 0-2569612 0-2600946

-

0-1947259 0-1894417 0-1840982 0-1786977 0-1732424

03144396

+ + + + +

0-2631145 0-2660201 0-2688106 0-2714853 0-2740434

- 0-1677348 - 0-1621770 - 0-1565714 - 0-1509204 - 0-1452262

03120324

£' 4

2 6-46 6-48 6-50 6-52 5

i

i 6-58 6-6o

PI 6-62 6-64 6-66 6-68 6-70

+ 0-2764843 + + + +

0-2788074 0-2810122 0-2830981 0-2850647

6-72

0-2869117

6-8o

+ + + + +

6-82 6-84 6-86 6-88 6-90

6-92 6-94 6-96 6-98 7-00

6$ 6-78

51

Ho(*)

X 6-02 6-04

11' 44*74 20' 43*43

- 0-1813339 - 0-1780085 - 0-1745809 - 0-1710529 - 0-1674264

Q-3I9937 1

304 305° 306° 307° 309

29' 38' 47' 56' 5

42*03 40*54 38*97 37*32 35*59

-

0-1637033 0-1598854 0-1559746 0-1519730 0-1478824

6-12 6-14 6-16 6-l8 6-20

0-3194255 0-3189162 0-3184094 0-3179050 0-3174029

310° 14' 33*77 311 23' 31*88 32' 29*90 31 313° 41' 27*85 314 5o' 25*73

-

0-1437050 0-1394427 0-1350977 0-1306719 0-1261676

6-22 6-24 6-26 6-28 6-3°

0-3169032

315° 59' 23*52 8' 21*24 317 318 17' 18*88 26' 16*44 319 320 35' 13*94

-

0-1215867 0-1169316 0-1122043 0-1074071 0-1025422

6-32 6*34 6-36 6-38 6-40

321 44' 11*36 322° 53' 8*71 324° 2' 5*98 325° 11' 3*18 326 20' 0*31

-

0-0976117 0-0926181 0-0875634 0-0824500 0-0772802

%\%

3 2 z: 28 ; 57-38 328° 37; 54*37 329° 46' 51*29 330° 55; 48'M 4' 44*93 332

-

0-0720564 0-0667807 0-0614556

6-52

00560834

6.58 6-6o

333° *3' 41*65 334° 22' 38*30 335° 3i' 34*88 336 40' 31*40 337° 49' 27*86

-

-

03230596 0-3225328

03214870 0-3209679

03204513

03164059 0-3159109 0-3154182 0-3149278 o-3i39537 0-3134701

03129887 0-3125094 o-3i 15575

0-3110848 0-3106143

03101458

-

0-1394913 0-1337179 0-1279085 0-1220655 0-1161911

0-3096795 0-3092152

0-1102879 0-1043582 0-0984043 0-0924287 0-0864339

03073788

0-2902449 0-2917307 0-2930956

-

+ + + + +

0-2943394 0-2954620 0-2964633 0-2973434 0-2981020

-

0-0804221 0-0743958 0-0683573 0-0623092 0-0562537

0-3051285

+ + + + +

0-2987395 0-2992557 0-2996510 0-2999254 0-3000793

-

0-0501933 0-0441303 0-0380671 0-0320062 0-0259497

o-'2886385

arg H"H( X )

298 299° 301° 302° 303

44' 48-19 53' 47*13 2'

45*98

0-0506665

6.-06

6-o8 6-io

6-42

6-48 6-50

tit

0-0285990 0-0229940

6-62 6-64 6-66 6-68 6-70

338° 58' 2 4 *2S 7' 20*58 340 341° 16' 16*84 342° 25' 13*04 343° 34 9 ? i8

0-0173587 0-0116953 0-0060063 0-0002941 + 0-0054389

6-72 7 7t 6-78 6-8o

0-3042421 0-3038017 0-3033633

344° 43' 5*26 345° 52' 1*27 34Z° ° 57*22 348° 9'53'i2 349 18' 48*95

+ + + + +

0-0111903 0-0169576 0-0227386 0-0285306

003433 1

6-82 6-84 6-86 6-88 6-90

0-3029268 0-3024921 0-3020593 0-3016283 0-3011992

350° 35i° 352° 353° 355

27' 44*73

+ + + + +

0-0401386 0-0459497 0-0517624 0-0575743 0-0633830

6-92 6-94 6-96 6-98 7-00

03087530 0-3082929

03078348 0-3069248 0-3064727 0-3060227 0-3055746

03046843

36 40*45 45 36*10 54' 31*70 3 27*25

0-0452073 0-0397080

00341 7 1

t

1

8

1


X

Y

/iW

I.

679


&rgH

{

H,W

X

(x)

\H \\*)\

0-1801479 0-1851985 0-1901605 0-1950322 0-1998122

0-3268325 0-3262804 o-32573ii 0-3251846 0-3246409

213° 214° 215° 216 217

26' 56*12 35' 0*27 43' 4*69 51' 9*37 59' 14*31

+ + + + +

0-4729337 0-4677800 0-4627160 o-457743i 0-4528629

6-02 6-04 6-o6

03240999

+ + + + +

0-4480769 0-4433866 0-4387935 0-4342988 0-4299040

6-12 6-14 6-16

0-3219630

7' 19*50 219 220° 15' 24*94 221° 23' 30*64 222° 31' 36*59 223 39' 4 2 *7 8

1

t

(x)

6-02 6-04 6-o6 6-o8 6-IO

-

0-2727017 0-2686269 0-2644612 0-2602066 0-2558648

-

6-12 6-14 6-16 6-i8 6-20

-

0-25I437 8 0-2469275

0-2376649 0-2329166

-

0-2044989 0-2090908 0-2135865 0-2179846 - 0-2222836

6-22 6-24 6-26 6-28 6-30

-

0-2280930 0-2231961 0-2182281 0-2131910 0-2080869

-

0-2264824 0-2305796 0-2345740 0-2384643 0-2422495

0-3214354 0-3209104 0-3203880 0-3198682 0-3193509

224° 225° 227° 228° 229°

47' 49*21 55' 55"89 4' 2*8l 12' 9*97 20' 17*35

+ + + + +

0-4256104 0-4214192 0-4I733I7 0-4133490 0-4094724

6-22 6-24 6-26 6-28 6-30

6-32 6-34 6-36 6-38 6-40

-

0-2029180 0-1976865 0-1870440 0-1816375

-

0-2459284 0-2494998 0-2529629 0-2563166 0-2595599

0-3188362 0-3183239 0-3178142 0-3173069 0-3168020

23O 28' 24*98 23I 36' 32"83 232° 44' 4°*9i 233° 52' 49*22 235° 0' 57*75

+ + + + +

0-4057028 0-4020415 0-3984894 0-3950474 0-3917166

6-32 6-34 6-36 6-38 6-40

6-42 6-44 6-46 6- 8 4 6-50

-

0-1761771 0-1706650 0-1651035 0-1594949 0-1538413

-

0-2626920 0-2657119 0-2686190 0-2714123 0-2740913

0-3162996 o-3i57996 0-3153020 0-3148067 0-3143138

236

9' 6*51 7' 15*48

+ + + +

0-3884978 0-3853919 0-3823996 0-3795218 + 0-3767591

6-42

652

0-1481451 0-1424086 0-1366341 0-1308238 0-1249802

- 0-2766551 - 0-2791032 - 0-2814349 - 0-2836498 - 0-2857473

03138232

241° 49' 53*56

0-313335° 0-3128490 0-3123653 0-3118839

2 +2

6-56 6-58 6-6o

-

13*86 244 245° 14' 24*33 246 22' 35*00

+ + + + +

0-3741123 0-3715819 0-3691685 0-3668728 0-3646951

%1 600

6-62 6-64 6-66 6-68 6-70

-

0-1191054 0-1132019 0-1072720 0-1013179

-

0-2877269 0-2895883 0-2913310 0-2929548 0-2944593

0-3114047 0-3169277 0-3104529 0-3099804

2

nl 3°; 45*88 248 38' 56*96 249° 47' 8*24 250 55' I9*7i 3' 3i*38 252

+ + + + +

0-3626360 0-3606958 0-3588749 o-357i737 o-3555923

6-62 6-64 6-66 6-68 6-70

6-72 6-74

0-2958444 0-2971098 0-2982554 0-2992811 0-3001869

0-3081116 0-3076^97 0-3071899

253° 11' 43*25 2 54° 19 55*31 255° 28' 7*56 256 36 20*01 257° 44' 32*64

+ + + + +

0-354I3 10 0-3527901 0-3515696 0-3504696 0-3494901

6-72

0-0773076 0-0712681 0-0652187

-

0-3090417

t%

6-8o

-

6-82 6-84 6-86 6-88 6-90

- 0-0591615 - 0-0530989 - 0-0470332 - 0-0409669 - 0-0349021

-

0-3009727 0-3016385 0-3021846 0-3026109 0-3029176

0-3067322 0-3062765 0-3058229 0-30537I3 0-3049217

258° 52' 45*45 260° 0' 58*45 261 ° 9 11*64 262 17' 25*01 263 25' 38*56

+ + + + +

0-3486313 0-3478930 0-3472751 o-3467775 0-3464001

6-82 6-84 6-86 6-88 6-90

6-92 6-94

-

-

0-3031051 0303 1 734

0-3044741 0-3040285

33' 52*29 42' 6*19

03031230 0302954

03035849

264 265 266 267° 269

+ + + + +

0-3461426 0-3460047 0-3459862 0-3460867 0-3463057

6-92 6-94 6-96

6-

54

676 678

696 6-98 7-00

02423358

01923944

0095342 0-0893469

00833346

0^0288412 0-0227066 0-0167404 0-0107051 0-0046828

0-3026672

0-3235616 0-3230261

03224932

03095099

03085756

0-3031432 0-3027035

2 3Z°

x

238 25' 24*68 239° 33' 34'og 240 41' 43*72

o

5!

6'

3 ;£J

50' 20*27

58' 34*53 6'

48*96

608 6-io

6- 1

6-20

t$ 6-48 6-50 6-52 6-54

8

6-78 6-8o

698 7-00


680

Table

X

;.w

I.


y.w

!«?(*)!

arg# oM

HoW

(

X

7'02 7-04 7-06 7-08 7-10

+ + + + +

0-3001128 0-3000264 0-2998204 0-2994953 0-2990514

0-0199002 0-0138600 0-0078314 0-0018167 + 0-0041818

0-3007719 0-3003464 0-2999227 0-2995008 0-2990806

356 12' 22*74 357° 21' 18*17 358° 30' 13-54 8*86 359° 39 360 48' 4*12

+ + + + +

0-0691861 0-0749812 0-0807662 0-0865383 0-0922958

7-02 7-04 7-06 7-08 7-10

7-12 7.14 7-16 7-18 7-20

+ + + + +

0-2984893 0-2970096 0-2970128 0-2960996 0-2950707

+ + + + +

0-0101617 0-0161208 0-0220568 0-0279674 0-0338504

0-2986622 0-2982456 0-2978307 0-2974175 0-2970060

36i° 56' 59*33 5' 3f3° 14' 54*49 364° 49*59 365° 23' 44*64 366 32' 39*63

+ 0-0980360 + 0-1037565 + 0-1094553 + 0-1151299 + 0-1207782

7-12 7-14 7-16 7-18 7-20

7-22 7-24 7-26 7-28 7*3°

+ + + + +

0-2939268 0-2926686 0-2912970 0-2898128 0-2882169

+ + + +

0-0397036 0-0455247 0-0513115 0-0570620 + 0-0627739

0-2965962 0-2961881 0-2957817 0-2953769 0-2949738

41' 34-57 3f7° 368 50' 29*46 369 59' 24*30 371° 8' 19*09 372° 17' 13*83

+ + + + +

0-1263979 0-1319868 o-i3754 2 7 0-1430634 0-1485467

7-22 7-24 7.26 7-28 7-30

7-32 7-34 7-36 7-38 7-40

+ + + + +

0-2865103 0-2846939 0-2827607 0-2807358 0-2785962

+ + + + +

0-0684451 0-0740734 0-0796569 0-0851934 0-0906809

0-2945724 0-2941726 o-2937743 0-2933777 0-2929827

373° 374° 375° 376° 378

+ 0-1539905 + 0-1593927

+ 0-1647511 + 0-1700638 + 0-1753286

7-32 7'34 7-36 7-38 7-40

7-42 7'44 7-46 7-48 7-50

+ + + + +

0-2763512 0-2740018 0-2715492 0-2689947 0-2663397

+ + + + +

0-0961173 0-1015007 0-1068292 0-1121007 0-1173133

0-2925893 0-2921975 0-2918072 0-2914185 0*2910313

379 380

I§* 5 2

+ + + + +

0-1805435 0-1857066 0-1908158 0-1958692 0-2008648

7-42 7-44 7-40 7-48 7-50

7*5^ 7'54 7 3 « 7-58 7 '60

+ + + + +

0-2635853 0-2607329 0-2577839 0-2547397 0-2516018

+ + + + +

0-1224652 0-1275545 0-1325793 0-I375379 0-1424285

0-2906457 0-2902616 0-2898790 0-2894979 0-2891183

384° 55' 12*72 ' 6?88 386 4 387° 13' 1*00 21' 388° 55*07 389 30 49*09

+ + + + +

0-2058008 0-2106753 0-2154865 0-2202325 0-2249115

7-52 7'54 7-56 7-58 7-60

7-62 7-64 7-66 7-68 7-70

+ 0-2483717

+ + + + +

0-1472494 0-1519988 0-1566751 0-1612765 0-1658016

0-2887401 0-2883635 0-2879883 0-2876146 0-2872424

39°! 391 392° 394° 395

39' 43*07 48' 37*00 57' 30*89

+ + + + +

0-2295219 0-2340620 0-2385299 0-2429241 0-2472429

7-62 7 6 7 7-68 7.70

772 774

+ + + + +

0-2308910 0-2271400 0-2233081 0-2193967 0-2154078

+ 0-1702488 + 0-1746164

+ 0-1789029

396 397° 398° 399° 4°o

24' 12*30 33' 6*02 41' 59*70 50' 53*33

+ 0-2514848 + 0-2556482 + 0-2597315

+ 0-1831070 + 0-1872272

0-2868715 0-2865021 0-2861341 0-2857676 0-2854024

7'72 7'74

7-76 7-78 7-80

+ 0-2637334 + 0-2676524

7-78 7-80

7-82 7-84 7-86 7-88 7.90

+ + + + +

0-2113430 0-2072042 0-2029932 0-1987118 0-1943618

+ + + + +

0-1912620 0-1952101 0-1990701 0-2028408 0-2065209

0-2850386 0-2846763 0-2843152 0-2839556 0-2835973

402 403° 404° 4°5° 406

+ + + + +

0-2714870 0-2752358 0-2788977 0-2824711 0-2859549

7-82

7-86 7-88 7.90

7-92 7-94 7-96 7-98 8-oo

4-

0-1899452

01 854639

+ + + + +

0-2101093 0-2136046 0-2170058 0-2203118 0-2235215

0-2832404

+ + + +

407° 53' 2 409

+ + + + +

0-2893479 0-2926488 0-2958566 0-2989700 0-3019881

7-92 7-94 7.96 8 z 9 8-00

'

+ 0-2450508 + 0-2416407 + 0-2381429

+ 0-2345591

0-1809198 0-1763147 0-1716508

-

02828848 02825305 0-2821776 0-2818259

26'

8*52 3-i6 35 43 57*75 52 52-29 1' 46-78

io' 19' 381 28' 382° 37' 383 46'

41*22 35"6i 29*96 24*26

6' 24*74

15

18*54

59 46*92 8' 40*47 17' 33*98

26 27*45 35 20*87 44' 14*26 7*6i

0*92 410 10' 54*19 4 JI ° x §' 47*41 4 1 2° 28 40*60

d

r7 6

7§4

-


X

Y

/iW

1

I.

681


(x)

iî'wi

7-02 7-04 7-06 7-08 7-10

+ + + + +

0-0013241 0-0073134 0-0132828 0-0192302 0-0251533

-

0-3022627 0-3017411 0-3011029 0-3003486 0-2994789

0-3022656 0-3018297 0-3013957 0-3009636

03005333

7-12 7-14 7-16 7-18 7-20

+ + + + +

0-0310498 0-0369177 0-0427547 0-0485586 0-0543274

-

0-2984943 0-2973957 0-2961837 0-2948590 0-2934226

7-22 7-24 7-26 7-28 7-30

+ + + + +

0-0600589 0-0657511 0-0714017 0-0770089 0-0825704

-

7-32 7*34 7-36 7-38 7-40

+ + + + +

0-0880844 0-0935488 0-0989617 0-1043211 0-1096251

-

7-42 7'44 7-46 7-48 7-50

+ + + + +

0-1148718 o- 1 200593 0-1251857 01 302494 0-1352484

7-52 7*54

0-1401811 0-1450456 0-1498404

7-58 7-60

+ + + + +

argtf, 270° 271° 272° 2 73°

H^*)

(x)

X

39 4»*37 274 48' 3*65

+ + + + +

0-3466429 0-3470978 0-3476699 0-3483585 0-3491631

7-02 7-04 7-06 7-08 7-10

0-3001049 0-2996704 0-2992536 0-2988307 0-2984096

275° 56' 19*08 2 7Z° 4' 34"69 278 12' 50*45 279 21 ' 6*37 280 29' 22*45

+ + + + +

0-3500830 0-3511175 0-3522659 0-3535275 0-3549013

7-12 7-14 7-16 7-18 7-20

0-2918752 0-2902177 0-2884511 0-2865763 0-2845944

0-2979903 0-2975727 0-2971569 0-2967429 0-2963306

281 ° 37' 38-69 282 45' 55*09 283° 54' 11*65 2' 28*36 285

+ + + +

286

10' 45*22

0-3563867 0-3579825 0-3596880 0-3615021 + 0-3634239

7-22 7-24 7-26 7-28 7-30

0-2825063 0-2803132 0-2780161 0-2756163 0-2731149

0-2959200 0-2951041 0-2946986 0-2942948

287° 288 289° 290 291 °

19' 27' 35' 43' 52'

+ + + + +

0-3654523 0-3675862 0-3698244 0-3721659 0-3746094

7*32 7-34 7-36 7-38 7-40

- 0-2705132 - 0-2678124

0-2938927 0-2934923 0-2930935 0-2926963 0-2923007

0' 29*54 293 294° 8' 47*44 295° 17' 5*48 296 25' 23*67 297° 33' 42-00

+ 0-3771537 + + + +

0-3797976 0-3825396 0-3853786 0-3883131

7-42 7-44 7-46 7-48 7-50

+ + + + +

0-3913417 0-3944630 0-3976756 0-4009779 0-4043684

7-52 7 54 7-56 7-58 7-60

+ 0-4078456 + 0-4114078 + 0-4150535

7-62 7-64 7-66 7-68 7.70

- 0-2650138 - 0-2621187 - 0-2591285

029551 12

15' 3*56 23' 18*33 31' 33-27

2*23 19*40 36*7i 54*17 11*78

0-2560446 0-2528684 0-2496015 0-2462451 0-2428010

0-2919068 0-2915145 0-2911237

0-1592138

-

0-2903469

298 42' 0-46 299 50' 19*07 300 58' 37*82 302°- 6' 56*70 303° 15' 15*72

7-62 7>6 i 7-66 7-68 7-70

+ 0-1637892 + 0-1682883 + 0-1727096 + 0-1770516 + 0-1813127

-

0-2392706 0-2356555 0-2319574 0-2281778 0-2243185

899609 0-2895764 0-2891935 0-2888I20 0-2884321

304° 305° 306° 307° 308

7-72 7-74

+ + + + +

0-1854916 0-1895868 0-1935970 0-1975208 0-2013569

- 0-2203810 - 0-2163672 - 0-2122788 - 0-2081175 - 0-2038851

0-2880537 0-2876768

0-2869274 0-2865549

5' 12*63 310 311° i 3 '32?58 312 21' 52*65 313° 30' 12*85 3M° 38' 33*17

+ + + + +

0-4264750 0-4304379 0-4344758 0-4385870 0-4427696

0-2051041 0-2087611 0-2123267 0-2157999 0-2191794

-

0-1995834 0-1952143 0-1907797 0-1862813 0-1817211

0-2861839 0-2858143 0-2854462 0-2850795 0-2847142

315° 316° 318° 319° 320

46' 53*62 55' 14*20 3 34-89

+ + + + +

0-4470219 0-4513419 0-4557279 0-4601780 0-4646902

7-82

7-86 7-88 7-90

+ + + + +

7-92 7-94 7-96 7-98 8-00

+ + + + +

0-2224642 0-2256533 0-2287457 0-2317403 0-2346363

-

0-1771010 0-1724229 0-1676888 0-1629007 0-1580605

0-2843503 0-2839878 0-2836267 0-2832670 0-2829087

321 28' 37*72 322° 3 6' 58-90 323 45' 20*20

+ + + + +

0-4692627 0-4738934 0-4785806 0-4833221 0-4881160

7*92 7 94 7-96 7-98 8-oo

Tb%

776 7-78 7-80 7-82

Tb

01545636

02907345 0-<2

O2873OI4

23' 34*87 31 54*16

40 13*58 48' 33*13 56' 52*82

55*71 20 16*66 11

/

324°53 4 I *62 326

2'

3-16

4-

0-4187811

+ 0-4225888

-

7'7 2

7-74 7-76 7-78 7-80

7-86 7-88 7-90

-


682

Table

7oW

X

I.

Y*{*)

+ + + +


iO)i

0-3049098 0-3077342 0-3104602 0-3130870 0-3156137

8-02 8-04 8-o6 8-o8 8-io

+ + + + +

0-3180394 0-3203635 0-3225852 0-3247036 0-3267183

8-12 8-14 8-i6 8-i8 8-20

+ + + + +

0-3286286 0-3304339 0-3321337 0-3337274 0-3352147

8-22 8-24 8-26 8-28 8-30

0-3365952 0-3378684 0-3390341 0-3400920 0-3410418

8-32 8-34

17 24*21 26' i6'<68

+ + + + +

0-2747321 0-2744075 0-2740840 0-2737617 0-2734405

436° 35' 9 -ii 437° 44 i;5J 438° 52' 53-88 46*22 r 440 441 io' 38*53

+ + + + +

0-3418834 0-3426166

0-2731204 0-2728015 0-2724836 0-2721669 0-2718513

442 443 444° 445° 446

19' 30*80 28' 23*04 37' 15*25

+ + + + +

0-3444644 0-3446550 o-3447373 0-3447114 o-3445775

8-52 8-54

0-2713818 0-2711977 0-2709056 0-2705060 0-2699992

0-2715368

448° 449° 450° 451° 452°

+ + + + +

0-3443357 0-3439865 o-343530i 0-3429669 0-3422972

8-62 8-64 8-66 8-68 8-70

0-2693857

0-2699803 0-2696722 0-2693651 0-2690591 0-2687541

453 48' 11*80 454° 57' 3*73 456° 5 55*64

0-3415216 0-3406404 + 0-3396543 + 0-3385638 + o-3373694

872

0-2684502 0-2681473 0-2678454 0-2675445 0-2672446

459° 32' 31*17 460 41' 22*95 461° 50' 14*71 462 59' 6*44 464° 7' 58-14

+ + + + 4-

0-3360719 0-3346718 0-3331700 0-3315672 0-3298642

8-82 8-84 8-86 8-88 8-90

0-2669458 0-2666479 0-2663510 0-2660552 0-2657603

465 16' 49*81 466 25' 41*45 467° 34' 33;07

+ + + + +

0-3280617 0-3261608 0-3241622 0-3220669 0-3198760

8-92 8-94 8-96 8-98 9-00

0-2814756 0-2811266 0-2807789 0-2804324 0-2800873

413° 414 4 I 5° 417 418

+ 0-1425423 + 0-1375223 + 0-1324598

+ 0-2407033 + 0-2432126 + 0-2456187

+ 0-1273568 + 0-1222153

4-

419 21' 58^95 420 30' .5 1*87 4 2I° 39/ 44*76 422 48' 37*62

+ 0-2501180

0-2797434 0-2794007 0-2790594 0^2787192 0-2783803

8-22 8-24 8-26 8-28 8-30

+ + + + +

0-1170375 0-1118256 0-1065816 0-1013077 0-0960061

+ + + + +

0-2522101 0-2541963 0-2560762 0-2578492 0-2595150

0-2780427 0-2777062 0-2773710 0-2770370 0-2767042

425 426 427° 428° 429

8-32

4-

8-36 8- 3 8 8-40

+ + + +

0-0906789 0-0853282 0-0799563 0-0745652 0-0691573

+ + + + +

0-2610730 0-2625230 0-2638647 0-2650977 0-2662219

0-2763725 0-2760421 0-2757128 0-2753847 0-2750578

430 431° 433° 434° 435

8-42 8-44 8-46 8-48 8-50

+ 0-0637345

+ + + + +

0-2672370 0-2681430 0-2689397 0-2696271 0-2702051

8-52 5i 8-56 8-58 8-6o

+ + + + +

0-0364752 0-0310094 0-0255440 0-02008I2 0-0146230

4-

0-2706738

+ + + +

02710333

8-62 8-64 8-66 8-68

+ 0-0091717 + OOO37293

4-

+ + + + 4-

+ + + + +

8-12 8-14 8-i6 8-i8 8-20

f

870

0-1669299 0*1621542 0-1573255 0-1524459 0-1475175

0-0582992 + 0-0528534 + 0-0473994 + 0-0419393 4-

- 0-OOI70I9 - 0-007I200 - 0-OI25227

8-8o

- 0-0179081 - 0-0232739 - 0-0286l82 - 0-0339388 - 0-0392338

8-82 8-84 8-86 8-88 8-90

-

872 §•74

876 878

8-92 8-94 8-96 8-98 9-00

-

4-

.

0-2479207

0-2712837 0-2714251 o-27i4577

+ 0-2686660 4-

0-^678405

+ 0-2669100 + 0-2658749

0-0445011 0-0497387 0-0549445 0-0601167

+ 0-2647360 + 0-2634939

O0652532

+ 0-2591558

0-0703522 0-0754116 0-0804295 0-0854042 0-0903336

4-

0-2621493

+ 0-2607030 + + + + +

X

+ + + + +

0-2266339 0-2296480 0-2325628 0-2353776 0-2380913

8-02 8-04 8-o6 8-o8 8-io

HW

argH^W

0-2575085 0-2557620 0-2539172 0-2519751 0-2499367

02712233 0-2709109 0-2705996 0-2702894

37' 33*75 46' 26*87 55' I9"94 4' 12*98

13'

5*98

423° 57' 30*43

4 5Z°

458

6' 23*21 15' 15*96 2 4 ' 8*67 33' 1*35

41' 53*99 50' 46*59 59' 39*i6 8'

31*70

7*43 54' 59*58

46

3'

51*69

12' 43*77

35*83 30 27*85 39 19*84 21

M

47*5i

23' 39*35

46»: 43 24 66 469 52 l6-22

03432414 o-3437576 0-3441653

4-

£' 3

S 8-38 8-40 8-42 8-44 8-46 8-48 8-50

I'

56 ,

8

%1 8-6o

4-

8-76 8-78 8-8o


X

Y

7iW

l

I.

683


(x)

HCtol

HiW

aig^Jw

X

8-02 8-04 8-o6 8-o8 8-io

+ + + + +

0-2374329 0-2401291 0-2427241 0-2452173 0-2476078

-

0-1531702 0-1482318 0-1432475 0-1382191 0-1331488

0-2825517 0-2821961 0-2818419 0-2814889 0-2811373

327 328 329 330° 33i

24*81 46*58 8*46 30-46 52*57

+ + + + +

0-4929604 0-4978531 0-5027922 0-5077756 0-5128012

812 8-14 8-i6 8-i8 8-20

+ + + + +

0-2498950 0-2520702 0-2541570 0-2561306 0-2579986

-

0-1280386 0-1228906 0-1177069 0-1124896 0-1072407

0-2807871 0-2804381 0-2800905 0-2797441 0-2793991

332 52' 14*79 0' 37*12 334 335° 8' 59*56 17' 22*11 336 337° 25' 44 T 78

+ + + + +

0-5178671 0-5229711 0-5281111 0-5332850 0-5384907

8-14 8-i6 8-i8 8-20

8-22 8-24 8-26 8-28 8-30

+ + + + +

0-2597605 0-2614159 0-2629644 0-2644056 0-2657393

-

0-1019624 0-0966569 0-0913261 0-0859723 0-0805975

0-2790553 0-2787128 0-2783716 0-2780316 0-2776929

338° 339° 340 o 341° 343°

34' 7 r55 42 3o r 42 50 53 ? 40 59' 16*49 7' 39*69

+ + + + +

0-5437262 0-5489893 0-5542779 0-5595898 0-5649229

8-22 8-24 8-26 8-28 8-30

8.32 §' 3 4

+ + + + +

0-2669651 0-2680829 0-2690924 0-2699936 0-2707863

-

0-0752040 0-0697937 0-0643690

0-2773554 0-2770191 0-2766841 0-2763503 0-2760177

344° 345 346 34 Z° 348°

16' 2*99 24' 26*39 32' 49 ?90

+ + + +

0-5702752 0-5756443 0-5810283 0-5864249 + 0-5918321

8-32 §'34 8-36 8-38 8-40

+ + + + +

0-2714704 0-2720460 0-2725131 0-2728717 0-2731220

-

0-0480290 0-0425676

0-2756864 o-2753562 0-2750272 0-2746994 0-2743728

349° 58' 351° 6; 352° 14 353 23', 354 °3i

+ + + + +

0-5972476 0-6026694 0-6080953 0-6135232 0-6189510

8-42

8-56 8*58 8-6o

+ + + + +

0-2732640 0-2732981 0-2732244 0-2730432 0-2727548

-

0-0097926 0-0043488 + 0-0010840

0-2740473 0-2737230 o-2733998 0-2730779 0-2727570

355° 356° 357 359° 360

56 50*46 5' 15*06 13' 39*74

+ + + + +

0-6243764 0-6297975 0-6352120 0-6406180 0-6460132

8-58 8-6o

8-62 8*64 8-66 8-68 8*70

+ + + + +

0-2723596 0-2718580 0-2712504 0-2705372 0-2697190

+ + + + +

0-0119084 0-0172953 0-0226640 0-0280110

0-2724373 0-2721187 0-2718012 0-2714849 0-2711696

361 22' 4*52 362 30' 29*39 363° 38' 54*35 364 47' 19*41 365° 55 44*57

+ + + + +

0-6513957 0-6567633 0-6621139 0-6674455 0-6727561

8-62 8-64 8-66 8-68 8-70

872

+ + + + +

0-2687964 0-2677699 0-2666402 0-2654079 0-2640737

+ + + + +

0033334 6 0-0386328

0-2708555 0-2705424

9*81

00439037

02702305

0-0491453

00543556

0-2699196 0-2696098

367° 368° 369° 370 37i°

+ + + + +

0-6780436 0-6833060 0-6885413 0-6937475 0-6989226

8-72 8*74 8-76 8-78 8-8o

+ 0-262638^ + 0-2577339 + 0-2559024

+ + + + +

0-0595326 0-0646744 0-0697790 0-0748447 0-0798694

0-2693011 0-2689934 0-2686868 0-2683813 0-2680768

372 373 375° 376° 377

20'

0*98

+ + + + +

0-7040647 0-7091718 0-7142420 0-7192734 0-7242641

8-82 8-84 8-86 8-88 8-90

+ + + + +

+ + + + +

0-2677733 0-2674709 0-2671694 0-2668691 0-2665697

378 28' 379° 36' 380 45' 38i° 53'

27*10 53*3° 19*59 45-96

+ + + + +

0-7292122 0-7341159 0-7389734 0-7437828 0-7485424

8-92

Pi 8.38 8-40 8-42

8-46 8-48 8-50 8.52

§•74 8-76 8-78 8-8o 8-82 8-84 8-86 8-88 8-90

8-92 i* 9 4

8-96 8-98 9»oo

+ 0-2611028 + 0-2594677

0-2539740 0-2519497 0-2498306 0-2476176 0-2453118

00589319 00534845

00371023 00316333

- 0-0261687

0-0207046

00152452

00065038

0-0848513 0-0897886 0-0946795 0-0995220 0-1043146

383

10' 18' 27' 35' 43'

*K

1

^l* 49 37*22 1*03

24*94 48*95 13*05 37 r 26

40' 1*56 48' 25*97

4

'

12' 35*14 21' 0*57 29' 26*08 37' 51*68

46' 17*36 54' 43*14 3, 9*oo 11 34*95

2' 12*41

8-02 8-04

806 8-o8 8-io

812

%\% 8- 8 4 8-50

8-52

tu

9*

i* 896

8-98 9-6o

2

1


684

Table

/oW

X

I.

Y*(*)


l«?(*)l.

4?I o 472° 473° 474° 475

j/

0-3175904 0-3152111 0-3127393 0-3101761 0-3075226

9-02

0-2628641

0-3047800 0-3019495 0-2990324 0-2960300 0-2929435

9-12

480 11' 59*05 481° 20' 50*35

+ + + + +

0-2214700 0-2183644 0-2151782 0-2119130 0-2085701

0-2625796 0-2622961 0-2620135 0-2617318 0-2614510

482 29' 41*62 483° 38' 32*86 484° 47 24*08 485° 56 15-28 487° 5' 6*44

+ + + + +

0-2897743 0-2865238 0-2831934 0-2797845 0-2762985

9-22 9-24 9-26 9-28 9-30

+ 0-2051510 + 0-2016573

0-2611711 0-2608921 0-2606140 0-2603368 0-2600604

488 489 490 491° 492

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932

0-1869673 0-1831240 0-1792157 0-1752440 0-1712106

0-2597850

493° 58' 12*95 495° 7, 3*94

0-2538498 0-2498675 0-2458208 0-2417112 0-2375406

9-42 9-44 9-46

+ + + + +

0-2117244 0-2143797 0-2169439 0-2194161 0-2217955

-

0-0952160 0-1000496 0-1048325 0-1095629 0-1142392

+ + + + +

0-2478029 0-2455748 0-2432536 0-2408402 0-2383360

0-2654663 0-2651734 0-2648814

9-12 9-14 9-16 9-18 9-20

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9-22 9-24 9-26 9-28

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0-1616274 0-1655265 0-1693511 0-1730999 0-1767716

9-42 9-44 9-46 9.48 9*5°

- 0-1803648 - 0-1838783 - 0-1873109 - 0-1906615 - 0-1939287

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952 9-54 9'56 9-58 9-60

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0-1971117 0-2002092 0-2032202 0-2061437 0-2089787

9-62 9-64 9-66 9-68 9.70

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9.72 9-74 9-76

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934

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0-2584206 0-2581503 0-2578809 0-2576123 0-2573445

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0-2333107 0-2290231 0-2246796 0-2202821 0-2158322

952 954

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0-1458137 0-1413982 0-1369352 0-1324268 0-1278748

0-2570775 0-2568114 0-2565461 0-2562816 0-2560180

505° 5o6° 507° 508°

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0-2113319 0-2067829 0-2021871 0-1975464 0-1928625

9-62

53 14*12 510 2' 4*82

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0-2240814 0-2262730 0-2283698 0-2303710 - 0-2322760

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0-1232810 0-1186475 0-1139760 0-1092686 0-1045271

0-2557551 0-2554931

02552318 0-2549714 0-2547117

5II IO 5 49 512 l 19' 46*15 513 28' 36-79 5 r 4° 37' 27*40 515 46' 18-00

+ + + + +

0-1881375 0-1833732 0-1785715 0-1737344 0-1688637

9-72 9-74 9-76 9-78 9-80

9-82 9-84 9-86 9-88 9.90

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5^:55; 8*57

518° 3 59*12 519° 12' 49-65 520°2l' 40*15 52 I ° 30' 30*64

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0-0949498 0-0901178 0-0852597 0-0803773

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- 0-2374090 - 02389243 - 0240341

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9-82 9-84 9-86 9-88 9.90

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+ + + + +

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0-2416590 0-2428777 0-2439968 0-2450163

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0-2429143 0-2404263 0-2378489 0-2351833 0-2324307

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0-1090553 0-1137425 0-1183744 0-1229495 0-1274659

0-2662713 0-2659740 0-2656776 0-2653823 0-2650879

384 io' 38*94 3|5° 19' 5-56 386 27' 32*26 387^35'59'o 4 388 44' 25*90

+ + + + +

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9-02

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9-12 9-14 9-16 9-18 9-20

+ + + + +

0-2295925 0-2266698 0-2236640 0-2205765 0-2174087

+ 0-1319221 + 0-1363164 + 0-1449133 + 0-1491128

0-2647945 0-2645021 0-2642107 0-2639202 0-2636307

389° 52' 52*84 391° i' 19-86 392° 9' 46 -95 393 18' 14*12 394° 26' 41*37

+ + + + +

07759567 07803195 07846192 07888540 07930226

9-12 9-14 9-16 9-18 9-20

9. 22

+ + + + +

0-2141618 0-2108375 0-2074370 0-2039620 0-2004139

+ + + + +

0-2633421 0-2630545 0-2627679 0-2624822 0-2621974

395° 35' 8*70 396° 43' 36*11 397° 52' 3*59 399 o' 31*15 400 8' 58*78

+ + + + +

07971234 0-8011549 0-8051156 0-8090043 0-8128195

9-22 9-24 9-26 9-28 9-30

+ + + + +

0-1967943 0-1931047 0-1893468 0-1855221 0-1816322

+ 0-1728314 + 0-1765251 + 0-1871357

0-2619135 0-2616306 0-2613486 0-2610675 0-2607873

401 ° 402° 403° 404 4 05

34 22*13 42 50*06 5i'i8*o6

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0-8165598 0-8202240 0-8238107 0-8273187 0-8307469

9.32 9-34 9-36 9-38 9-40

+ + + + +

0-1776789 0-1736637 0-1695884 0-1654548 0-1612644

+ + + + +

0-1905116 0-1938050 0-1970150 0-2001403 0-2031799

0-2605080 0-2602297 0*2599522 0-2596756 0-2593999

406 59' 46*14 408 8' 14*28 409 ° 16' 42*50 410 25' 10*79 4"° 33' 39-15

+ + + + +

0-8340939 0-8373587 0-8405401 0-8436371 0-8466485

9H2

+ + + + +

0-1570192 0-1527208 0-1483711 0-1439718 0-1395248

+ + + + +

0-2061329 0-2089982 0-2117749 0-2I4402I 0-2170590

0-2591250 0-2588511 0-2585780 0-2583058 0-2580344

412° 42' 7*58 413° 5o' 36*08

+ + + + +

0-8495735 0-8524110 0-8551601 0-8578198 0-8603894

9-5£ 9-58 9-60

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0-1350319 0-1304950 0-1259159 0-1212965 0-1166386

+ + + + +

0-2195646 0-2219783 0-2242992 0-2265267 0-2286600

0-2577639 0-2574943 0-2572255 0-2569576 0-2566904

418 24' 30*76 4i9°32'59"6i 420 41' 28*52 422 58' 26*53

+ + + + +

0-8628679 0-8652546 0-8675487 0-8697495 0-8718563

9-62 9-64 9-66 9-68 9.70

+ + + + +

0-1119443 0-1072154 0-1024537 0-0976613 0-0928401

+ + + + +

0-2306986 0-2326417 0-2344889 0-2362395 0-2378932

0-2564242 0-2561587 0-255894 1 0-2556303 0-2553673

424° 6' 55*64 425° 15' 24*82 426 23' 54*06 427° 32' 23^36 428 40' 52*73

+ + + + +

0-8738685 0-8757855 0-8776066 0-8793314 0-8809594

9-74 9-76

+ + + + +

0-0879920 00831 189 0-0782229 0-0733059 0-0683698

+ + + +

0-2394495 0-2409079 0-2422681 0-2435297 + 0-2446924

0-2551052 0-2548438 0-2545833

43<>° 57'

429 49' 22*16 52*66 6' 21*22 432 433°i4'5o"8 4 434 23' 20*52

+ + + + +

0-8824901 0-8839230 0-8852579

0-8876322

9-82 9-8 4 9-86 9-88 9-90

+ + + + +

0-0634167 0-0584484 0-0534670 °'°484745 -0434727

+ + + + +

0-2538064 0-2535491 0-2532925 0-2530367 0-2527816

435°3i'50 ? 27 436 40' 20*08

+ + + + +

0-8886710 0-8896106 0-8904508 0-8911914 0-8918325

9.92 9-94 9-96 9.98 IO-OO

9-02

904 9-06

908

9-24 9-26 9-28 9.30

932 9'34 9-36 9-38 9'4° 9-42 9-44

946 948 9'5°

952 9-54 9-50 9-58 9-60 9-62 9-64 9-66 9-68

970 9.72 9-74

976 978 9-80 9-82

9 'h

9-86 9-88 9.90 9.92

9-94

996 9.98 IO-OO

4-

0-1406474

o- 1532443

0-1573063 0-1612974 0-1652162 0-1690613

+ 0-1801413 + 0:1836785

0-2457560 0-2467203 0-2475850 0-2483501 0-2490154

02543235 0-2540646

17' 26*49 25' 54*27

416°

7'33*2§ 4i7°i6' 1*99

4 2I °

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437 48' 49*95 438° 57' 19-88 440° 5' 49*87

08864944

904 9-06 9-10

9-44 9-46 9-48 9-50 9-52 9*5j

972

978 9-80


686

Table

X -

0-2467551 0-2474743 0-2480931 0-2486116 0-2490297

IO-I2 10-14 IO-IO 10-18 10-20

-

0-2493474 0-2495649 0-2496822 0-2496996 0-2496171

!«?(*)

+ 0-0506850 + 0-0456806

0-2519069 0-2516564 0-2514068 0-2511578 0-2509096

4-

0-0406838

+ 0-0356728 + 0-0306574 + 0-0256397 + 0-0206216 0-0156052 0-0105924 + 0-0055852

4-

4-

-

0-2494350 0-2491536 0-2487732 0-2482942 0-2477108

4-

0-2470416 0-2462690 0-2453994 0-2444335 0-2433718

-

0-0242303 0-0291435

10-40

-

10-42 10-44 10-46 10-48 10-50

-

0-2422148 0-2409633 0-2396178 0-2381792 0-2366482

-

1052

- 02350255 - 02333120 - 0-2315085 - 0-2296158 - 0-2276350

-

10-70

-

10-72 10-74 10-76 10-78 io-8o 10-82 10-84

IO-22

IC24 IO-2D IO-28

IO3O


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10-02 10-04 IO-OO 10-08 IO-IO

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0-0005856 0-0044044 0-0093830 0-0143481

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1

arg H^(x)

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528° 23' 33*15 529 32' 23*49

+ 0-1136338 + 0-1085142

53i° 50' 4'i3 532° 58 54'4i

4-

10-02 10-04 10-06 10-08 IO-IO

530°4i'i 3 ^2

0-2506622 0-2504154 0-2501694 0-2499241 0-2496795

534° 7' 44 -68 535° 16 34*93 536° 25 '25*16

0-2494357

539° 5* '55*73 541° 0' 45*88 542 9' 36*02 543 18' 26*13 544 27' 16*23

02491925 0-2489501 0-2487084 0-2484674

537°34'i5'37 538° 43 5'56

0-0389045 0-0437486

0-2482270 0-2479874 0-2477484 0-2475102 0-2472726

545° 546° 547° 549 o

0-0485659

0-2470357

00533546

02467995

0-0581128

0-1033867 0-0982533 + 0-093II62 4-

0-0879771 0-0828382 + 0-07770I4 4- 0-0725687 + 0-0674420

IO-I2 10-14 10-16

0-0623234 0-0572148 O-O52II0I + 0-0470353 4- 0-0419684

IO-22 10-24 IO-26 IO-28 IO-30

4-

4-

4-

44-

OO369I92

IO-32

1034 1036 1038

0-OI20062 0-007I034 0-00223I5 - 0-0026077 - 0-0074I23

10-42 10-44 10-46 10-48

-

0-0121806 0-OI69I08 0-02I60I2 0-0262499 0-0308553

10-52 10-54 10-56 10-58 IO-OO

- 0-0354157 - 0-0399295 - 0-0531741

10-62 10-64 io-66 io-68 10-70

- 0-0574847 - 0-0617406 - 0-0659402 - 0-070082I - 0-0741646

10-72 10-74 10-76 10-78 io-8o

-

0-0781864 0-0821461 0-086042I 0-0898731 0-0936378

10-82

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OO973349

10-92 10-94 10-96 10-98 II-OO

4-

44 56*37 53 46*41 2 3 44 £I 550 11' 26*44

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0-0675304

0-2465640 0-2463291 0-2460949

55i°2o' 16*43 552° 29' 6*41 553° 37 5<>"36 554* 46' 46*30 555 55 36 ? 22

0-0721862 0-0768043 0-0813830 0-0859205 - 0-0904152

0-2458614 0-2456285 0-2453963 0-2451648 0-2449339

557° 558 559° 560° 561 °

4'26?i2

0-2255670 0-2234127 0-2211732 0-2188495 0-2164427

-

0-0948652 0-0992689 0-1036247 0-1079309 0-1121859

0-2447037 0-2444741 0-2442451 0-2440168 0-2437891

562° 563° 565° 566° 567

-

0-2139539 0-2113843 0-2087349 0-2060071 0-2032020

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0-1163881 0-1205360 0-1246281 0-1206627

0-2435621 0-2433357 0-2431099 0-2428847 0-2426602

568 32' 44*20 569° 41' 33*91 570 50' 23*61 57 I °59'i3 ? 30 573 8' 2*97

0-2003208

Q-I943357 0-1912343 0-1880622

-

0-2424363 0-2422130

io-88 10-90

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574° 575° 576° 577°

10-92 10-94 10-96 10-98 II'OO

- 0-1848208 - 0-1815115 - 0-1781356 - 0-1746947 - 0-1711903

-

1038

10-54 10-50 10-58

1060 10-62 10-64 io-66

1068

1086

01973650

00340355

00628386

01326384 0-1365537

01 404073

02419903

0-1441977 0-1479234 0-1515832

0-2417683 0-2415468

0-1551757 0-1586996 0-1621537 0-1655367 0-1688473

0-2413260 0-2411057 0-2408861 0-2406671 0-2404486

13 16*00 22' 5*87 30' 55*72 39' 45*55 48' 35*37

57 25*17 6' 14*95 15 4*71 23' 54*46

16' 52*62 25; 42 -26

34 31-88 43' 21*48

578°52'n*07 580 581

1'

0*64

9' 50*20 18' 39*75 27' 29*28

582 583 584 36' 18*79

10-20

0-0318896 + 0-02688I7 4- 0-02I8972 + O-OI6930I

36' 6*31

IO-32 10-34 10-36

1018

44-

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0-1009630 0-I045209 0-1080073 0-III42IO

10-40

1050

1084 io-86 io-88 10-90


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IO-I2 10-14 IO-IO 10-18 10-20

+ 0-0133801 + 00083694 + 0-0033652

IO-22 IO-24 IO-26 IO-28

-

IO3O

0-0384638 0-0334497 0-0284322

00234135 00183955

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0-0115886 0-0165471 0-0214895 0-0264137 0-0313178

I.

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0-2495809 0-2500465 0-2504122 0-2506782 0-2508444

0-2525274 0-2522739 0-2520212 0-2517692 0-2515180

44i°i4'i9'93 442 22' 50*04 443 31' 20*21 444°39'5o'44 445 48' 20*73

+ + + + +

0-2509111 0-2508783 0-2507464 0-2505154 0-2501858

0-2512676 0-2510179 0-2507690 0-2505208 0-2502733

446 56' 51*08 448° 5' 21-49 449° 13, 51 "95 450 22 22*48

+ 0-2497578

0-2500266 0-2497806 0-2492907 0-2490469

452° 453° 454° 456° 457

0-8923738 0-8928155 0-8931574 0-8933996 0-8935423

[0-02 [0-04 to-oo 10-08 [O-IO

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08935856

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0-8935295 0-8933744 0-8931204 0-8927679

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0-8923172 0-8917685 0-8911224

13' 26*80

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45i°3o'53 ?o6 39' 23*69 47' 54*39 56' 25*14

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0-8886036 0-8875722 0-8864459 0-8852252 0-8839107

[0-32 [0-34 [0-36 10-38 10-40

464° 4'33 ? i2 465° 13 4*36 466° 21 35*65 467°3o' 7-00 468 38' 38*40

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47' 9 r 8 5 55' 4i*35 4' 12*91 12 44*52 21' 16*18

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475° 29' 47*89 476° 38 19-65 477° 46 51*46 55 23*32 *lK 4«o° 3 '55'23

+ 0-8634870 + 0-8561136 + 0-8534944

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0-8507964 0-8480203 0-8451689 0-8422418 0-8392408

[0-72 [o-74 10-76 [0-78 co-8o

0-1942349 0-1911093 0-1879138 0-1846498 0-1813185

0-2429492 0-2427236 0-2424986 0-2422742 0-2420505

55; 7-71 4fto 488° 3 39-96 489 12 12-26 490 20' 44*61 491 29' 17*00

+ 0-8361673 + 0-8330226 + 0-8298080

CO-82 [0-84 to-86 co-88 [0-90

0-1779215 0-1744602 0-1709362 0-1673507 0-1637055

0-2418273 0-2416048 0-2413829 0-2411616 0-2409410

492° 37' 49 -43 493 46 21*92 494°54'54 ?45 496° 3 27*03 497 11 59 T 65

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0-2492319 + 0-2486082 + 0-2478874 + 0-2470699 4-

02495353

4,55*94

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0-2461562 0-2451468 0-2440423 0-2428434 0-2415506

0-2488038 0-2485614 0-2483197 0-2480787 0-2478385

45 8 ° 21' 57*72 459° 30' 28-69 460 38' 59*71 461° 47' 30*79 462 56' 1*93

1050

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0-0602176 0-0649296 0-0696068 0-0742475 0-0788500

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0-2401646 0-2386862 0-2371162 0-2354552 0-2337042

0-2475989 0-2473600 0-2471218 0-2468843 0-2466475

10-52 10-54 10-56 10-58 10-60

-

0-0834125

+ + + +

0-2318640 0-2299355 0-2279195 0-2258171 + 0-2236293

0-2464114 0-2461760 0-2459412 0-2457071 0-2454736

469° 47o° 472° 473° 474

10-62 10-64 io-6o io-68 10-70

- 0-1055659 - o- 1098532 - 0-1140889 - 01182715 - 01 223994

+ + + + +

0-2213570 0-2190013 0-2165633 0-2140441 0-2114448

0-2452409 0-2450088 0-2447773 0-2445465 0-2443164

- 0-1264711 0-1304852 0-1344401

+ + + + +

0-2087666 0-2066107 0-2031783 0-2002707 0-1972891

0-2440869 0-2438581 0-2436299 0-2434024

1090

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o- 1496394 0-1532774 0-1568479 0-1603497

+ + + + +

10-92 10-94 10-96 10-98 ii-oo

- 0-1637815 - 0-1671422 - 01 704305 - 01 736452 - 0-1767853

+ + + + +

1038 10-40 10-42 10-44 10-46 10-48

10-72

1074 10-76 IO-78 io-oo 10-82 10-84 io-86 io-88

- 0-0362001 - 0-0410586

-

00879333 0-0924107 0-0960431 0-1012287

01 383343 0-1421666

01459354

H,M + + + + +

+ + + + +

IO-32 10-34 10-36

687

08903793

+ o-86li 1 10 + 0-8586529

+ 0-8265250

+ 0-8231749 0-8197592 0-8162792 0-8127366 0-8091327 0-8054691

10-92 10-94

[096 [0-98 :i-oo


688

Table

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5^45,' 8*29

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0-1676238 0-1639968 0-1603109 0-1565675 0-1527683

-

0-1720845 0-1752470 0-1783338 0-1813437 0-1842758

0-2402308 0-2400135 0-2397968 0-2395807 0-2393652

11-12 11-14 ii-i6 ii-i8 II-20

-

0-1489149 0-1450089 0-1410520 0-1370458 0-1329919

-

0-1871289 0-1899021 0-1925944 0-1952050 0-1977329

02391503

11-22 11-24 11-26 11-28

-

0-1288922 0-1247483 0-1205618 0-1163346 0-1120685

- 0-2001772 - 0-2025372 - 0-2048121 - 0-2070011 - 0-2091034

0-2380843 0-2378728 0-2376618 0-2374514 0-2372416

II-02

WZt n-o8

1130 11-32

\m 1 1 -38

n

-

4o

H-42 \\.\\ 11.48 11-50 11-52

\m 11-58

n-6o 11-62

w-it u-68 11-70

-

0-2389359 0-2387222 0-2385090 0-2382963

57-77 5fto 53 588° 2 47*23 589 11' 36^69 590°2o' 26*13

11-22

-

0-1552926 0-1573169 0-1610982 0-1628541

0-1645196 0-1660942 0-1675773 0-1689687 0-1702681

11-42

8' 2*73 16' 51*89

-

n-52

22*44 11*78 1*10 50*41

39*70

0-2370323 0-2368236 0-2366155 0-2364078 0-2362008

602 604 605° 606 607

- 0-0857517 - 0-0812623 - 0-0767484 - 0-0722117 - 0-0676539

-

0-2198634 0-2359942 0-2213425 -0-2357882 0-2227306 0-2355828 0-2240273 0-2353779 0-2252321 0-2351735

608 609 6io° 612 613

-

0-0630771 0-0584830 0-0536735 0-0492505 0-0446157

-

0-2263449 0-2273652 0-2282930 0-2291278 0-2298697

0-2349696 0-2347663 0-2345635 0-2343612 0-2341595

614 25' 41*04 615 34' 30*17 616 43' 19*28 6i 7 52' 8*39 619° 0' 57*48

-

0-1714749 0-1725891 0-1736103 0-1745384 o- 1 75373 1

0-0399711 0-0353184 0-0306597 0-0259967 - 0-0213313

-

0-2305185 0-2310740

0-2319050

0-2339582 0-2337575 0-2335573 0-2333576

C2321806

02331584

620 621° 622 623 624

46*56 35*63 24*68 13*72 45' 2*75

-

0-1761144 0-1767622 0-1773163 0-1777768 0-1781436

-

0-2323620 0-2324526 0-2324481 0-2323513 0-2321618

02329598 0-2323668 0-2321701

625°53'5i"76 2 40*77 627 628 11' 29*75 629 20' 18*73 63O 29' 7*70

-

0-1784169 0-1785968 0-1786832 0-1786765 0-1785768

-

0-1783843 0-1780994 0*1777222 0-1772532 0-1766928

-

0-1760413 0-1752992 0-1744669 0-1735451 0-1725341

57' 28*98 6' 18*25 15' 7*51 23' 56*75

32' 45*97 41' 35*18 50' 24*38 59' 13*56

9' 18' 27' 36'

0-0166653 0-0120006 0-0073391 0-0026825 + 0-0019672

11-82

1 1 -90

+ + + + +

0-0066082 0-0112388 0-0158571 0-0204612 0-0250494

-

0-2318798 0-2315056 0-2310396 0-2304820 0-2298332

0-2319740 0-2317783 0-2315831 0-2313884 0-2311942

631° 37' 56-65 632° 46 45*59 6 33° 55 34*52 635° 4 23*44 636 13' 12*34

11-92

+ 0-0296200 -+

-

0-2290937 0-2282638 0-2273441 0-2263351 0-2252373

0-2310005

n-94

637° 638 639° 640 641

1 1

-96

11-98 I2-00

01531801

0-2111185 0-2130457 0-2148843 0-2166338 0-2182937

0-0341710 + 0-0387007 + 0-0432074 + 0-0476893

0-2327616

02325639

02308073 0-2306146

02304223 0-2302306

11-08 II-IO

0-1438608 0-1463196 0-1486929 0-1509799

13' 22' 31' 39' 48'

-

02315362

\\&

-

593° 46' 54*35 594° 55, 43 r 73 590° 4 33-°9

0-1077650 0-1034261 0-0990535 0-0946491 0-0902145

-

II-02

11-12

11-72 11-74 11-76 11-78 II-OO

n-88

-

0-1147608 0-1180257 0-1212144 0-1243260 0-1273593 0-1303133 0-1331870 0-1359795 0-1386899 0-1413173

-

Hit

-

X

-

59i°29'i5 ?55 592 38' 4*96

597° 598 599 6oo° 601 °

HoW

22' 30' 39' 48' 57'

1*24 50*12 38 -99 27*84 16*69

01592522

\\-\l ii-i8 II-20

1 1

-24

11-26 11-28 II-30 11-32

%%

11-38 II-40

n-44 11-46 1 1

-48

1150

mt 11-58

n-6o 11-62 11-64 11-66

n-68 11-70 11-72

n-74 11-76

\\t 11-82

wit n-88 11-90 11-92

W-tt 11-98 12-00

1


X

i\0)

7iW -

0-1798496 0-1828371 0-1857467 0-1885774 0-1913283

+ + + + +

11-18 II-20

-

0-1939984 0-1965863 0-1990926 0-2015150 0-2038531

+ + + +

11-22 II-2 4 II-26 11-28 II-30

-

11-32

II-02

\1% 11

08

II-IO 11-12

III4 IIl6

n-34 11-36 11-38

1140 11-42 11-44 11-46 11

-48

1150 11-52

n-54 1 1 -56

1158 ii-6o

1162

\!U

n -68

11-70 11-72

"'%

ml 11-82

wit n-88 11

90

1192

l!U 11-98 I2'00

I.

o- 1 60002

0-1562419 0-1524266 0-1485578 0-1446371

689

Functions of order unitv

\H

(x)\ 1

arg

0-2407209 0-2405014 0-2402826 0-2400643 0-2398466

498° 499° 500° 501 ° 502

0-2396296

504° 505 506 507 508°

H

j

HW

(x)

32*32 5*03 37 '-79 10*59 43*44

+ + + + +

0-8017474 0-7970691 0-794I357 0-7902489 0-7863103

3'i6*34

+ + + + +

0-7823215 0-7782842 0-7742001 0-7700707 0-7658979 0-7616832 0-7574286 o-753i356 0-7488060 0-7444416

11-22 II-2 4

n-32 n-34

20' 29' 37' 46' 54'

+

0-1406661 0-1366465 0-1325799 0-1284681 0-1243127

0-2391972 0-2389819 0-2387671

0-2061063 0-2082738 0-2103549 0-2123488 0-2142550

+ + + + +

0-1201154 0-1158779 0-1116021 0-1072896 0-1029422

0-2385530 0-2383394 0-2381264 0-2379140 0-2377021

509°46' 1*46 5io°54'3 4 '-6i 512° 3' 7-8i 513 11 41-05 514 20 14*33

+ + + + +

-

0-2160729 0-2178019 0-2194415 0-2209912 0-2224506

+ + + + +

0-0985617 0-0941498 0-0897083 0-0852391 0-0807440

0-2374909 0-2372801 0-2370700 0-2368604 0-2366513

515° 28' 47*65 516° 37' 21*01 ? 5 I Z°45'54 42 518° 54 520 3' 1*36

+ + + + +

0-7400442 0-7356155 o-73ii573 0-7266715 0-7221598

-

0-2238192 0-2250966 0-2262825 0-2273766 0-2283786

+ + + + +

0-0762247 0-0716830 0-0671209 0-0625402 0-0579425

0-2364428 0-2362349 0-2360275 0-2358207 0-2356144

521

n'34*89

522°20' 8*46 523° 28' 42*07 524° 37' 15*72 525 45 49"4i

+ + + + +

0-7176241 0-7130661 0-7084877 0-7038907 0-6992769

0-2292883

0-0533299 0-0487042 0-0440671 0-0394206 0-0347665

0-2354086 0-2352034 0-2349987 0-2347946 o- 2 3459io

526° 54' 23*14 2' 56*91 528 529 11' 30*72 530 20' 4*58 531° 28' 38*47

+ + + + +

0-6946483 0-6900066 0-6853536 0-6806913 0-6760215

02394131

X

X

ii' 49^27

2o'22-25 28' 55*28 37' 28*35

27%

I I

-02

1 1

04

n-o6 n-o8 II'IO 11-12

;;:;* ii-i8 II-20

11-26 11-28 II- 3

11.36 11-38 II- 4 o

1142 11-44 11-46 1 1

-48

11-50

-

0*2320005

+ + + + +

-

0-2324463 0-2327992 0-2330591 0-2332261 0-2333002

+ + + + +

0-0301065 0-0254427 0-0207768 0-0161106 0-0114460

0-2343879 0-2341854 0-2339833 0-2337818 0-2335809

532 37' 12*39 533° 45' 46-3° 534° '54 20*37 536° 2 54*41 537 ii' 28*49

+ + + + +

0-6713459 0-6666665 0-6619851 0-6573035 0-6526236

-

0-2332817 0-2331707 0-2329674 0-2326720 0-2322847

+ 0-0067849 + 0-0021209 - 0-0025199 - 0-0071598 - 0-0117890

0-2333804 0-2331804 0-2329810 0-2327821 0-2325837

538 20' 2*61 539° 28' 36*77 540 37' 10*96 54i°45'45'i9 542° 54 I9 ?46

+ + + + +

0-6479472 0-6432761 0-6386122 0-6339572 0-6293131

11-72 11-74 11-76 11-78 II-OO

-

0-2318060 0-2312361 0-2305754 0-2298243 0-2289832

-

0-0164057 0-02IOOOI 0-0255944 0-0301628 0-0347115

0-2323858 0-2321884 0-2319915 0-2317951 0-2315993

544° 2'53'77 545° 11 28*11 546°2o' 2*49 547 28' 36*91 548 37' 11*36

+ + + + +

0-6246815 0-6200643 0-6154633 0-6108802 0-6063169

11-82 Ii.84

- 0-2280528 - 0-2270334

-

0-0392388 0-0437430 0-0482223 0-0526749 0-0570992

02314039

549° 45' 45-85 550° 54, 20*37 552° 2 54*93 553° 1 1' 29*53 554 20 4*16

+ + + + +

0-6017751 0-5972565 0-5927628 0-5882959 0-5838573

11-92

02301055 0-2308300

02314617

- 0-2259255 - 0-2247299 - 0-2234471

0-2312090 0-2310146 0-2308207 0-2306273

11-52

n-54 11-56

n-58 n-6o 11-62 ll-6 4

n-66 H-68 11-70

n-86 n-88 1190 n-94 11-96 11-98 I2-00


690

Table

X

Y

;.w

I.


(x)

i^Cwi 02300393

axgH

HoW

(x)

X

0-2298485 0-2296581 0-2294682 0-2292788

643° 6' 5*52 644° M' 54-35 645° 23 43-16 646 32' 31*96 647 41' 20*74

-

0-1714348 0-1702475 0-1689731 0-1676121 0-1661654

12-02 12-04 12-06 12-08 I2-IO

0-2114698 0-2095218

0-2290899 0-2289014 0-2287134 0-2285259 0-2283388

648°5o' 9*52 649 58' 58*29 651° 7'47'-°4 652 16' 35-78 653 25' 24^51

-

0-1646336 0-1630175 0-1613180 0-1595359 0-1576720

12-12 I2-IA 12-16 12-18 12-20

-

0-2074933 0-2053852 0-2031985 0-2009341 0-1985931

0-2281522 0-2279660 0-2277803 0-2275950 0-2274102

654° 34' 13 -23 655° 43 i*94 656 51' 50*64 658 o' 39*32 659 9' 28-00

-

0-1557274 0-1537028 0-1515994 0-1494180 0-1471598

12-22 12-24 12-26 12-28 I2-30

0-1146576 0-1184651 0-1222191 0-1259182 0-1295610

-

0-1961765 0-1936855 0-1911210 0-1884844 0-1857766

0-2272258 0-2270419 0-2268585 0-2266754 0-2264928

66o° 18' 16*67 66i°27' 5-32 662 35' 53*96 663 44' 42*60 664 53' 31*22

-

0-1448257 0-1424169 0-1399344 0-1373794 0-1347532

12-32 12-34 12-36 12-38 12-40

0-1331462 0-1366724 0-1401382 0-1435426 0-1460841

- 0-1829990

-

01320567

0-1801527 0-1772390 0-1742591 0-1712143

0-2263107 0-2261290 0-2259477 0-2257669 0-2255865

666'

-

0-1292914 0-1264583 0-1235588 - 0-1205943

12-42 12-44 12-46 12-48 12-50

-

0-1681060 0-1649354 0-1617040 0-1584131 0-1550641

0-2254065 0-2252270 0-2250479 0-2248692 0-2246909

67I 672 6 74° 675°

-

0-1175659 01 144750 0-1113230 0-1048412

12-52 12-54 12-56 12-58 12-60

12-02 12-04 I2-00 12-08 I2-IO

+ + + + +

0-0521447 0-0565718 0-0609690 0-0653346 0-0696668

-

0-2240513 0-2227778

12-12 12-14 I2-I6 12-18 12-20

+ + + + +

0-0739640 0-078224^ 0-0824468 0-0866292 0-0907701

-

0-2168214 0-2151204

12-22 12-24 12-26 12-28 12-30

+ .0-0948680 + 0-0909212 + 0-1029283 + 0-1068877 + 0-1107980

12-32 12-34 12-36 12-38 12-40

+ + + + +

12-42 12-44 12-46 12-48 12-50

+ + + + +

12-52 12-54 12-56 12-58 12-60

+ + + + +

0-1501615

01533737 0-1565195 0-1595977 0-1626073

02214173 0-2199706 0-2184384

02133362

2' 19-83

667°ii' 8*43 668° 19' 57*02 669 28' 45*60 °70°37'34* I 7

676

46' 22*73 55' 11*28 3' 59 -82 12' 48*35 21 '36*86

o-io8ni2

12-62 12-64 12-66 12-68 12-70

+ + + + +

0-1655471 0-1684160 0-1712131 0-1739374 0-1765879

-

0-1516585 0-1481976 0-1446830 0-1411161 0-1374984

0-2245131 0-2243357 0-2241587 0-2239821 0-2238059

677° 30' 25*37

678° 679° 680° 682

39 13-87 4 8' 2*36 56' 50*84 5' 39 -30

-

0-1015142 0-0981318 0-0946953 0-0912064 0-0876664

12-62 12-64 12-66 12-68 12-70

12-72 12-74 12-76 12-78 12-80

+ + + + +

0-1791636 0-1816637 0-1840872 0-1864334 0-1887014

-

-

0-1338314 0-1301168 0-1263559 0-1225504 0-1187019

0-2236302 0-2234548 0-2232799 0-2229313

683° 684 68 5 ° 686° 687

I4 ' 27*76 23' 16*21 3 2' 4*65 40' 53*08 49' 41*50

-

0-0840769 0-0804395 0-0767556 0-0730269 0-0692549

12-72 12-74 12-76 12-78 12-80

0-1148120. 0-1108823 0-1069143 0-1029098 0-0988704

0-2227576 0-2225843 0-2224114 0-2222389 0-2220668

688° 58' 29*91 690 7' 18-31 691° 16' 6*70 692 24' 55*08 693°33'43''45

- 0-0654413 - 0-0615076 - 0-0576954 - 0-0537665 — 0-0498024

12-82 12-84 12-86 12-88 12-90

0-2218951 0-2217238 0-2215529 0-2213824 0-2212123

694 695 697 698° 699°

42' 31*81 51' 20*16 0' 8*51

- 0-0458049 - 0-0417755 - 0-0377160 - 0-0336280 - 00295 1 33

12-92 12-94 12-96 12-98 13-00

12-82 12-84 12-86 12-88 12-90

+ 0-1908904 + o- 1929997 + 0-1950286 0-1969764 + 0-1988424

-

12-92 12-94 12-96 12-98

+ + + + +

- 0-0906934 - 0-0865592 - 0-0823968 - 0-0782079

1300

4-

0-200626I 0-2\023269

0-2039441 0-2054773 0-2069261

- 00947977

02231054

8' 56*85

17' 45*17

1


X

I.

Y1 (x)

Ji(*)

691


I^WI

arg

^V)

H^*)

X

12-02 I2-04 12-00 12-08 I2-IO

-

0-2220777 0-2206225 0-2190821 °-2i74574 0-2157490

-

0-0614935 0-0658561 0-0701853 0-0744794 0-0787369

0-2304343 0-2302419 0-2300499 0-2298584 0-2296674

555° 28' 38*83 556° 37' 13*53 557° 45 48-27 558° 54 23*04 560 2' 57*84

+ + + + +

0-5794489 0-5750722 0-5707290 0-5664209 0-5621495

I2-IO

12-12 12-14 12-16 12-18 12-20

-

02139578 0-2120846 0-2101303 *» X 0-2080958 0-2059820

-

0-0829561 0-0871355

0-0953682 0-0994184

0-2294769 0-2292868 0-2290973 0-2289082 0-2287195

561 11' 32*68 562°2o' 7 55 563 28' 42*46 564°37'i7 ? 40 565° 45' 52^38

+ + + + +

0-5579164 0-5537234 0-5495718 0-5454634 0-5413995

12-12 12-14 I2-IO 12-18 12-20

12-22 12-24 12-26 12-28 I2-30

-

0-2037900 0-2015206 0-1991749 0-1967540 0-1942588

-

0-1034226 0-1073791 0-1112866 0-1151435 0-1189484

0-2285313 0-2283436 0-2281564 0-2279696 0-2277833

566° 54' 27*39 568° 3 2*43

+ + + + +

0-5373819 0-5334119 0-5294911 0-5256209 0-5218027

12-22 12-24 12-26 12-28 I2-30

12-32 12-34

-

0-1916907 0-1890506 0-1863397 0-1835591 0-1807102

-

0-1226999 0-1263966 0-1300372 0-1336202 0-1371444

0-2275974 0-2274120 0-2272270 0-2270425 0-2268585

45 58-13 54 33/37 3' 8*64 11 43*95

+ + + + +

0-5180381 0-5143282 0-5106746 0-5070786 0-5035415

I2-32

573° 574° 576° 577

0-1777942 0-1748122 0-1717656 0-1686557 0-1654838

-

0-1406084 0-1440111 0-1473511 0-1506272 0-1538383

0-2266749 0-2264917 0-2263090 0-2261267 0-2259449

578 579° 5°o° 581° 582

20' 19*28 28' 54*65

1250

-

+ + + + +

0-5000646 0-4966492 0-4932964 0-4900076 0-4867839

12-42 12-44 12-46 12-48 12-50

12-52 12-54 12-56 12-58 12-60

-

0-1622513 0-1589594 0-1556097

0-1569831 0-1600606 0-1630696 0-1660091 0-1688779

0-2257635 0-2255826 0-2254020 0-2252220 0-2250423

584 3 'i6"43 585 11' 51*95 586 20' 27*51

0-1487423

-

587° 29' 3-09 588° 37' 38*71

+ + + + +

0-4836265 0-4805365 0-4775151 0-4745632 0-4716820

12-52 I2 \54 12-56 12-58 12-60

12-62

12-66 12-68 12-70

-

0-1452276 0-1416606 01 38043 0-I343765 0-1306622

-

0-1716752 0-1743998 0-1770509 0-1796274 0-1821280

0-2248631 0-2246843 0-2245059 0-2243280

02241505

589 590° 592° 593 594°

46' 14*35 54' 50*03 3 25*74 12' 1*47 20' 37*24

+ + + + +

0-4688725 0-4661356 0-4634724 0-4608838 0-4583706

12-62 12-64 12-66 12-68 12-70

12-72 12-74 12-76 12-78 I2-80

-

0-1269019 0-1230971 0-1192494 0-1153604 0-1114310

-

0-1845534 0-1869012 0-1891710 0-1913621 o-i934738

0-2239734 0-2237967 0-2236204 0-2234446 0-2232692

595° 29' 13*04 596 37'' 48*86 59 Zo 46 2 4*72 598°55' 0*60 600 3' 36*51

+ + + + +

o-4559337 0-4535740

0-4490893 0-4469659

12-72 12-74 12-76 12-78

1280

12-82 12-84 12-86 12-88 12-90

- 0-1074646 - 0-1034612 - 0-0994229

01955054

- 00953513 - 0-0912483

-

0-2011125 0-2028170

0-2230942 0-2229196 0-2227454 0-2225716 0-2223982

6oi° 12' 12*45 602 20' 48*43 603 29' 24*43

+ + + + +

0-4449226 0-4429602 0-4410794 0-4392807 0-4375647

12-82 12-84 12-86 12-88 12-90

-

-

0-2044382 0-2059758 0-2074291 0-2087978 0-2100814

0-2222253 0-2220527 0-2218805 0-2217088 0-2215374

6o6°55'l2*6o 608 3' 48*71 609 12' 24*85 6io°2i' 1*02

+ + + + +

0-4359320 0-4343830 0-4329183 0-43I5383

12-92

04302435

1300

I236 12-38 12-40 12-42 12-44 12-46 12-48

12

64

12-92

1294 12-96 12-98 13-00

•"*

01522036

0-0871153 0-0829541 0-0787603 0-0745538 0-0703181

00912733

0-1974561

01993253

"-

569°n'37-50 570 20' 12*61 57i°28' 4 7?75

572°37'22"93

37 30*05 5*48 54' 40*94

46

6o4°38' 0*46 605 46' 36*51

61 i° 29' 37*22

04512923

12

02

12-04 12-06

1208

1234 12-36 12-38 12-40

1294 12-96 12-98


692

Table

AW

X

1302 1304 1306 13-08 13-10 13-12 13-14 13-16 13-18

1320

Y

I.


(x)

i*C(*)i

arg

H

26' 33-48 35; 21*79

+ + + + +

0-2082899 0-2095684 0-2107612 0-2118679 0-2128882

-

0-0739941 0-0697573 0-0654990 0-0612211 0-0569253

0-2210426 0-2208732 0-2207043 0-2205357 0-2203676

700 701° 702 703° 705

+ + + + +

0-2138219 0-2146687 0-2154284 0-2161009 0-2166859

-

0-0526132 0-0482867

706 10' 34*92 707 19' 23*18

0-0395973 0-0352379

0-2201998 0-2200324 0-2198654 0-2196987 0-2195325

00439475

H„(*)

(x)

44 10*08 52' 58 -37 1' 46*65

708° 28' 11*43

709°36'59-68 710 45' 47*91

-

X

00253735

1302

0-0212104 0-0170257 0-0128211 0-0085984

13-04 13-06 13-08 13-10

- 0-0043592 - 0-0001054 + 0-0041614 + 0-0084393 + 0-0127268

13-12 13-14 13-16 13-18 13-2°

+ + + + +

0-0170219 0-0213229 0-0256282 0-0299359 0-0342443

13-22

+ + + + +

0-0385517 0-0428564 0-0471565 0-0514504 0-0557363

I3-32 13-34

0-0600126 0-0642775 0-0605292 0-0727662 0-0769866

1342

1352

1322 1324 1326 1328 1330

+ + + +

0-2171835 0-2175935 0-2179159 0-2181508 + 0-2182981

-

0-0308710 0-0264983 0-022I2I7 0-0177428 0-0133634

0-2193666 0-2192010 0-2190359 0-2188711 0-2187067

7H° 54' 36"i4

13-32

+ + + + +

0-2183579 0-2183304 0-2182156 0-2180138 0-2177252

- 0-0089853 - 0-0046I0I - 0-0002396 + 0-0041244 + 0-0084802

0-2185427 0-2183790 0-2182158 0-2180528 0-2178903

717° 718° 719° 721° 722

38' 47' 56' 5 13'

+ + + + +

0-2173499 0-2168884 0-2163409 0-2157076 0-2149892

+ 0-OI28262 + 0-0214816 + 0-0257876 + 0-0300770

0-2177281 0-2175662 0-2174047 0-2172436 0-2170029

723 724 725° 726

22' 37*92 31' 26*05 40' 14*17

49 2*29 727°57'5o"39

+ + + + +

0-0343480 0-0385990 0-0428282 0-05I2I50

0-2169225 0-2167624 0-2166027 0-2164434 0-2162044

729 6' 38*49 730 15' 26*58 731 24' 14*66 732° 33' 2*73 733 41 '50*80

+ + + + +

0-0811889 0-0853714

+ 0-2123263 + 0-2112712 + 0-2101332

+ + + + +

0-0936702 0-0977832

13-54 I3-56 I3-58 13-60

+ + + + +

0-2089128 0-2076107 0-2062276 0-2047641 0-2032208

+ + + + +

0-0553693 0-0594954 0-0635916 0-0676565 0-0716883

0-2161257 0-2159674 0-2158095 0-2156519 0-2154946

734° 735 737° 738° 739

50' 38"86

+ + + + +

0-1018698 0-1059283 o- 1099573 01 139550 0-1179199

13-62 13-64 13-66 13-68 13-70

+ + + + +

0-2015986 0-1998982 0-1981203 0-1962659

+ 0-0756856 0-0796469 0-0835705

0-2153377 0-2151811 0-2150249 0-2148690 0-2147134

740 34' 39*02 741° 43' 27*03 742° 52' 15*03

+ + + + +

0-1218505 0-1257452 0-1296025 0-1334209 0-1371989

13-72 13-74 13-76 I3-78 13-80

+ + + + +

01 409350

13-82 13-84

!334 1336 I3-38 13-40

1342 J3-44 13-46 I3-48 13-5°

1352 1354 1356 I3-58 13-60 13-62

1364 13-66 13-68

1370 13-72

1374 13-76

1378 1380

+ 0-2141858

+ 0-2132981

01943356

+ 0-0171605

746 18' 38*99 747 27' 26*96 748 36' 14*92

02140945

749° 45' 2*87 750° 53 50*82

0-1134294 OI I69492 0-I204I72 0-1238321 0-I27I926

0-2137871 0-2136339 0-2134010

13-92 13-94

+ + + + +

+ + + + +

0-1710735

59 26*90 8' 14*94 17 2*98 25' 51*00

0-2145582 0-2144033 0-2142488

+ 0-1835799

01737085

49*78

0-0951009 0-0988593 0-I025727 0-1062398 0-1098592

+ + + + +

14-00

1*63

0-0874551 0-09I2990

+ 01923305 + 0-1902515 + 0-1880993 + 0-1858751

1396 1398

37*14 25*31 13*47

+ + + +

1382 1384 Îf, 13-88 1390

0-1812145 0-1787801 0-1762777

OO47034I

713° 3 24*35 714 12 12*56 7i5 2i' 0*76 716° 29*48*96

0-2139407

02133284 0-2131762

3"o3 744° 1 95i-oi 745

752 753 754° 755° 756°

2'

38*76

11' 26*70 20' 14*63 29' 2*54

37 50*45

00895324

0-1446278 0-1482758 0-1518777 0-1554320

+ 0-1589374 + 0-1623925 + 0-1657960

+ 0-1691466 + 0-1724429

1324 13-26 13-28 13-3°

1336 I3-38 13-40

I3-44 I3-46

1348 I3-50

6 ^•f 13-88 1390 ,

1392 13-94

1396 1398 14-00

7 3

TABLES OF BESSEL FUNCTIONS Table X

Y

/iW

1

I.


(x)

itf'l'wi

13-06 13-08 13-10

-

0-0660609 0-0617841 0-0574892 0-0531781 0-0488525

-

0-2112796 0-2123920 0-2134183 0-2143582 0-2152115

0-2213665 0-2211959 0-2210257 0-2208559 0-2206066

13-12 13-14 13-16 13-18 13-20

-

00445140

-

0-2159780 0-2166575 0-2172499 0-2177551 0-2181729

1322 1328

-

13-30

- 0-0051775

-

13-02

1304

13-24 13-26

1332 1334 1336 1338 13-40 13-42 13-44 i3-4| 13-48

1350

0-0401645

00358056 0-0314391 0-0270667

0-0226902 0-0183113 0-0139317

00095532

- 0-0008063 0-0035587 0-0079157 0-0122630

+ + + +

0-0165990

-

+ + + + +

0-0209219 0-0252301 0-0295218 0-0337954 0-0380493

-

693

U

axgH^x) 612° 38' 13*45

t

(x)

X

614° 55' 25*99 616 4' 2*30 1 2' 38*63 61

+ + + + +

0-4290341 0-4279106 0-4268732 0-4259222 0-4250579

0-2205176 0-2203490 0-2201007 0-2200129 0-2198455

618 21' 14*99 6i9°29 , 5i*38 620 38' 27*80 621° 47' 4*24 622 55' 40*71

+ + + + +

0-4242805 0-4235900 0-4229868 0-4224709 0-4220422

13-12 13-14 13-16 13-18

0-2185034 0-2187466 0-2189025 0-2189712 0-2189527

0-2196784 0-2195117 0-2193454 0-2191795 0-2190139

624 4' 17*21 625° 12' 53-73 626 21' 30*28 627°3o' 6*85 628° 38' 43*45

+ + + + +

0-4217010 0-4214472 0-4212807 0-4212014 0-4212094

1322

0-2188473 0-2186550 0-2183761 0-2180108 0-2175595

0-2188487 0-2186839 0-2185195 0-2183555 0-2181918

629 630° 632° 633 634

47' 20*08 55' 56 ? 73 4 33 ? 4i 13 10*12 21 '46*85

+ + + + +

0-4213044 0-4214062 0-4217547 0-4221096 0-4225507

0-2170223 0-2163997 0-2156920 0-2148996 0-2140229

0-2180285 0-2178655 0-2177029 0-2175407 0-2173788

635 30' 23*60 636 39' 0*38 6 37°47'37* I 9 638 56' 14*02 640 4' 50*87

+ + + + +

0-4230776 0-4236901 0-4243876 0-4251699 0-4260365 0-4269869 0-4280206 0-4291371 0-4303358 0-4316161

6i3°46'49 ? 7i

°

1302 1304 1306 1308 13-10

1320 13-24

1326 13-28 13-30

1332 13-34

1336 I3-38 13-40

1342 13-44 I3-46 I3-48

1350

13-52 13-54 13-56 I3-58 13-60

+ + + + +

0-0422817 0-0464911 0-0506758 0-0548341 0-0589646

-

0-2130625 0-2120188 0-2108924 0-2096838 0-2083936

0-2172174 0-2170562 0-2168954 0-2167350 0-2165750

644° 39' 18*54 645 47 55*52

+ + + + +

1362 13-64 13-66 13-68 13-70

+ + + + +

0-0630655 0-0671353 0-0711725 0-0751755 0-0791428

-

0-2070225 0-2055711 0-2040400 0-2024302 0-2007421

0-2164153 0-2162559 0-2160969 0-2159382 0-2157799

6 4^° 648° 649 650 651°

32*52 9*55 46*60 23*67 0*77

+ + + + +

0-4329775 0-4344191 0'4359404 0-4375406 0-4392190

13-62 13*64 13-66 13-68

13-72 13-74 13-76 13-78 13-80

+ + + + +

0-0830728 0-0869640 0-0908150 0-0946243 0-0983905

- 0-1989768 - 0-1971349 - 0-1952173 - 0-19322^9 - 0-1911585

0-2156220 0-2154644 0-2153071 0-2151502

+ + + + +

0-4409748 0-4428071 o-4447i52 0-4466981 0-4487550

1372

02149936

652 39' 37*89 653° 48' 15*04 654° 56 52*21 656 5 29*41 657°i 4 ' 6*62

13-82 13-84 I3 1f I3'88 13-90

+ + + + +

0-I02II2I 0-1057877 0-1094160 o-ii 29955 0-1165249

-

0-1890191 0-1868077 0-1845252 0-1821726 0-1797510

0-2148374 0-2146815 0-2145260 0-2143708 0-2142159

658 659° 66o° 661° 662

+ 0-4508850 + 0-4530871

13-82 13-84 13-86 13-88

1392

+ + + + +

0-I200029 0-1234282 0-1267995 0-1301156

-

0-1772613 0-1747048 0-1720824 0-1693954 0-1666448

0-2140614 0-2139072

664° 665° 666° 667 668°

13-94

1396 1398 14-00

OI333752

02137533 02135998 0-2134466

6 4 I I 3'27 ? 75 642°22' 4*66

643°3o'4i-59

56' 5 13 22' 31'

22' 43*86

21*12 39' 58*41 48' 35*72 57' 13*05 1

'

5' 50*40 14' 27*78

23' 5-18 31' 42*60 40' 20*64

+ o-45536o3 + 0-4577036 + 0-4601160

1352 13-54

1356 I3-58 13-60

1370 13-74 13-76 I3-78

1380

1390

+ 0-4625965

1392

+ + + +

1396 1398 1400

0-4651439 0-4677571 0'4704350 0-4731766

13-94


694

Table

X

Y

;•(*)

I.


(x)

iÎ'wi 02130243

axgH

Ho(*)

(x)

46' 38^36

4-

55 26*25 4' 14*14 13 2*02 21' 49*89

+ 0-1788681 + 0-1819944 + 0-1850617

X

14-02 14-04 14-06 14-08 14-10

+ + + + +

0-1683739 0-1656108 0-1627855 0-1598991 0-1569529

+ + + + +

0-1304974 0-1337454 0-1369354 0-1400660 0-1431362

0-2128727 0-2127214 0-2125705 0-2124198

757° 758° 760° 761 762

I 4 -I2

+ + + + +

0-1539^81 0-1508861 0-1477681 0-1445954 0-1413694

+ + + + +

0-1461449 0-1490909 0-1519731 0-1547905 0-1575421

0-2122695 0-2121195 0-2119699 0-2118205 0-2116715

763° 30' 37*76 764 39' 25*62 765° 48' 13*47 766°57' 1*32 768 5' 49-16

+ + + + +

0-1910143 0-1938974 0-1967169 0-1994718 O-2O2I0IO

14-14 14-16 14-18 14-20

+ + + + +

0-1380914 0-1347629 0-1313851 0-1279596 0-1244877

+ + + + +

0-1602268 0-1628437 0-1653919 0-1678704 0-1702783

0-2115227 0-2113743 0-2112262 0-2110784 0-2109310

769° 14' 36*99 770 23' 24*81 771 32' 12*63 77 2 °4i' 0*44 773 49 48*24

+ + + + +

0-2047836 0-2073385 0-2098248 0-2I224I6 0-2145880

14-22 14-24 14-26 14-28 14-30

0-1209709 0-1174107 o- 1 1 3808 s 0-1101658 0-1064841

+ + + + +

0-1726147 0-1748790 0-1770701 0-1791875 0-1812302

0-2107838 0-2106369 0-2104904 0-2103441 0-2101982

774° 770° 777 778° 779°

58' 36*03 7 23*82

14-36 I4-38 14-40

+ + + + +

+ + + + +

0-2168632 0-2190663 0-2211964 0-2232530 0-225235I

14-32 14-34 14-36 I4-38 14-40

14-42 14-44 14-46 14-48 14-50

+ + + + +

0-1027650 0-0990100 0-0952206 0-0913984 0-0875449

+ + + + +

0-1831977 0-1850893 0-1869042 0-1886420 0-1903019

0-2100525 0-2099072 0-2097621 0-2096174 0-2094729

780° 42; 34*91 781 51' 22*66 8 58 6 7', 45 ll -89 1

+ + + + +

0-2271421 0-2289733 0-2307281 0-2324058 0-2340059

14-42 14-44 14-46 14.48 14*50

I4-52 14-54 14-56 I4-58 14-60

+ + + + +

0-0836617 0-0797504 0-0758127 0-0718500 0-0678641

+ + + + +

0-1918835 0-1933862 0-1948095 0-1961530 0-1974163

0-2093288 0-2091849 0-2090414 0-2088981 0-2087552

786 26' 33*61 787° 35' 21*33 7£8°44 9 -o 5 789° 52' 56*76 791 ° 1' 44*46

+ + + + +

0-2355279 0-2369710 0-2383350

I4-52 14-54 14-56 I4-58 14-60

14-62 14-64 14-66 14-68 14-70

+ + + + +

0-0638565 0-0598280 6-0557827 0-0517198 0-0476418

+ + + + +

0-1985989 0-1997005 0-2007208 0-2016595 0-2025163

0-2086125 0-2084701 0-2083280 0-2081862 0-2080447

792° 10' 32*15 7°3° I g /' x 9;84 794° 28' 7*52 795°36'55 ? 20 796 45' 42*87

+ + + + +

0-2419474 0-2429903 0-2439521

14-72

14-76 14-78 14-80

+ + + + +

0-0435503 0-0394470 0-0353334 0-0312113 0-0270823

+ + + + +

0-2032910 0-2039834 0-2045933 0-2051206 0-2055652

0-2079035 0-2077626 0-2076219 0-2074816 0-2073415

797°54'3o"53 799° 3 18*19 8oo°i2' 5*84 8oi°2o'53*4 g 802 29' 41*12

+ + + + +

0-2463475 0-2469820 0-2475341 0-2480038 0-2483909

14-72 I4'74 14-76 14-78

14-82 14-84 14-86 14-88 14-90

+ + + + +

0-0229481 0-0188102 0-0146704 0-0105303 0-0063915

+ 0-2059270

0-2072017 0-2070622 0-2069229 0-2067840 0-2066453

803° 804 ° 8o 5 807 808

+ + + + +

0-2486955 0-2489173 0-2490565 0-2491132 0-2490872

14-82

14-92 14-94 14-96 14-98

+ 0-0022558

809 22' 26*8l 8io° 31' 14*40

1500

00142245

14-14 14-16 14-18 14-20 14-22

1424 14-26 14-28

1430 14-32

J434

M-74

-

0-0018753 0-0060002 0-0101171

+ + + +

0-2062060 0-2064022 0-2065157 0-2065464

+ + + + +

0-2064946 0-2063603 0-2061436 0-2058449

0-2065069

02054643

02059561

02063688 0-2062309 0-2060934

?K ?K 785°

16 11*60 24' 59*38

33 47*15

o;

1

^

1

38' 28*75 47' 16*37 5 6' 3*98

51*60 13' 39*21 4'

8n°4o'

1*99

812° 48' 49*57 8i3° 57' 37*M

0-1756839

+ 0-1880687

02396194 0-2408236

O2448324 O2456309

+ 0-2489788

+ + + +

0-2487881 0-2485152 0-2481604 0-2477238

14-02 14-04 14-06 14-08 14-10 I 4 -I2

14-62 14-64 14-66 14-68 14-70

1480

14-86 14-88 14-90

1492 1494 14-96 14-98

1500


X

I.

Y1 (x)

7i(*)


i<'wi

arg

0-1365770 0-1397201 0-1428030 0-1458248 0-1487844

-

0-1638320 0-1609579 0-1580240 0-1550314 0-1519813

0-2132937 0-2131411 0-2129889 0-2128370 0-2126855

669° 670° 672 673° 674

+ + + +

0-1516805 0-1545122 0-1572785 0-1599783 + 0-1626107

-

0-1488752 0-1457142 0-1424998

0-2125342

14-22 14-24 14-26 14-28 14-30

+ + + + +

0-1651747 0-1676695 0-1700940 0-1724475 0-1747291

-

1432 1434 14-36 14-38 14-40

+ + + + +

14-42 14-44 14-46 14-48 14-50

14-02 14-04 14-00 14-08 14-10

+ + + + +

14-12 14-14 14-16 14-18 14-20

695

H

1 ]

HiW

(x)

4 8'57 ?5i 57 35'oo

X

14' 5o'o4 23' 27*60

+ + + + +

0-4759804 0-4788455 0-4817705 0-4847542 0-4877954

14-02 14-04 14-06 14-08 14-10

0-1359159

0-2122327 0-2120824 0-2119325

5;i7 f75°32' 6 7°° 4o' 42*77 677° 49' 20*39 678° 57' 58 -03 68o° 6' 35*69

+ + + + +

0-4908927 0-4940449 0-4972506 0-5005085 0-5038172

14-12 14-14 14-16 14-18 14-20

0-1325491 0-1291344 0-1256731 0-1221667 0-1186166

0-2117828 0-2116335 0-2114845 0-2113358 0-2111874

681° 682 683 68 4 685

+ + + + +

0-507I753 0-5105814 0-5140341 0-5175320 0-5210736

14-22 14-24 14-26 14-28

0-1769380 0-1790734 0-1811346 0-1831209 0-1850317

- 0-1150243 - 0-1113912 - 0-1077189 - 0-1040089 - 0-1002626

0-2110394 0-2108916 0-2107442 0-2105971 0-2104502

686° 688° 689° 690 691°

+ + + + +

0-5246575 0-5282821 0-5319460 0-5356477 0-5393857

+ + + + +

0-1868661 0-1886237 0-1903038 0-1919059 0-1934295

-

0-0964816 0-0926676 0-0888219 0-0849462 0-0810421

0-2103037 0-2101575 0-2100116 0-2098660 0-2097207

692 41' 31*37

693°5o' 9*28 694 58' 47 -22 696 7' 25*18 697°i6' 3"i6

+ + + + +

0-543I583 0-5469642 0-5508016 0-5546691 0-5585651

14-42 14-44 14-46 14-48 14-50

14*52

4

1454

+ + + +

0-1948740 0-1962389 0-1975240 0-1987287 0-1998527

-

0-0771111 0-0731549 0-0691749 0-0651730 0-0611506

0-2095757 0-2094310 0-2092866 0-2091425 0-2089987

698 699° 700° 701° 702

59' 13*31

+ + + + +

0-5624879 0-5664361 0-5704080 0-5744019 0-5784163

14-52 J4-54 14-56 I4-58 14-60

0-2008956 0-2018572 0-2027371

0-0571093 0-0530509 0-0489769 0-0448890 0-0407888

0-2088552 0-2087120 0-2085691 0-2084265 0-2082842

704°

7' 51*40 705°i6'29*5i 7o6°25' 7*64 7°Z°33'45"8o 708 42' 23*97

+ + + + +

0-5824496 0-5865001 0-5905662 0-5946463 0-5987387

14-62

1S&

0-0366779 0-0325580 0-0284307 0-0242978 0-0201607

0-2081422 0-2080004 0-2078590 0-2077178 0-2075709

709°5i' 2*16 7 IO !59'40-36 8' 18*59 712 713° 16' 56*84 714 25' 35*10

+ + + + +

0-6028418

14-72

OOO69539 0-6110734 0-6151987 0-6193280

M'74 14-76 14.78 14-80

34' 13*39 42' 51*69 51' 30*01

0-6234599 0-6275926 0-6317244 0-635853$ 0-6399791

14-82 14-84 14-86 14-88 14-90

0-6440987 0-6482109 0-6523142 0-6564068 0-6604873

14-92 14-94 14-96 14-98

14-56 14-58 14-60

01392332

02123833

6' 12*51

15' 13*38 23' 51*08 32' 28*81

4 i' 6*55 49' 44*32 58' 22"ii

59*92 15' 37 -75 24 15*60 32' 53 -47 6'

24' 41*14 33' 19*15 41' 57*18

50 35*23

1470

+ + + + +

0-2042513

-

14-72 14-74 14-76 14-78 14-80

+ + + + +

0-2048851 0-2054365 0-2059054 0-2062910 0-2065956

-

14-82 14-84

+ + + + +

0-2068167 0-2069553 0-2070113 0-2069849 0-2068702

-

0-OI002I2 0-OII8809 0-0077415 0-0036045 + 0-0005283

0-2074363 0-2072960 0-2071560 0-2070163 0-2068768

715 716 717 719° 720

0' 8*36 8' 46*72

+ + + + +

+ + + + +

0-2066853 0-2064124 0-2060577 0-2056215 0-2051040

+ + + +

0-2067377 0-2065988 0-2064602 0-2063219 0-2061838

17' 25*09 722°26' 3*49 723° 34' 41*90 724° 43' 20*33 725 5 I' 58*79

+ + + + +

14-62

1$ 14-68

i486 14-88

1490 1492 14-94 14-96

1498 1500

02035352

0-0046553 0-0087750 0-0128857 0-0169858 + 0-02I0736

721

1430 1432 14-34 14-36

1438 14-40

14-68 14-70

1500


696

Table

/oW

X

I.


nw

1*0*)

1

argH

15-02 15-04 15-06 15-08 15-10

-

0-0183207 0-0224042 0-0264732 0-0305263 0-0345619

4 0-205002I 4- 0-2044585 + 0-2038339 4 0-2031287 4- 0-2023432

0-2058191 0-2056823 0-2055459 0-2054097 0-2052737

815°

15-12 15-14 15-16 15-18 15-20

- 0-0385782 - 0-0425738 - 0-0465472 - 0-0504967 - 0-0544208

4 0-2014779 4 0-2005332 4- 0-1995096 4- 0-1984076 4- 0-1972277

15-22 15-24 15-20 15-28 15-30

- 0-0583180 - 0-0621868 - 0-0660256 - 0-0698330 - 0-0736075

+ 0-1959705 + 0-1946367 + 0-1932268

15-32 15-34 15-36 15-38 15-40

-

0-0773477 0-0810521 0-0847192 0-0883477 0-0919362

4-

15-42 15-44 15-46 15-48 15-50

-

15-52 15-54 15-56 15-58 15-60

-

15-62 15-64 15-66 15-68 15-70

-

15-72 15-74 15-76 15-78 15-80

- 0-1428446 - 0-1455583 - 0-1482104 - 0-1507998 - o-i533257

15-82

0-1557872 0-1581832 0-1605130 0-1627757 0-1649705

4-

15-88 15-90

-

15-92 15-94 15-96 15-98 16-00

-

0-1670966 0-1691532 0-1711396 o-i73055i 0-1748991

4-

mt

Ho(*)

{x)

6; 24*71

X

0-2472058 0-2466066 0-2459265 0-2451659 0-2443252

I5-02 I5-04 15-06 I 5 -o8 I5-IO

0-2434048 0-2424052 4- 0-2413268 4- 0-2401701 + 0-2389357

15-12 I5-I4 15-16 I5-I8 I5-20

0-2376242 0-2347721 0-2332329

15-22 I5-24 I5-26 15-28

+ 0-2316191

I53O

4-

816 15' 12*27

4-

8i7° 23' 59-83 818° 32' 47*38 8l 9 4i 34-93

4-

0-2051381 0-2050027 0-2048675 0-2047327 0-2045981

820 50' 22*47 82i°59' 10*01 §23° 7'57 ? 54 824°i6'45?o6 825 25' 32*57

4-

826 34' 20*08 7-58 l 828 51' 55*08 830 0' 42*58 831° 9'3o''o7

+ 0-2362361

+ 0-1901815

0-2044638 0-2043297 0-2041959 0-2040624 0-2039291

+ + + 4

0-1885475 0-1868403 0-1850607 0-1832093 0-1812872

0-2037961 0-2036633 0-2035308 0-2033986 0-2032666

832 18' 17*55 8 33° 27' 5 ? 03 834° 35, 52 -50 835° 44, 39*96 8 36°53 27*42

4 0-2299315 0-2281707 4- 0-2263377 4- 0-2244331 + 0-2224578

15-32 15-34

0-0954833 0-0989876 0-1024478 0-1058626 0-1092307

4-

0-1792950

0-2031349 0-2030034 0-2028722 0-2027412 0-2026105

0-1125507 0-1158213 0-1190418 0-1222103 0-1253260

4 0-1683167 + 0-1659242 + 0-1634685

0-1283875 0-1313938 0,I 34343 8 0-1372363 0-1400702

4-

0-1917415

+ 0-1772338 4-

0-1751045

+ 0-1729078 + 0-1706449

2

ll<

838°

44-

4-

4-

840 19' 49*76 841° 28; 37*20 842 37' 24*63

0-I557325 0-1530346 0-1502790 0-1474668 0-1445992

0-2018315 0-2017026 0-2015739 0-2014454 0-2013172

0-1416775 0-1387029 4- 0-1356766 + 0-1326000 + 0-1294742

0-2011892 0-2010615 0-2009340 0-2008067 0-2006797

85 5

0-1263006 0-1230805 0-1198153 0-1165064 0-1131550

4-

4-

4-

849°3o' 9*11 850°38'56 ?50 5I f o4 ^ 43 ? 9 852 56' 31*27 854 5'i8 ? 6 5

+ + + + +

4-

14*87

2;

843°46' 1 2 "06 844° 54' 59-48 846 3' 46*89 ?47° I2 '34 ? 30 848 21' 21*71

0-1609506 + 0-1583715

4-

839°n' 2*32

0-2024801 0-2023499 0-2022199 0-2020902 0-2019607

4-

4-

i4

'

6*02

44-

4-

0-1963278

15-62

+ 0-1935784 4- OT907714 44-

86o°58' 2*81 6' 50*15 862 863° 15' 37-48

4-

0-1097626 0-1063305 0-1028603

0-1999227 o- 1997974 0-1996723

+ OO993533

OI995474

0-0958110

0-1994228

4-

4-

4-

0-1879079 0-1849893

+ 0-l820l66

0-2005530 0-2004264 0-2003001 0-2001741 0-2000483

4-

I5-50

4-

4-

4-

4-

I5H8

4-

4-

?57°3i',40"75 858 40' 28*11 859° 49' 15-46

4-

I5H2 15-44 I5-46

I5-52 15-54 I5-50 I5-58 I5-60

4-

4-

I5-38 I5-40

0-2091715 0-2067269 0-2042191 0-2016493 0-1990185

4-

4-

856 22' 53*39

4-

0-2204127 0-2182987 0-2161166 0-2138674 0-2115520

1536

44-

0-1789911 0-1759141 0-1727868 0-1696105

4-

0-1663866 0-1631163 0-1598009 0-1564420

865 33' 12^14

4-

OI530407

866° 41' 59*46 867 50' 46*78 868° 59' 34*09 870 8' 21*39 87i°i7' 8*69

4-

0-1495986

44-

+ 0-1461170 4-

4-

0-1425972 0-1390409

+ 0-1354493

15-66 15-68

1570 15-72 15-74 15-76 I5-78 15-80 15-82 15-84

^•is 1590 15-88

15-92 15-94 15-96

1598 16-00


Y

/iW 15-02 15-04 15-06 15-08 15-10

+ + + + +

0-2045057 0-2038267 0-2030675 0-2022286 0-2013102

+ + + + +

15-12 15-14 15-16 15-18 15-20

+ + + + +

0-2003130 0-1992373 0-1980838 0-1968530 0-1955454

+ + + +

15-22 15-24 15-26 15-28

+ + + + + + + + + +

l

I.

(x)

697


1*0*)

I

arg

H

{

HiW

]

(x) \

0-0251476 0-0292062 0-0332478 0-0372707 0-0412735

0-2060460 0-2059085 0-2057713 0-2056344 0-2054977

727 728° 729° 73o° 73i

0-0452546 0-0492124 0-0531454 0-0570521 + 0-0609309

0-2053613 0-2052252 0-2050893 0-2049537 0-2048184

732° 733° 735° 736° 737°

18' 24-28

4-

0-7002724

0-1941618 0-1927027 0-1911688 0-1895608 0-1878794

+ + + + +

0-2046834 0-2045486 0-2044141 0-2042798 0-2041459

738=27' 2*93 739° 35 4**59 740 44 20*27 741*52 58*97 1 37-69 743

+ + + + +

0-7041148 0-7079263 0-7117056

0-1861255 0-1842998 0-1824032 0-1804363 0-1784003

+ + + + +

0-0835358 0-0871785 0-0907816 0-0943440 0-0978642

0-2040121 0-2038787 0-2037455 0-2036125 0-2034799

744° 10' 16*42 745° i8' 55*17

746° 27 33*94 747° 36' 12*72 748° 44 51-52

+ + + + +

0-7228353 0-7264711 0-7300674 0-7336229

+ 0-1762958

0-1013409 0-1047727 0-1081584 0-1114960 0-1147861

0-2033475 0-2032153 0-2030834 0-2029518 0-2028204

749° 751° 752° 753° 754°

+ + + + +

0-7406061

I5-42

07440312

I5'4^

+ 0-1695816 + 0-1672132

+ + + + +

0-7474101 0-7507416 0-7540245

15 -4« I5-48 15-50

15-52 15-54 I5-56 I5-58 15-60

+ + + + +

+ + + + +

0-1180258 0-1212142 0-1243503 0-1274329 0-1304608

0-2026892 0-2025584 0-2024278 0-2022974 0-2021673

755° 36' 44 -67

+ + + + +

0-7572574 0-7604392 0-7635687 0-7666447 0-7696660

1552

756° 45 23*59 757° 54 2*52 2 4i*47 759 760 11 20*43

15-62 15-64 15-66 15-68 15-70

+ 0-1517062 + 0-1489160

+ + + + +

0-1334329 0-1363480 0-1392052 0-1420033 o-i4474i3

0-2020374 0-2019078 0-2017784 0-2016493 0-2015204

761° 762° 763° 764° 765

15-72 15-74 15-76 15-78 15-80

+ 0-1372099 + 0-1341533 + 0-1310471 + 0-1278927 + 0-1246913

+ + + + +

0-1474181 0-1500329 0-1525847 0-1550724 o-i5 74953

0-2013918 0-2012634 0-2011353 0-2010074 0-2008798

767° 768° 769° 770° 77*°

15-82 15-84 15-86 15-88 15-90

+ + + + +

0-1214444 0-1181532 0-1148192 0-1114436 0-1080279

+ + + + +

0-1598524 0-1621429 0-1643659 0-1665207 0-1686064

0-2007524 0-2006252 0-2004983 0-2003717 0-2002452

772 773° 775° 776 777°

15-92 15-94 I5-96 15-98 16-00

+ + + + +

0-1045735 0-1010818 0-0975542 0-0939922 0-0903972

+ + + + +

0-1706224 0-1725680 0-1744424 0-1762450 0-1779752

0-2001190 0-1999931 0-1998674 0-1997419 0-1996167

1530 15-32 15-34 15-36 15-38

1540 15-42 15-44 15-46 15-48 15-50

+ 0-1741239 + 0-1718855

0-1647812 0-1622868 0-1597310 0-1571149 O' 1 544396

+ 0-1460700 + 0-1431695 + 0-1402157

-

j

0-0647803 0-0685990 0-0723853 0-0761370 0-0798551

9'i5 7 74 1 7 54*25 26 32^77 ? 3i 35

+ + + + +

0-6645540 0-6606052 0-6726395 0-6766552 0-6806508

I5-02 I5-04 15-06 I5-08 I5-IO

43' 49-87 52' 28*45

+ 0-6846246 + 0-6885753

I5'I2 15-14 I5-l6 I5-I8 I5-20

°' 37*25

n

7-04 1 9 45-65

53' 30 -34

9*17 2 io' 48*02 19' 26*89 28'

5 "77

+ 0-6925011 + 0-6964006

07I545I2 0-7191616

07371363

19' 59*41 28' 38 -41 37' 17-43

+ 0-7726316 + 07755402 + O7783909

45 56-46 54 35*5<>

+ 0-7811825 + 0-7839140

+ + + + +

0-7865845 0-7891929 0-7917383 0-7942197 0-7966362

6-8i

+ + + + +

O-0OI27I2 0-8034880 0-8056365 0-8077161

778° 29' 46 -03 779° 38' 25*26 7«o 47' 4*50 7§i° 55 43*76 783 4 23-04

+ + + + +

3'i4''56 11' 53 ''64 20' 32 "73

29' 11-84 37' 50-96 46' 30*10 55' 9*25 3' 48 -42 12' 27*61

21'

O7989870

0-8097259 0-8116653

08135336 0-8153300 0-817054!

I5-22 I5-24 I5-26 I5-28 I5-30

1532 r 5-34 15-36 I5-38 15-40

15-54 I5-56 I5-58 15-60 15-62 15-64 15-66 15-68 15-70

1572 15-74 15.76 15-78 15-80

15-82 15-84 15-86 15-88 15-90 15-92 15-94 15-96 15-98 16-00

23


698

Table

X

e~ x /„(*)

II.

Functions of imaginary argument, and

e-*I x {*)

ex

K

(x)

e*

ex

gX

K^x)

X

o

1

-ooooooo

o-ooooooo

0-02 0-04 0-06 0-08 O-IO

0-9802967 0-9611738 0-9426123 0-9245939 0-9071009

0-0098025 0-0192196 0*0282657 0-0369542 0-0452984

4-1098376 3-4727083 3-1142387 2-8679911 2-6823261

50-9638701 25-9404241 17-5879738 13-4048206 10-8901827

I -0202013 1-0408108 1-0618365 1-0832871 1-1051709

0-02 0-04 o-66 0-08 O-IO

0-12 0-14 0-16 0-18 0-20

0-8901162 0-8736233 0-8576062 0-8420496 0-8269386

0-0533111 0-0610043 0-0683899 0-0754792 0-0822831

2-5344522 2-4123173 2-3087874 2-2192980 2-1407573

9-2102792 8-0076794 7-1036124 6-3987260 5-8333860

1-1274969 1-1502738 1-1735109 1-1972174 1-2214028

0-12 0-14 o-i6 0-18 0-20

0-22 0-24 0-26 0-28

030

0-8122587 0-7979961 0-7841375 0-7706698 0-7575806

0-0888122 0-0950766 0-1010861 0-1068501 0-1123776

2-0709767 2-0083522 1-9516748 1-9000114 1-8526273

5-3696274 4-9821285 4-6533504 4-3707591 4-1251578

1-2460767 1-2712492 1-2969301 1-3231298 1-3498588

0-22 0-24 0-20 0-28 O-30

0-32 o-34 0-36 0-38 0-40

0-7448578 0-7324896 0-7204648 0-7087725 0-6974022

0-1176774 0-1227579 0-1276272 0-1322931 0-1367632

1-8089345 1-7684552 1-7307961 1-6956301 1-6626821

3-9096449 3-7189398 3-5489328 3-3963772 3-2586739

1-3771278 1-4049476 1-4333294 1-4622846 1-4918247

0-32 o-34 0-36 0-38 0-40

0-42 0-44 0-46 0-48 0-50

0-6863436 0-6755870 0-6651228 0-6549419 0-6450353

0-1410447 0-1451446 0-1490697 0-1528263 0-1564208

1-6317188 1-6025406 I-5749758 1-5488754 1-5241094

3-1337176 3-0197845 2-9154495 2-8195242 2-7310097

1-5219616 -5527072 1-5840740 1-6160744 1-6487213

0-42 o-44 0-46 0-48 0-50

0-52 o-54 0-56 0-58 o-6o

0-6353945 0-6260111 0-6168773 0-6079851 0-5993272

0-1598592 0-1631473 0-1662906 0-1692946 0-1721644

1-5005638 1-4781381 I-4567432 1-4363000 1-4167376

2-6490599 2-5729537 2-5020724 2-4358821 2-3739200

1-6820276 1-7160069 1-7506725 1-7860384 1-8221188

0-52 °-54 0-56 0-58 o-6o

0-62 0-64 o-66 o-68 0-70

0-5908962 0-5826853 0-5746875 0-5668963 0-5593055

0-1749051 0-1775214 0-1800181 0-1823995 0-1846700

1-3979927 1-3800080 1-3627320 1-3461180 1-3301237

2-3157825 2-2611160 2-2096099 2-1609894 2-1150113

1-8589280 1-8964809 1-9347923 1 '9738777 2-0137527

0-62 0-64 o-66 o-68 0-70

0-72 0-74 0-76 0-78 o-8o

0-5519089 0-5447006 0-5376748 0-5308260 0-5241489

0-1868337 0-1888946 0-1908567 0-1927235 0-1944987

1-3147102 1-2998425 1-2854881 1-2716174 1-2582031

2-0714590 2-0301393 1-9908791 1-9535227 1-9179303

2-0544332 2-0959355 2-1382762 2-1814723 2-2255409

0-72 o-74 0-76 0-78 o-8o

0-82 0-84 o-86 o-88 0-90

0-5176383 0-5112892 0-5050967 0-4990561 o-493 J 630

0-1961857 0-1977879 0-1993083 0-2007502 0-2021165

1-2452202 1-2326455 1-2204575 1-2086362 1-1971634

1-8839752 1-8515429 1-8205294 1-7908399 1-7623882

2-2704998 2-3163670 2-3631607 2-4108997 2-4596031

0-82 o-8 4 o-86 o-88 0-90

0-92 0-94 0-96 0-98 I-OO

0-4874128 0-4818015 0-4763248 0-4709788 o-4657596

0-2034101 0-2046336 0-2057898 0-2068813 0-2079104

1-1860217 I-I75I953 1*1646692 1-1544294 1-1444631

1

-7350954 1-7088892 1-6837033 1-6594768 J -636i535

2-5092904 2-55998I4 2-6116965 2-6644562 2-7182818

0-92 0-94 0-96 0-98 I-OO

00

00

1

-ooooooo

1


X 1-02 1-04 i-oo

(x)


e-*I x {x)

e*K

ex

(x)

K

1

{x)

0-4606636 0-4556871 0-4508266 0-4460788 0-4414404

0-2088796 0-2097912 0-2106473 0-2114501 0-2I220I6

1 -ii 60860

1-1070984 1-0983303

1-6136817 1-5920135 1-5711046 1-5509141 1-5314038

i-i6 i-i8 1-20

0-4369082 0-4324792 0-4281504 0-4239190 0-4197821

0-2I29039 0-2135587 0-2141680 0-2147335 0-2152569

1-0897726 1-0814169 1-0732553 1-0652802 1-0574845

1-5125382 1-4942846 1-4766120 x-4594919 1-4428976

1-22 I-2A 1-26 1-28 i-3°

o-4i5737i 0-4117813 0-4079123 0-4041277 0-4004249

0-2I57398 0-2161838 0-2165905 0-2169613 0-2172976

1-0498615 1-0424048 1-0351082 1-0279662 1-0209732

1-4268038 1-4111872 1-3960258 1-3812990 1-3669873

132

0-3968018 0-3932561 0-3897857 0-3863885 0-3830625

0-2176008 0-2178721 0-2IOII29

1-0141239

1-34 i- 3 6 1-38 1-40

1 -00741 36

1-42 1-44 1-46 1-48 1-50

108 I-IO

112

'

e~* I

II.

699

I-I347579 1-1253024

ex

e*

X

2-7731948 2-8292170 2-8863710 2-9446796 3-0041660

1-02 1-04 1-06 1-08 I-IO

3-0648542 3-1267684 3-I899333 3-2543742 33201 169

I-I2 1-14 1-16 1-18 1-20

3-3871877

1-22 1-24 1-26 1-28

3H556I35 3-5254215 3*5966397 3-6692967

3'74342i4 3-8190435 3-8961933 3-97490I6

I

30

O2183243

09943910

0-2185076

0-9880700

1-3530725 1-3395375 1-3263660 1-3135429 1-3010537

40552000

1-32 1-34 1-36 1-38 1-40

0-3798057 0-3766162 0-3734922 0-37043 !9 0-3674336

0-2186638 0-2187941 0-2188994 0-2189809

O2190394

0-9818703 o-975788i 0-9698197 0-9639615 0-9582101

1-2888849 1-2770235 1-2654575 1-2541752 1-2431659

4-1371204 4-2206958 4-3059595 4-3929457 4-4816891

1-42 1-44 1-46 1-48 1-50

1-52 i'54 l- 5 6 1-58 l-6o

0-3644956 0-3616163 0-3587941 0-3560275 0-353*150

0-2190759 0-2190913 0-2190865 0-2190623 0-2190195

0-9525622 0-9470148 0-9415648 0-9362095 0-9309460

1-2324190 1-2219249 1-2116741 1-2016578 1-1918676

4-5722252 4-6645903 4-7588212 4-8549558 4-9530324

152 1.58 l-6o

1-62 1-64 1-66 1-68 1-70

0-3506552 0-3480467 0-3454881 0-3429782 0-3405157

0-2189589 0-2188812 0-2187872 0-2186775 0-2185528

0-9257717 0-9206842 0-9156810 0-9107597 0-9059181

1-1822954 I I 72 9335 1-1637748 1-1460392

5-0530903 5-1551695 5-2593108 5-365556o 5-4739474

1-62 1-64 1-66 1-68 1-70

1-72 1-74 1-76 1-78 i-8o

0-3380993 0-3357279 0-3334003

0-2184138 0-2182610 0-2180952 0-2179167 0-2177263

0-9011541 0-8964656 0-8918506 0-8873072

08828335

r-1374494 1-1290368 1-1207955 1-1127201 1-1048054

5-5845285 5-6973434 5-8124374 5-9298564 6-0496475

1-72 1-74 1-76 1-78 i-8o

0-8784278 0-8740882 0-8698132 0-8656012 0-8614506

1-0970461 1-0894376 1-0819752 1-0746544 1-0674709

6-1718584 6-2965383 6-4237368 6-5535049 6-6858944

1-82 1-84 1-86 1-88 1-90

1-0604208

6-8209585 6-9587510 7-0993271 7-2427430 7-3890561

1-92 1-94 1-96 1-98 2-00

1-82 1-84 1-86 1-88 1-90 1-92 1-94 1-96 1-98 2-00

o*33«i53 0-3288719 0-3266691

0-2175244

03245058

02173115

0-3223810 0-3202938

0-2170881 0-2168548

03182432

o-2i66iij9

0-3162282 0-3142481 0-3123020 0-3103890 0-3085083

0-2163599 0-2160993 0-2158304 0-2155536

02152693

1-0008375

o-8573599 0-8533277 o-8493526 o-8454332 0-8415682

'

11548123

1

0535000

1-0467048 1-0400316 I-0334768

1-54

156


700 Table

II.


K

"x

ex

X

e~*h{x)

er*I x (x)

2-02 2-04 2 OO 2-08 2-IO

0-3066592 0-3048408 0-3030525 0-3012935 0-2995631

0-2149779 0-2146797 0-2143750 0-2140643 0-2137477

0-8377564 0-8339966 0-8302875 0-8266281 0-8230172

1-0270373 1-0207097 1-0144909 1-0083780 1-0023681

7-5383249 7-6906092 7-8459698 8*0044689 8-1661699

2-02 2-04 2-00 2-o8 2-IO

2-12 2-14 2-16 2-l8 2-20

0-2978606 0-2961855 0-2945369 0-2929144 0-2913173

0-2134256 0-2130983 0-2127660 0-2124291 0-2120877

o-8i94537 0-8159366 0-8124650 0-8090377 0-8056540

0-9964584 0-9906463 0-9849292 0-9793046 0-9737702

8-33II375 8-4994376 8-6711377 8-8463063

90250135

2-12 2-14 2-16 2-18 2-20

2-22 2-24 2-26 2-28 2-30

0-2897451 0-2881970 0-2866727 0-2851715 0-2836930

0-2117422 0-2113927 0-2110396 0-2106829 0-2103230

0-8023128 0-7990133 o-7957545 o-792535$ 0-7893561

0-9683236 0-9629626 0-957685I 0-9524890 0-9473722

9-2073308 9-3933313 9-5830892 9-7766804 9-9741825

2-22 2-24 2-26 2-28 2-30

2-32 2-34 2-36 2-38 2-40

0-2822366 0-2808018 0-2793881 0-2779951 0-2766223

0-2099600 0-2095941 0-2092256 0-2088545 0-2084811

0-7862149 0-7831112 0-7800443 0-7770135 0-7740181

o-94 2 3329

10-1756743 10-3812366 10-5909515 10-8049029 11-0231764

232

0-9373692 0-9324793 0-9276613 0-9229137

2-42 2-44 2-46 2-48 2-50

0-2752693 0-2739356 0-2726209 0-2713246 0-2700464

0-2081055 0-2077279 0-2073485 0-2069674 0-2065846

0-7710575 0-7681308 0-7652376 0-7623771 o-7595487

0-9182347 0-9136228 0-9090764 0-9045941 0-9001744

11-2458593 11-4730407 11-7048115 11-9412644 12-1824940

2-42 2-44 2-46 2-48

2-52 2'54 2-56 2-58 2-60

0-2687860 0-2675429 0-2663168 0-2651072 0-2639140

0-2062005 0-2058151 0-2054285 0-2050408 0-2046523

o-75675i8 o-7539859 0-7512504 o-7485447 0-7458682

0-8958159 0-8915172 0-8872771 0-8830942 0-8789673

12-4285967 12-6796710 12-9358173 13-1971382

134637380

2-52 2-54 2-56 2-58 2-60

2-62 2-64 2-66 2-68 2-70

0-2627367 0-2615749 0-2604285 0-2592970 0-2581801

0-2042628 0-2038727 0-2034820 0-2030907 0-2026991

0-7432205 0-7406011 0-7380094 o-7354449 0-7329072

0-8748952 0-8708767 0-8669107 0-8629961 0-8591319

I3-7357236 14-0132036 14-2962891 14-5850933 14-8797317

2-62 2-64 2-66 2-68 2-70

2-72 2-74 2-76 2-78 2-80

0-2570776 0-2559892 0-2549146 0-2538534 0-2528055

0-2023071 0-2019148 0-2015224 0-2011299 0-2007374

0-7303957 0-7279102 0-7254500 0-7230148 0-7206041

0-8553169 0-8515502 0-8478308 0-8441577 0-8405301

15-1803222 15-4869851 15-7998429 16-1190209 16-4446468

2-72 2-74 2-76 2-78 2-80

2-82 2-84 2-86 2-88 2-90

0-2517706 0-2507484 0-2497387 0-2487412 0-2477557

0-2003450 0-1999527 0-1995606 0-1991688 0-1987773

0-7182176 0-7158548 o-7i35 r 54 0-7111989 0-7089050

0-8369469 0-8334074 0-8299106 0-8264557 0-8230420

16-7768507

2-82 2-84 2-86 2-88 2-90

2-92 2-94 2-96 2-98 3-00

0-2467820 0-2458198 0-2448690 0-2439292 0-2430004

0-1983862

0-7066333 0-7043834

0-8196687 0-8163349 0-8130399 0-8097830 0-8065635

01979955 0-1976053 0-1972157 0-1968267

e*

(x)

07021551 0-6999479 0-6977616

if,W

ex

I7-H57655 17-4615269 17-8142732 18-1741454 18-5412875 18-9158463 19-2979718 19-6878167 20-0855369

X

»

2-34 2 "36 2-38 2-40

250

2-92 2-94 2-96 2-98

3°o


II.

701

Functions of imaginary argument, and ex

KQ (x)

ex

K

ex

X

20-491292

3-02 3-04

X

e~*h(x)

3.02 3-04 3-06 3-08 3-10

0-2420822 0-2411745 0-2402772 0-2393899 0-2385126

o- 1964383

0-1960506 0-1956637 0-1952775 0-1948921

3-12 3-14 3 -i6

0-2376451 0-2367871 0-2359385 0-2350991 0-2342688

0-1941240 0-1937412 0-1933594 0-1929786

0-6850631 0-6830138 0-6809829 0-6789701 0-6769751

0-7879938 0-7850176 0-7820736 0-7791613 0-7762803

22-646380 23-103867 23-570596 24-046754

314

24532530

320

0-2334475 0-2326348 0-2318308 0-2310352 0-2302480

0-1925988 0-1922200 0-1918423 0-1914657 0-1910902

0-6749978 0-6730377 0-6710948 0-6691687 0-6672592

o-7734299 0-7706096 0-7678189 °- 765057 3 0-7623243

25-028120 25-533722 26-049537 2 °-575773 27-112639

3*22 3-24 3-26 3-28 3-30

0-2294689 0-2286978 0-2279346 0-2271792 0-2264314

0-1907158 0-1903425 0-1899704 0-1895995 0-1892299

0-6653660 0-6634890 0-6616278 0-6597823 0-6579523

0-7596194 0-7569422 0-7542922 0-7516690 0-7490721

27-660351 28-219127 28-789191 29-370771 29-964100

3-34 3-36 3-38

0-2256911 0-2249582

0-6561375 0-6543377 0-6525527 0-6507823 0-6490263

0-7465010 0-7439555

30-569415 31-186958 31-816977 32-459722

318 320

3*22 3*24 3.26 3-28

330 332 3*34 3-36 3-38

3HO

01945076

e*

0-6955958 0-6934501 0-6913243 0-6892181 0-6871311

1

{x)

0-8033807 0-8002339 0-7971224

07940457 0-7910030

20905243 21-327557 21-758402 22-197951

306 3-o8 3-10 3-12

3-16 3-18

332

340

3-44 3-46 3-48 3-5°

0-2235140 0-2228024

0-1888614 0-1884941 0-1881282 0-1877634 0-1874000

3-52 3\54 3-56 3-58 3-60

0-2220978 0-2214000 0-2207089 0-2200243 0-2193462

0-1870378 0-1866770 0-1863174 0-1859592 0-1856022

0-6472846 °-°455569 0-6438430 0-6421427 0-6404560

o-73i5957 0-7291947 0-7268165 0-7244607

33-784428 34-466919 35-163197 35-873541 36-598234

3-52 3\54 3-56

3-62 3-64 3-66 3-68

370

0-2186745 0-2180091 0-2173498 0-2166966 0-2160494

0-1852467 0-1848924 0-1845396 0-1841880 0-1838379

0-6387825 0-6371221 '6354747 0-6338401 0-6322181

0-7221270 0-7198150 0-7175245 0-7152551 0-7130065

37-337568 38-091837 38-861343 39-646394 40-447304

3-62 3-64 3-66 3-68 3-7o

372 374 376 378 380

0-2154081 0-2147726 0-2141429 0*2135187 0-2129001

0-1834891 0-1831416 0-1827956 0-1824509 0-1821076

0-6306085 0-6290112 0-6274261 0-6258529 0-6242916

0-7107784 0-7085704 0-7063823 0-7042139 0-7020647

41-264394 42-097990 42-948426 43-816042 44-701184

3-72 3-74 3-76

378 380

3-82 3-84 3-86 3-88

0-2122870 0-2116793 0-2110768 0-2104796 0-2098875

0-1817657 0-1814251 0-1810860 0-1807482 0-1804119

0-6227419 0-6212038 0-6196771 0-6181617 0-6166573

06999345 0-6978232 0-6957302 0-6936555 0-6915988

45-604208 46-525474 47-46535I 48-424215 49-402449

3-82 3-84 3-86 3-88 3-90

0-2093005 0-2087186 0-2081415 0-2075693 0-2070019

0-1800769 o-i797433 0-1794111 0-1790803 0-1787508

0-6151640 0-6136814 0-6122096 0-6107484 0-6092977

0-6895598 0-6875382 o-6855339 0-6835466 0-6815759

50-400445 51-418601 52-457326 53*517034 54-598I50

3-92 3-94 7 3-96 3-98 4-00

3H2

390 392 394 396 3-98 4-00

02242325

07414350 o-738939i 0-7364675

07340199

33115452

3-42 3*44 3-46 3-48 3-5o

358 360


702 Table

X

e-*I

(x)

II.

Functions of imaginary argument, and e x

e~ x h{x)

ex

K

(x)

e*K x {x)

ex

x

4-02 4-04 4-06 4-08 4-10

0-2064393 0-2058812 0-2053278 0-2047789 0-2042345

0-1784228 0-1780961 0-1777709 0-1774470 0-1771245

0-6078573 0-6064270 0-6050069 0-6035968 0-6021965

0-6796219 0-6776840 0-6757623 0-6738564 0-6719662

55-701106 56-826343 57-974311 59-I45470 60-340288

4-02 4-04 4-06 4-08 4-10

4-12 4-14 4-16 4-18 4-20

0-2036945 0-2031589 0-2026275 0-2021003 0-2015774

0-1768033 0-1764836 0-1761652 0-1758482 o-i755325

0-6008060 0-5994251 0-5980537 0-5966917

61-559242 62-802821

05953390

0-6700914 0-6682318 0-6663872 0-6645575 0-6627424

4-12 4-14 4-16 4-18 4-20

4-22 4-24 4-26 4-28 4-30

0-2010585 0-2005438 0-2000330 0-1995262 0-1990233

0-1752182 0-1749053 o-i745937 0-1742835 o-i739746

Q-5939955 6-5926611 o-59i3357 0-5900192 0-5887114

0-6609418 o-659i553 0-6573830 0-6556246 0-6538798

68-033484 69-407852 70-809983 72-240440 73-699794

4-22 4-24 4-26 4-28

4-32 4-34 4-36 4-38 4-40

0-1985242 0-1980290 o-i975375 0-1970497 0-1965656

0-1736671 0-1733609

0-1727525 0-1724502

0-5874124 0-5861220 0-5848400 0-5835665 0-5823013

0-6521486 0-6504308 0-6487262 0-6470346 0-6453559

75-188628 76-707539 78-257134 79-838033 81-450869

4-32 4*34 4-36 4-38 4-40

4-42 4-44 4-46 4-48 4-50

0-1960851 0-1956081 0-1951347 0-1946648 0-1941983

0-1721493 0-1718497 0-1715515 0-1712545 0-1709588

0-5810443 0-5797954 o-5785546 0-5773218 0-5760968

0-6436899 0-6420364 0-6403953 0-6387665 0-6371498

83-096285 84-774942 86-487509 88-234673

4-42 4*44 4-46 4-48 4-50

4-52 4'54 4-56 4-58 4-60

0-I93735 2 0-1932754 0-1928190 0-1923658 0-1919159

0-1706644 0-1703713 0-1700795 0-1697890 0-1694997

o-5748796 0-5736701 0-5724683 0-5712740 0-5700872

0-6355450

9I-835598 93-690800 95-58348o

4-62 4-64 4-66 4-68

470

0-1914692 0-1910256 0-1905851 0-1901478 0-1897134

0-1692117 0-1689250 0-1686395 0-1683553 0-1680723

0-5689078 o-5677357 0-5665708 0-5654131 0-5642625

4-72 4-74 4-76 4-78 4-80

0-1892821 0-1888538 0-1884283 0-1880058 0-1875862

0-1677905 0-1675100 0-1672307 0-1669526 0-1666757

4-82 4-84 4-86 4-88 4-90

0-1871694 0-1867554 0-1863442 0-1859357 0-1855300

4-92 4-94 4-96 4-98

0-1851269 0-1847265 0-1843287

00

0-1835408

5

0183933s

01 730560

64071523 65-365853 66-686331

90017131

430

99-484316

4-52 4*54 4-56 4-58 4-60

0-6276955 0-6261595 0-6246345 0-6231203 0-6216169

101-494032 103-544348 105-636082 107-770073 109-947172

4-62 4-64 4-66 4-68 4-70

0-5631189 0-5619823 0-5608525 o-5597295 0-5586133

0-6201241 0-6186418 0-6171699 0-6157082 0-6142566

112-168253 114-434202 116-745926 1 19- 104350 121-510418

4-72 4-74 4-76 4-78 4-80

0-1664000 0-1661256 0-1658523 0-1655802 0-1653093

0-5575038 0-5564008 0-5553045 o-5542i45 o-553i3io

0-6128151 0-6113834 0-6099616 0-6085494 0-6071468

123-965091 126-469352 129-024202 131-630664 134-289780

4-82

0-1650396 0-1647710 0-1645036 0-1642374 0-1639723

0-5520539 0-5509830 0-5499184 0-5488599 0-5478076

0-6057537 0-6043699 0-6029955 0-6016301 0-6002739

137-002613 139-770250 I42-593796

4-92 4-94 4-96 4-98

06339521 0-6323708 0-6308010 0-6292426

97-5M394

M5-474382 148-413159

4-f4

41^ 4-88 4-90

500


hi*) 5-02 5-04 5-06 5-08 5-10

0-1831507 0-1827631 0-1823780 o-i8i9953 0-1816151

5-12 5-14

II.

703


e~ x h(x)

0-1637083

K

(x)

ex

K

x

{x)

0-1631838 0-1629233 0-1626639

0-5467613 0-5457209 0-5446865 0-5436580 0-5426354

0-^989266 o-597588i 0-5962584 o-5949375 0-5936250

151-41130 154-47002 I57-59052 160-77406 164-02191

5-02 5-04 5-06 5-08 5-10

0-1812373 0-1808618 0-1804887 0-1801180 °-i 797495

0-1624055 0-1621483 0-1618922 0-1616372 0-1613833

0-5416184 0-5406072 0-5396017 0-5386017 0-5376074

0-5923211 0-5910256 0-5897384 0-5884594 0-5871886

I67-33537 170-71577 174-16446 177-68281 181-27224

5-12 5-14 5-16 5-i8 S-20

0-1793833 0-1790194 0-1786577 0-1782982 0-1779409

0-1611304 0-1608787 0-1606280 0-1603784 0-1601298

0-5366185 0-535635° 0-5346570 0-5336843 0-5327170

0-5859258 0-5846710 0-5834241 0-5821850 0-5809536

184-93418 188-67010 192-48149 196-36988 200-33681

5-22 5-24 5-26 5-28 5-3o

o-i775857 0-1772327 0-1768818 o-i76533i 0-1761863

0-1598823 0-1596358 0-1593904 0-1591460 0-1589026

o-53i7549 0-5307980 0-5298462 0-5288996 0-5279580

0-5797299 0-5785I37 0-5773050 0-5761038 0-5749099

204-38388 208-51271 212-72495 217-02228 221-40642

5-32 5-34

0-1758417

0-1586603 0-1584189 0-1581786

0-1577010

0-5270215 0-5260899 0-5251633 0-5242416 0-5233247

o-5737233 o-5725438 0-57I37I5 0-5702062 0-5690480

225-87912 230-44218 235-09742 239-84671 244-69193

5-42 5*44 5-46 5-48 5-50

0-1734850 0-1731561 0-1728291

o-i574637 0-1572274 0-1569920 0-1567576 0-1565242

0-5224127 0-5215053 0-5206028 0-5197049 0-5188116

0-5078966 0-5667521 0-5656144 0-5644834 0-5633590

249-63504 254-67800 259-82284 265-07161 270-42641

5-52 5*54 5-56 5-58 5-60

0-1725039 0-1721806 0-1718591 0-I7I5394 0-1712215

0-1562918 0-1560603 0-1558298 0-1556002 o-i5537i6

0-5179230 0-5170389 0-5161593 0-5152842 0-5144136

0-5622413 0-5611300 0-5600253 0-5589269 o-5578348

275-88938 281-46272 287-14864 292-94943 298-86740

5-62 5-64 5-66 5-68 5-70

0,I 55i439

o-5i35474 0-5126855 0-5118280 0-5109748 0-5101258

o-556749i 0-5556695 o-5545962 o-5535289 0-5524676

304-90492 311-06441 3i7-34«33 323-759I9 330-29956

5-72 5-74 5-76

5-8o

0-1709054 0-1705911 0-1702785 0-1699676 0-1696584

5-82 5-84 5-86 5-88 5-90

0-1693509 0-1690451 0-1687410 0-1684385 0-1681377

o-i535758 o -I 533554 0-I53I359

0-5092811 0-5084406 0-5076042 0-5067719 0-5059438

0-5514124 0-5503631 o-5493i97 0-5482821 0-5472503

336-97205 343-77934 350-72414 357-80924 365-03747

5-92

0-1678384 0-1675408 0-1672448 0-1669503 0-1666574

0-1529172 0-1526995 0-1524826 0-1522666 0-1520515

0-5051197 0-5042996 0-5034835 0-5026713 0-5018631

0-5462242 0-5452037 0-5441889 o-543i796 o-542i759

5-i8 5-20 22

24 26 28 30 5-32 5'34 5-36 5-38

5HO 5-42 5'44 5'46 5-48 5-5° 5-52 5'54 5-56 5-58 5-60 5-62 5-64 5-66 5-68

57° 572 574 576 578

594 5-96 5-98 6-oo

01 754991 0-1751585 0-1748199 0-1744833

0-1741486

T738i59

o- I 634455

OT579393

0-1549171 0-1546913 0-1544664 0-1542424

0-1540193 °- I 53797i

536 5-38 5-4°

578 580 5-82 5-8 4

5-86 5-88 5-9o

372-41171

5 92

37993493

5-94 5-96 5-98 6-oo

387-61012 395-44037 403-42879


704 Table

X

e-*I

(x)

II.


e-'I^xy

e*

K

(x)

ex

K

1

(x)

ex

ex

x

6-02 6-04 6-o6 6-o8 6-io

0-1663661 0-1660763 0-1657880 0-1655012 0-1652159

0-1518372 0-1516237 0-1514111 0-1511994 0-1509885

0-5010588 0-5002584 0-4994618 0-4980689 0-4978799

0-5411776 0-5401848 o-539i973 0-5382151 o-537 2 382

411-57860 419-89303 428-37544 437-02919 445-85777

6-02 6-04 6-o6 6-o8 6-io

6-12 6-14 6-16 6-i8 6-20

0-1649321 0-1646498 0-1643689 0-1640894 0-1638114

0-1507784 0-1505691 0-1503607 0-1501531 0-1499463

0-4970946 04963 1 30 0-495535J 0-4947608 0-4939902

0-5362666 0-535300I o-5343387 0-5333825 o-53243i3

454-86469 464-05357 473-42807 482-99196 492-74904

6-12 6-14 6-i6 6-i8 6-20

6-22 6-24 6-26 6-28 6-30

0-1635348 0-1632596 0-1629858 0-1627134 0-1624424

0-1497403 0-1495351 0-1493307 0-1491271 0-1489243

0-4932232 0-4924597 0-4916998 0-4909434 0-4901905

0-5314851 0-5305438 0-5296075 0-5286761 o-5277494

502-70323 512-85851 523-21894 533-78866 544-57191

6-22 6-24 6-26 6-28 6-30

6-32 6-34 6-36 6-38 6-40

0-1621727 0-1619044 0-1616374 0-1613717 0-1611073

0-1487223 0-1485211 0-1483206 0-1481209 0-1479220

0-4894411 0-4886950 0-4879524 0-4872132 0-4864773

0-5268276 0-5259105 0-5249982 0-5240905 0-5231874

555-57299 566-79631 578-24636 589-92771 601-84504

6-32 6-34 6-36 y 6-38 6-40

6-42 6-44 6-46 6-48 6-50

0-1608443 0-1605825 0-1603220 0-1600628 0-1598048

0-1477238 0-1475264 0-1473297 0-1471338 0-1469386

o-4857448 0-4850156 0-4842896 0-4835669 0-4828474

0-5222889 0-5213950 0-5205056 0-5196207 0-5187402

614-00311 626-40680 639-06106 651-97095 665-14163

6-42 6-44 6-46 6-48 6-50

6-52 6'54 6-56 6-58 6-6o

0-1595481 0-1592927 0-1590385 0-1587855 o-i585337

0-1467442 0-1465505 0-1463576 0-1461653 o-i459738

0-4821312 0-4814181 0-4807082 0-4800014 0-4792978

0-5178642 0-5169925 0-5161251 0-5152620 0-5144032

678-57839 692-28658 706-27169 720-53933

73509519

6-52 6-54 6-56 6-58 6-6o

6-62 6-64 6-66 6-68 6-70

0-1582831 0-1580336 0-1577854 o-i5753 8 4 0-1572925

0-1457830 0-I455930 0-1454036 0-1452149 0-1450270

0-4785972 0-4778997 0-4772053 0-4765138 0-4758254

0-5135486 0-5126982 0-5118520 0-5110099 0-5101719

749-945 10 765-09499 780-55094 796-31911 812-40583

6-62 6-64 6-66 6-68 6-70

6-72

0-1570477 0-1568042 0-1565617

0-1448397 0-1446532 0-1444673 0-1442821 0-1440976

0-4751400 o-4744575 o-4737779 0-4731013 0-4724276

0-5093380 0-5085080 0-5076821 0-5068602 0-5060421

828-81751 845-56074 862-64220 880-06872 897-84729

6-72 7 f' 2 6-76 6-78 6-8o

0-5052280 0-5044178 0-5036114 0-5028088 0-5020099

915-98501 934-489I3 953-36707 972-62636 992-27472

6-82 6-84 6-86 6-88 6-90

0-5012149 0-5004235 o-4996359

1012-31999 1032-77021 1053-63356 1074-91837 1090-63316

6-92 6-94 6-96 6-98 7-00

t% 6-78 6-8o

01563204

6-82 6-84 6-86 6-88 6-90

0-1558411 0-1556031 0-1553662 0-1551304 0-1548956

0-1439138 0-1437306 0-1435481

0-4717567 0-4710887

01433663 0-1431852

0-4697612 0-4691016

692

0-1546619 0-1544293 0-1541978 0-1539672 0-1537377

0-1430047 0-1428248 0-1426457 0-1424671 0-1422892

0-4684449 0-4677908 0-4671395 0-4664910 0-4658451

6-94 6-96 6-98 7-00

0-1560802

04704235

0-498851.9

0-4980716

TABLES OP BESSEL FUNCTIONS Table

II.

705


ex

e*K x {x)

ex

X

0-4652019 0-4645614 0-4639235 0-4632882 0-4626556

0-4972948 0-4965217 o-49575 21 0-4949860 0-4942235

1118-7866 1141-3876 1164-4452 1187-9685 1211-9671

7-02 7-04 7-06 7-08 7-10

0-1412352 0-1410617 0-1408889 0-1407166 0-1405450

0-4620255 0-4613980 0-4607731 0-4601507 o-45953o8

o-4934644 0-4927087 0-4919565 0-4912077 0-4904623

1236-4504 1261-4284 1286-9109 1312-9083 I339-4308

7-12 7-14 7-16

0-1512796 0-1510620 0-1508453 0-1506295 0-1504147

0-1403739 0-1402035 0-1400337 0-1398644 0-1396958

0-4589134 0-4582985 0-4576861 0-4570761 0-4564686

0-4897202 0-4889814 0-4882459 0-4875137 0-4867848

1366-4891 1394-0940 1422-2565 1450-9880 1480-2999

7-22 7-24 7-26 7-28 7'30

7*32 7-34 7'30 7'38 7-40

0-1502007 0-1499877 o-i497756 0-1495644 0-1493541

0-1395277 0-1393603 0-1391934 0-1390271 0-1388613

0-4558634 0-4552607 0-4546604 0-4540625 0-4534669

0-4860591 0-4853365 0-4846172 0-4839010 0-4831880

1510-2040 1540-7121 1571-8366 1603-5898 1635-9844

7.32 7-34 7-36 7-38 7-40

7-42 7'44 7-4° 7 48

7SO

0-1491447 0-1489362 0-1487285 0-1485218 0-1483158

0-1386962 0-1385316 0-1383676 0-1382041 0-1380412

0-4528736 0-4522827 0-4516941 0-4511077 0-4505237

0-4824780 0-4817712 0-4810674 0-4803667 0-4796689

1669-0335 1702-7502 1737-1481 1772-2408 1808-0424

7-42 7-44 7-46 7-48 7-50

7'52 7-54 7'56 7-58 7-60

0-1481108 0-1479066 0-1477032 0-1475007 0-1472990

0-1378789 0-1377171 Q-I375559 o-i373952 0,I 37 2 350

04499419 0-4493624 0-4487851 0-4482101 0-4476372

0-4789742 0-4782825 o-4775937 0-4769079 0-4762249

1844-5673 1881-8300 1919-8455 1958-6290 1998-1959

7-52 7-54 7-50 7-58 7-60

7-62 7 7 tt 7-68 7 7o

0-1470981 0-1468981 0-1466988 0-1465004 0-1463028

0-I370754 0-1369164 o-i367579 0-1365999 0-1364424

0-4470665 0-4464981 o-44593i7 o-4453676 0-4448056

o-4755449 0-4748678 o-474i935 0-4735220 0-4728534

2038-5621 2079-7438 2121-7574 2164-6198 2208-3480

7-62 7-64 7-66 7-68 7-70

7-72 7'74 7.76 7-78 7-80

0-1461060 0-1459100 0-1457148 0-1455203 0-1453267

0-1362855 0-1361291 0-1359732 0-1358179 0-1356630

0-4442457 0-4436879 0-4431322 0-4425786 0-4420271

0-4721876 o-47!5245 0-4708642 0-4702066 0-4695518

2252-9596 2298-4724 2344-9046 2392-2748 2440-6020

7-72 7'74 7.76 7-78 7-80

7-82 7-84 7-86 7-88 7.90

0-1451338 0-1449417 0-1447503 o-i445597 0-1443699

0-1355087 o-i353549 0-1352016 0-1350488 0-1348965

o-44i4776 0-4409302 0-4403848 0-4398414

0-4688997 0-4682502 0-4676034 0-4669593

7-82 7-84 7-86 7-88 7.90

7.92 7-94 7.96 7-98 8-oo

0-1441808 o-i439924

o-i347447 o-i345934 0-1344426 0-1342923 0-1341425

7-92 7-94 7-96 7-98 8-oo

X

e~x h{x)

t-Itix)

7'02

o- 1535093 0-1532819 0-I530554 0-1528300 0-1526056

0-1421120 0-1419354 0-1417594 0-1415840 0-1414093

7'io 7 -i8 7'20

0-1523822 0-1521597 0-1519382 0-1517177 0-1514982

7"22 7*24 7'2D 7-28 7-30

7«o6 7-08 7-10 7'12

T1 t

-

.

.0-1438048

0-1436179 0-1434318

e*K

(x)

04393001

04663178

2489-9054 2540-2048 2591-5204 2643-8726 2697-2823

0-4387607 0-4382234 0-4376880 o-437i545

0-4656789 0-4650426 0-4644089 o-4637777 0-4631491

2751-7710 2807-3605 2864-0730 2921-9311 2980-9580

04366230

7 -i8

7-20


706 Table

II.


X

e~* /„(*)

e~* I^x)

8-02 8-04 8-o6 8-o8 8-io

CV1432464 0-1430617 0-1428777 0-1426944 0-1425118

OI 33993 2

8-12 8-14 8-i6 8-i8 8-20

01423299 0-1421488 0-1419683 0-1417885 0-1416094

e*

K

(x)

ex

K

t

{x)

ea

ex

X

3041-1773 3102-6132 3165-2901

o-4345i63 o-4339944

0-4625230 0-4618994 0-4612783 0-4606597 0-4600436

32292332

8-02 8-04 8-o6 8-o8

3294-4681

810

0-1332538 0-1331073 0-1329613 0-1328158 0-1326708

o-4334743 0-4329562 o-4324398 0-4319254 0-4314127

o-4594299 0-4588186 0-4582097 0-4576033 o-4569992

3361-0207 3428-9179 3498-1866 3568-8547 3640-9503

8-12 8-14 8-i6 8-i8 8-20

3o

° -I 4 I 43°9 0-1412532 0-1410761 0-1408997 0-1407239

0-1325262 0-1323821 0-1322384 0-1320952 0-1319524

0-4309019 0-4303929 0-4298857 0-4293803 0-4288766

o-4563974 o-455798i 0-4552010 0-4546063 0-454° l 39

3714-5024 3789-5403 3866-0941 3944-1944 4023-8724

8-22 8-24 8-26 8-28 8-30

32 §•34 8-36 8- 8 3 8-40

0-1405488 0-1403744 0-1402006 0-1400274 0-1398549

0-1318101 0-1316683 0-1315269 0-1313859 0-1312454

0-4283748 0-4278747 0-4273763 0-4268797 0-4263848

0-4534238 0-4528359 0-4522504 0-4516670 0-4510859

4105-1600 4188-0897 4272-6948 4359-oo89 4447-0667

832

8-42

8-48 8-50

0-1396830 0-1395118 0-1393412 0-1391712 0-1390018

0-1311053 0-1309657 0-1308265 0-1306877 0-1305494

0-4258917 0-4254002 0-4249104 0-4244224 0-4239360

0-4505070 o-44993 3 4493559 0-4487835 0-4482134

4536-9035 4628-5550 4722-0580 4817-4499 4914-7688

8-52 8-54 8-56 8 l"5 8-6o

0-1388331 0-1386650 0-1384975 0-1383306 0-1381642

0-1304114 0-1302740 0-1301369 0-1300003 0-1298641

04234513 0-4229682 0-4224868 0-4220071 0-4215289

0-4476454 0-4470795 0-4465158 0-445954 2 0-445394 6

5014-0538 5115-3444 5218-6812

8-62 8-64 8-66 8-68

o- 1379985

0-1297283 0-1295929 0-1294579

0-4210524 0-4205776 0-4201043 0-4196326 0-4191625

0-4448372 0-4442818 0-4437285 0-4431772 0-4426280

5541-3864,

8-22 8-24 8-26 8-28 88-

l$t

870 872 8-74

876 878 8-8o

0-I378334 0-1376689 0>I 375050 o-i3734i7

0-1338443 0-1336960 0-1335481 o- 1334007

01293234 0-1291892

0-4360935 o-4355658

04350401

53241055 5431-6596 5653-3298 5767-5347 5884-0466 6002-9122

8-38 8-40 8-42

S:« 8-48 8-50 8-52 5l f 8-58 8-6o

8-62 8-64 8-66 8-68 8-70 8-72

0-4409923 0-4404511 0-4399119

6124-1791 6247-8957 6374-1116 6502-8772 6634-2440

0-1371789 0-1370167 0-1368551 0-1366941 0-1365336

0-1290555 0-1289222 0-1287892 0-1286567 0-1285246

0-4186940 0-4182270

0-4420808

04177616 0-4172978 0-4168355

04415356

l' 7 i

S"7f 8-78 8-8o

8-82 8-84 8-86 8-88 8-90

°- I

363737 0-1362144 0-1360556 o-i358974 o-i357397

0-1283929 0-1282615 0-1281306 0-1280001 0-1278699

0-4163747 0-4159155 °-4 I 54578 0-4150016 0-4145468

0-439374 6 0-4388392 0-4383058 o-4377743 0-4372448

6768-2646 6904-9926 7044-4827 7186-7907 7331-9735

8-82 8-84 8-86 8-88 8-90

8-92 8-94 8-96 8-98 9-00

0-1355826 0-1354260 0-1352700 0-1351145 o-i349595

0-1277402 0-1276108 0-1274818

0-4140936 0-4136419 0-4131917 0-4127429 0-4122955

0-4367171 0-4361913 0-4356674 o-435i454 0-4346252

7480-0892 7631-1971 7785-3575 7942-6321 8103-0839

8-92 8-94 8-96 8-98 9-00

01273532 0-1272250

1


X

9-02

904 9-06 9-08 9-10 9-12

914 9-16 9-18 9-20 9-22 9-24 9-26 9-28

930 932 9-34 9-36 9-38

II.


K

e-*IM

0-1348051 0-1346512 -i344978 0-1343450 0-1341927

0-1270971 0-1269697 0-1268426 0-1267159 0-1265895

0-4118497 0-4114053 0-4109623 0-4105207 0-4100806

0-4341069 0-4335904 0-4330758 0-4325629

0-1340409 0-1338896 o-i337388 0-1335885 0-1334388

0-1264636 0-1263380 0-1262127 0-1260879 0-1259634

0-4096419 0-4092045 0-4087686 0-4083341 0-4079010

o-43!5427

0-1332895 0-1331408 0-1329925 0-1328447 0-1326975

0-1258392 0-1257154

0-4290230 0-4285243 0-4280273

0-1254689 0-1253462

0-4074692 0-4070388 0-4066098 0-4061821 0-4057558

0-1252239 0-1251018 0-1249802 0-1248589 0-1247379

01255920

(x)

9-12 9-14 9-18 9-20

0-4270385

10097-0643 10301-0386 10509-1333 10721-4319 10938-0192

9-22 9-24 9-26 9-28 9-30

0-4053308 0-4049071 0-4044848 0-4040638 0-403644

0-4265467 0-4260565 0-4255680 0-4250811 0-4245960

11158-9819 11384-4082 11614-3885 11849-0148 12088-3807

932

0-4032257 0-4028087 0-4023929 0-4019784 0-4015651

0-4241124 0-4236305

12332-5822 12581-7169 12835-8844 13095-1864 x 3359-7268

01319684

01312513

0-1246173 0-1244970 0-1243771 0-1242575 0-1241382

952

0-1311092 0-1309677 0-1308266 0-1306859 0-I305457

0-1240193 0-1239008 0-1237825 0-1236646 0-1235470

0-4011532 0-4007425 0-4003331 0-3999249 0-3995180

968

0-1304060 0-1302667 0-1301278 0-1299894

9-70

01298514

0-1234298 0-1233128 0-1231962 0-1230800 0-1229640

9.72

0-1297139 0-1295768 0-1294401 0-1293039 0-1291681

0-1228484 0-1227331 0-1226181 0-1225034 0-1223891

9-86 9-88 9-90

0-1290328 0-1288978 0-1287633 0-1286292 0-1284955

0-1222751 0-1221613 0-1220479

0-1218220

o-3947 2 99 o-3943386 o-3939485 o-3935596

9 92 9-94 9-96 9-98 io-oo

0-1283623 0-1282294 0-1280970 0-1279650 0-1278333

0-1217096 0.12 15974 0-1214855 0-1213739 0-1212627

o-393r7i7 0-3927851 o-3923996 0-3920152 0-3916319

9-76

978 9-80 9-82

9*4

01219348

X

9136-2016 9320-7651 9509-0571 9701-1528 9897-1291

0-1318240 0-1316801 0-1315367 0-1313938

974

ex

9-02 9-04 9-06 9-08 9-10

9-42 9'44 9-46 9-48 9-50

9-62 9-64 9-66

K,[x)

ea

8266-7771 8433-777I 8604-1507 8777-9660 8955-2927

940

9*54 9-56 9-58 9-60

ex

e-*I 9 {x)

0*1325507 0-1324044 0-1322580 0-1321133

707

04320519 04310352 04305295 04300256 04295234

04275321

04231502 0-4226716 0-4221945

916

9-34 9-36 9-38 9-40 9-42

944 9-46 9-48 9-50

0-4198332

13629-6112 13904-9476 14185-8462 14472-4193 14764-7816

9-56 9-58 9-60

0-3991123 0-3987078 0-3983046 0-3979026 o-39750i8

0-4193656 0-4188996 0-4184351 0-4179721 0-4175107

15063-0499 I5367-3437 15677-7847 I5994-4969 16317-6072

9-62 9-64 9-66 9-68 9-70

0-3971023 0-3967039 0-3963067 o-3959i67 o-3955!59

0-4170508 0-4165924 0-4161355 0-4156801 0-4152261

16647-2447

972

i69»354 I 4 17326-6317 17676-6529 18033-7449

9-74 9-76 9-78 9-80

03951223

o-4i47737 0-4143227 0-4138731

04134250 04129784

18398-0507 18769-7160 19148-8894 I9535-7227 I9930-3704

9-82 9-84 9-86 9-88 9-90

0-4125332 0-4120894 0-4116471 0-4112061 0-4107666

20332-9906 20743-7443 21162-7957 21590-3125 22026-4658

9-92 9-94 9-96 9-98 io-oo

0-4217191 0-4212452 0-4207730

04203023

9-52

954


708 Table

X

e~*I

(x)

II.


e~*I x {x)

e*

K

(x)

ex

e'K^x)

ex

X 10-02 IO-04

10-02 IO-04 10-06 10-08 IO-IO

0-1277021 0-1275713 0-1274409 0-1273109 0-1271813

0-1211517 0-1210411 0-1209307 0-1208206 0-1207109

0-3912498 0-3908688 0-3904889 0-3901101 o-38973 2 4

0-4103284 0-4098917 0-4090223 0-4085897

22471-430 22925-383 23388-506 23860-986 24343-oo9

IO-I2 10-14 IO-IO 10-18 10-20

0-1270521 0-1269233 0-1267948 0-1266608 0-1265392

0-1206014 0-1204922 0-1203833 0-1202747 0-1201664

o-3893558 0-3889803 0-3886059 0-3882325 0-3878603

0-4081584 0-4077285 0-4073000 0-4068727 0-4064468

24834-771 25336-466 25848-297 26370-467 26903-186

IO-I2 10-14 IO-IO 10-18 10-20

IO-22 IO-24 IO-26 IO-28 IO-30

0-1264119 0-1262850 0-1261585 0-1260324 0-1259067

0-1200584 0-1199500 0-1198432 0-1197360 0-1196292

0-3874891 0-3871189 0-3867498 0-3863818 0-3860149

0-4060223 0-4055990 0-4051771 0-4047565 0-4043372

27446-666 28001-126 28566-786 29143-874 29732-619

IO-22 10-24 IO-26 IO-28 IO-30

IO32 10-34 10-36 IO- 3 8 10-40

0-1257813 0-1256563 0-1255317 0-1254075 0-1252836

0-1195226 0-1194162 0-1193102 0-1192044 0-1190990

0-3856489 0-3852841 0-3849202 o-3845574 0-3841956

0-4039191 0-4035024 0-4030869 0-4026728 0-4022598

30333-258 30946-030 31571-181 32208-961 32859-626

IO-32 io-34 10-36 10.38 10-40

IO-42 10-44 10-46 10-48 IO-50

0-1251601 0-1250369 0-1249141 0-1247917 0-1246697

0-1189938 0-1188888 0-1187842 0-1186798 0-1185757

0-3838348 o-383475o 0-3831163 0-3827586 0-3824018

0-4018482 0-4014378 0-4010286 0-4006207 0-4002140

33523-434 34200-652 3489I-55I 35596-4o8 36315-503

10-42 10-44 10-46 10-48 10-50

IO-52 10-54 10-56 10-58 io-6o

0-1245480 0-1244266 0-1243056 0-1241850 0-1240647

0-1184718 0-1183682 0-1182649 0-1181619 0-1180591

0-3820461 0-3816913 o-38i3375 0-3809848 0-3806330

0-3998085 0-3994043 0-3990013 o-3985995 0-3981989

37049-124 37797.566 38561-128 39340-114 40134-837

10-52 IO-54 10-56 10-58 io-6o

10-62 10-64 IO-66 10-68 10-70

0-1239448 0-1238252 0-1237059 0-1235870 0-1234685

0-1179566 0-1178544 0-1177524 0-1176507 0-1175492

0-3802821 o-3799323 o-3795834 o-3792354 0-3788884

Q-3977995 0-3974013 0-3970043 0-3966084 0-3962137

40945-615 41772-771 42616-637 43477-550 44355-855

10-62 10-64 10-66 io-68 10-70

10-72 10-74 10-76 10-78 io-8o

0-1233503 0-1232324 0-1231149 0-1229977 0-1228808

0-1174480 0-1173471 0-1172464 0-1171459 0-1170458

0-3785424 0-3781973 o-3778532 o-3775 IO ° 0-3771677

0-3958202 o-3954 2 79 0-3950367 0-3946467 o-3942578

45251-903 46166-052 47098-668 48050-124 49020-801

10-72 10-74 10-76 10-78 io-8o

10-82 10-84 io-86 io-88 10-90

0-1227642 0-1226480 0-1225322 0-1224166 0-1223014

0-1169458 0-1168462 0-1167467 0-1166476 0-1165487

0-3768264 0-3764860 0-3761465 0-3758079 0-3754702

0-3938701 o-3934835

0-3927137 0-3923305

50011-087 51021-378 52052-078 53103-600 54176-364

10-82 10-84 10-86 io-88 10-90

10-92 10-94 10-96

0-1221865 0-1220719 0-1219577 0-1218438 0-1217302

0-1164500 0-1163516 0-1162534 0-1161554 0-1160570

o-375i335 o-3747976 0-3744627 0-3741287 o-3737955

0-3919484 0-39I5673 0-3911874 0-3908086 0-3904309

55270-799 56387-343 5Z526-443 58688-554 59874-142

10-92 10-94 10-96 10-98 II-OO

1098 II-OO

04094563

03930980

IOO6 10-08 IO-IO

11


II.


X

«"*/,(*)

«-"A(*)

II-02

0-I2l6l69

0-1159603

OI2I5039

01158631

VSot n-o8

0-1149990 0-1149042 0-1148096 0-1147152

1132 "•34

o-

11-36 11-38

0-1197382 0-1196303 0-1195228

0-1145272 01 144335 0-1143401 0-1142468 0-1141538

0-1194156 0-1193086 0-1192020 0-1190956 0-1189895

0-1140610 0-1139685 0-1138762 0-1137841 0-1136922

0-1188837 0-1187782 0-1186729 0-1185680 0-1184633

0-1136005 0-1135090 0-1134178 0-1133268 0-1132360

0-1183589 0-1182548 0-1181509

0-1131454 01 1 3055 0-1129649 0-1128750 0-1127852

Wit 11-28 11-30

1140 11-42

n-44 11.46 11-48

1150 n-52

"54 11-56 11-58

1160 11-62 11-64

1

199547

01 198463

n-66 n-68

01180473

11-70

0-1179440

11-72

0-1178410 0-1177382 0-1176357 OII 75335 0-1174315

n-74 11-76 11-78 II-OO

01146211

01122513 0-1121630 0-1120750 0-1119871 01 1 18995

0-1168252 0-1167251 0-1166252

0-IIl8l20 0-1117248 0-1116378 0-1115509 0-1114643

11-92

j;u ii- 9 8

01165256

12-00

0-1164262

1112

0-3863468 0-3859818 0-3856178

03852548 03848929

74607-775 76114-952 77652-576 79221-262 80821-638

II-2 4 11-26 11-28

03685833

03845320

82454343

11-32

0-3682648

0-3841721 0-3838132

84120-031 85819-368 87553-035 89321-723

1138 1140

0-3701886 0-3698659 °-369544° 0-3692229 0-3689027

03679470 0-3676301 0-3673140

03889309

03834553 0-3830984

3669987

03827425

03666843

0-3823875 0-3820336 0-3816806 0-3813286

0-3663706 0-3660578 o-3^57457 o-3654344

03651240

0-3809775 0-3806275

0-3648143 0-3645054 o-364i973

03802783

0-3638900

0-3792367 0-3788914 0-378547°

03635834 0-3632777 0-3629727 0-3626684

0-3608590 0-3605600 0-3602618

01173298 0-1172284 0-1171272 0-1170263 0-1169256

-90

67507-906 68871-656 70262-956 71682-362 73130-442

0-3623650 0-3620623 0-3617603 o-36i459i 0-3611587

'Ait n-88 11

0-3881873 0-3878171 0-3874480 0-3870799 0-3867128

0-3714881 0-3711619 0-3708367 0-3705122

0-1126957 0-1126064 0-1125173 01 124284 0-1123398

11-82

X

0-3885586

03718151

0-1151894 01 1 5094

ex

11

03721430

01152849

ex

61083-680 62317-652 63576-552 64860-883 66171-160

01155730

0-1205011 0-1203912 0-1202817 0-1201724 0-1200634

II-20 11-22

e*Kt(x)

0-1157662 0-1156694 0-1154767 0-1153807

1118

(x)

0-3900543 0-3896788 0-3893043

0-I2I055I 0-1209437 0-1208326 0-1207218 0-1206113

11-12 11-14 ii-i6

e*K

o-3734632 0-373I3I9 0-3728014 0-3724717

0-I2I39I2 0-I2I2789 0-I2II669

II-IO

709

03599643 03596676 o-35937i6 0-3590763 °' 3587»i» 0-3584880 0-3581949

0-3799302 0-3795830

03782035 03778610 o-3775i94 0-3771787 0-3768389

02 04

1 1

11-06 11-08 II-IO

A-lt n-i8 II-20 11-22

II30

z&

91126-142 92967-012 94845-070 96761-068 98715-771

1142

100709-962 102744-438 104820-013 106937-518 109097-799

1152

11-44 11-46 1 1 48 11-50 -

xx-lo 11-58 11-60 11-62 11-64

111301-721 113550-165 115844-030 118184-235 120571-715

n-66 n-68

W%

11-70

1172

03765001

123007-425 1 25492 340 128027-453 130613-780

0-3761621

133252353

0-3758251

135944229

03754890

1 38690-48

o-375i537 0-3748194 o-3744859

141492-218 I44350-55I 147266-625

n-86 n-88

o-374i533 0-3738216 o-37349o8 0-3731608

150241-608 153276-690 156373-085 159532-031 162754-791

1192 n-94

03728318

<>

11-78 II-OO 11-82 11-84

11-90

1 1

-96

11-98 I2-00


710 Table

X

e-I

(x)

II.

Functions of imaginary argument, and e*

e-*Ii(x)

e*

K

(x)

e*K x {x)

ex

X

0-3725035 0-3721762 0-3718497 0-3715240 0-3711992

166042-66 169396-94 172818-99 176310-16 179871-86

12-02 I2-04 I2-00 12-08 I2-IO

183505-51 187212-57 190994-52 194852-86 198789-15

12-12 12-14 I2-IO

12-22 12-24 12-26 12-28 1 2 30

12-02 I2-04 12-00 12-08 I2-IO

0-1163271 0-1162283 0-1161296 0-1160313 0-1159332

0-1113779 0-1112916 0-1112056 0-1111197 0-1110341

0357902s

12-12 I2-IA I2-IO 12-18 12-20

o-ii58353

0-1109487 0-1108634 0-1107783 0-1106935 o- 1 106088

0-3564513 0-3561631 o-3558757 0-3553029

o-37o8753 0-3705522 0-3702299 0-3699085 0-3695879

0-H57377 0-1156404 0-1155432 0-1154464

0-3576108 o-3573i99 o-357o 2 96 0-3567401

03555890

I2l8 12-20

12-22 12-24 12-26 12-28 I2-30

o- 1 o- 1

153497 152533 0-1151572 0-1150613 0-1149656

0-1105243 0-1104400 0-1103559 o-i 102720 0-1101883

0-3550176 o-3547329 0-3544489 0-3541656 0-3538830

0-3692681 0-3689492 0-3686311 0-3683138 o-3679973

202804-96 206901-89 211081-59 2I5345-72 219695-99

12-32 12-34 12-36 12-38 12-40

0-1148702 0-1147750 0-1146801 0-1145853 0-1144909

0-1101048 0-1100215 0-1099383 0-1098553 0-1097726

o-3536oio o-3533i98 0-3530392 o-35 2 7592 0-3524800

0-3676816 0-3673667 0-3670527 0-3667394 0-3664269

224134-14 228661-95 233281-23 237993-82 242801-62

12-42 12-44 12-46 12-48 12-50

0-1143966 0-1143026 0-1142088 0-1141153 0-1140219

0-1096900 0-1096076 0-1095253 0-1094433 0-1093614

03522014 o-35!9234 0-3516461 0-35I3695 0-3510935

0-3661152 0-3658044 o-3654943 0-3651849 0-3648764

12-52 12-54 12-56 12-58 12-60

0-1139288 0-1138360

0-1136509 0-1135587

0-1092798 0-1091983 0-1091169 01 090358 o- 1089549

0-3508182 0-3505435 0-3502694 o-349996o o-3497233

0-3645687 0-3642617 o-3639555 0-363650° o-3633453

273758-o6 279288-34 284930-34 290686-31

29655857

12-54 12-56 12-58 12-60

12-62 12-64 12-66 12-68 12-70

0-1134668 0-1133750 0-1132835 0-1131922 0-1131011

0-1088741 0-1087935 0-1087131 0-1086328 0-1085527

0-3494512 0-349I797 0-3489088 0-3486386 0-3483690

0-3630414 0-3627383 0-3624359 0-3621342 0-3618333

302549-45 308661-35 314896-72 321258-06 327747-90

12-62 12-64 12-66 12-68 12-70

12-72 12-74 12-76 12-78 I2-0O

0-1130103 0-1129196 0-1128292 .0-1127390 0-1126490

0-1084728 0-1083931 0-1083136 0-1082342 0-1081550

0-3481000 0-3478317 o-3475639

0'36i5332

33436885

03612337

341123-55 348014-70 355045-06 362217-45

12-72 12-74 12-76 12-78 I2-80

12-82 12-84 12-86 12-88 12-90

0-1125592 0-1124697 0-1123803 0-1122912 0-1122023

0-1080760 0-1079971 0-1079184 0-1078399 0-1077616

12-92

0-1121136 1120251 0-1119368 0-1118487 0-1117608

0-1076834 0-1076054 0-1075276 o- 1074499 0-1073724

1294 12-96 12-98

1300

0-H37433

247706-54

25271054 257815-63 263023-85 268337-29

03472968

0-3609351 0-3606371

0-3470303

03603399

0-3467644 0-3464991 0-3462345 0-3459704 0-3457070

0-3600434 o-3597477 o-3594527 0-359I584 0-3588648

36953473

o-345444i 0-3451818 0-3449202 o-344659i o-3443986

0-35857I9 0-3582798 0-3579883 o-3576976 0-3574076

408399-03 416649-24 425066-11

376999-82 384615-73 392385-48 400312-19

43365302 44241339

12-32

1234 12-36 I2- 3 8 12-40 12-42 12-44 12-46 12-48

1250 1252

12-82 12-84 12-86

1288 12-90 12-92 12-94 12 12

96 98

I3-00

11


II.

711


ex

X

*"*/„(*)

e-'I^x)

e*KQ [x)

*Ki{*)

13-02 13-04

0-III6732 0-III5857 0-III4985 0-III4II4 0-III3246

0-1072951 0-1072179 0-1071409 0-1070640 0-1069874

0-3441388 o-3438795 0-3436208 0-3433626 o-343io5i

0-3571182 0-3568296 o-35654!7 o-3562544 o-3559679

45I350-74 460468-63 469770-71 479260-71 488942-41

13-06 13-08 13-10

OIII2379 OIIII5I5

0-1069109 0-1068345 0-1067583 0-1066823 0-1066064

0-3428481 0-3425917

0-3556820 o-3553968 03551 123 0-3548285 Q-3545454

498819-71 508896-53 519176-92 529664-99 540364-94

13-12 I3-I4 13-16 13-18 13-20

03542629

1306 13-08 13-10 13-12 13-14 13-16 13-18

1320 13-22 13-24 13-26 13-28

1330 1332 1334 13-36 I3-38

1340 1342 3-44 13-46 I3-48 J

1350 I3-52 13-54 I3-50 13*58

1360 13-62

1364 1366 13-68

I37O I372 13-74 13.76

1378 1380 13-82 13-84

1386 on

13-88 I3-90 13-92 13-94 I3-96 I3-98

I400

0-III0652 0-II09792 0-II08934

0-II08077 0-II07223 0-II06370 0-II05520 0-II0467I

01065307

o-34 2 3359

0-3420807 0*3418260

0-3415719 0-3413184

ex

X 13-02

1304

1322

03410654

o-35398ii o-3537°oo

0-3408130

03534195

551281-03 562417-65 573779-24 585370-35

0340561

o-353i398

59719561

0-3403098

0-3528606 0-3525821 0-3523043 0-3520272 0-35I7506

609259-77 621567-63 634124-13 646934- 29 660003-22

1332

0-3398089 3395593 0-3393 102

01057824

03390616

o-35i4748

Q-35H995

01056346

0-3388137 0-3385662

0-1055609 0-1054874

03383193 03380729

03506510

673336-I7 686938-47 700815-54 714972-96 729416-37

1342

0-1057084

0-1095465 0-1094639 0-1093816 0-1092994 0-1092174

0-1054140 0-1053408 0-1052677 0-1051948 0-1051221

0-3378271 0-3375818 o-337337i 0-3370928

0-3501051

03368491

03490207

744151-56 759184-42 774520-96 790167-32 806129-76

x 3-52

0-3495616 0-3492909

0-1091356 0-1090540 0-1089725 0-1088912 0-1088102

0-1050495 0-1049770 0-1049047 0-1048325 0-1047605

0-3366060 o-3363633 0-3361212 o-3358796 0*3356385

0-3487512 0-3484823 0-3482140 0-3479463 0-3476793

822414-66 839028-54 855978-04 873269-94 890911-17

13-62

0-1087293 0-1086485 0-1085680 0-1084876 0-1084074

0-1046886 0-1046169 0-1045453 0-1044739 0-1044026

o-335398o o-335i579

0-3474128

908908-77 927269-94 946002-04 965112-54 984609-11

0-1083274 0-1082476 0-1081679 0-1080885 0-1080092

0-1043315 0-1042605 0-1041896 0-1041189 0-1040484

0-1079300 0-1078511 0-1077723

0-1039779 0-1039077 0-1038375 0-1037675 0-1036977

0-II03825 0-II02980 0-II02I38 0-IIOI297 0-II00458 0-I09962I 0-I098786 0-1097953 0-1097122 0-1096292

01076937 0-1076153

°- io 64552

0-1063798 0-1063046 0-1062295

0-1061546 0-1060798 0-1060052 0-1059308 0-1058565

03400591

0-3509250 0-3503777

03498330

03349184

0-347H70 03468818

-3346794 0'3344409

03463532

03342029 0-33396^4 0-3337285 o-333492o 0-3332560

03330206 0-3327856

0332551 0-3323171 0-3320836

0-3466172 0-3460897 0-3458269 0-3455647

0-3450420

1004499-53 1024791-77 1045493-94 1066614-32 1088161-36

0-3447816 0-3445217 0-3442624 0-3440037 o-3437456

1110143-67 1132570-06 II55449-50 1178791-12 1202604-28

03453031

13-24 13-26 13-28 I3-30

x

3-34

1336 1338 1340 3-44 3-46 I3-48 I3-50 x J

13-54 I3-56 13-58 13-60

1364 13-66

1368 13-70 13-72 J3-74 13-76

1378 1380 13-82

1384 1386 13-88

1390 1392 13-94

1396 1398 14-00


712 Table

II.


K

X

e-*h(x)

e-*Ii(*)

14-02 I4-04 14-06 14-08 14-10

0-1075370 0-1074589 0.1073810 0-1073032 0-1072256

0.1036279 0-1035564 0.1034889 0-1034196 0-1033505

0-3318506 0.3316181 0-3313861 0-331 1546 0-3309235

14-12 14-14 14-16 14-18 14-20

0-1071482 0.1070710 0-1069939 0-1069169 0-1068402

0-1032814 0-1032126 0-1031438 0.1030752 0-1030067

0-3306930 0-3304629

14-22 14-24 14-26 14-28 14-30

0-1067636 0-1066872 0-1066109 0-1065348 0-1064589

0-1029384 0-1028702 0-1028021 0-1027342 o- 1 020663

14-32 14-34 14-36 14-38 14-40

0-1063831 0-1063075 0.1062321 0-1061568 0-1060817

0-1060067

«*!(*)

e*

X

0-3434881

1226898-5 1251683-5 1276969-2 1302765-7 1329083-3

14-02 14-04 14-06 14-08 14-10

o-34"959

I355932-5 1383324-2 1411269-2 I439778-7 1468864-2

14-12 14-14 14-16 14-18 14-20

o-3295474 0-3293197 0-3290924 0-3288657 0-3286394

0-3409441 0-3406927 0-3404420 0-3401918 o-339942i

J 498s37-2 1528809-7 I559693-7 1591201-6 1623346-0

14-22 14-24 14-26 14-28

0-1025987 0-1025311 0-1024637 0-1023965 0-1023293

0-3284136 0-3281882 0-3279633 0-3277389 0-3275149

0-3396930 0-3394444 0-3391964 0-3389489 0-3387020

1656139-7 1609596-0 1723728-1 I758549-7 1794074-8

1432

0-3272914 0-3270684 0-3268458 0-3266236 0-3264019

o-3384555 0-3382097 o-3379643 0-3377I95 o-3374752

18303175

1442

0-1058572 0-1057827 0-1057084

0-1022623 0-1021954 0-1021287 0-1020621 0-1019956

1867292-4 1905014-2 I943498-0 1982759-3

14-44 14-46 14.48 14-50

14-52 14-54 14-50 14-58 14-60

0-1056342 0-1055602 0-1054863 0-1054126 0-1053391

0-1019292 0-1018630 0-1017969 0-1017309 0-1016650

0-3261807 o-3259599 o-3257396 0-3255197

o-33723i5 0-3369883 o-3367455

03253002

0-3362617

2022813-7 2063677-2 2105366-2 2147897.5 2191287.9

14-52 14-54 i4-5o I4-58 14-60

14-62 14-64 14-66 14-68 14-70

0-1052657 0-1051924 0-1051193 0-1050464

0-1015993 0-1015337 0-1014682 0-1014029 0-1013377

0-3250812 0-3248626 0-3246445 0-3244268 0-3242096

0-3360206 o-3357799 o-3355398 0-3353002 o-335o6i 1

2235554-8 2280716-0 2326789-6 2373793-8 2421747-6

14-62

14-72 14-74 14-76 14.78 14.80

0-1049009 0-1048284

0-1012726 0-1012076 0.1011428 0.1010780 0-1010135

0-3239928 0-3237764 0-3235604 o-3233449 0-3231298

0-3348226 o-3345845 o-3343469 0-3341098 o-3338733

2470670-2 2520581-0 2571500-1 2623447-9 2676445-1

1472 1474 1476

14-82 14-84 14.86 14.88 14.90

0-1045400 0-1044682 0-1043966 0-1043252

o> 1 009490

01042539

0-3229152 0-3227010 0-3224872 0-3222738 0-3220608

o-3336372 O-33340I7 0-3331666 0-3329320 0-3326979

2730512-8 2785672-8 2841947-2 2899358-3 2957929-2

14-82

0.1008846 0-1008204 0-1007563 0-1006923

14.92 14-94 14.96 14.98 15-00

0-1041827 0-1041117 0-1040408 0-1039701 01 038995

0-1006284 0-1005647 0-1005011 0-1004376 0-1003742

0-3218483 0-3216362

0-3324644 0-3322313 0-3319987

1492 1494

03317665

3017683-4 3078644-6 3140837-4 3204286-5

0-33I5349

32690174

1500

14-42 14.44 14-46 14-48 14-50

01059319

01049736

01047561 0-1046839 0-1046119

e"

(x)

03302333 0-3300042 o-3297755

03214245 0-3212132 0-3210024

0-34323" 0-3429747 0-3427189 0-3424637 0-3422090 o-34 r 9549 0-3417013 0-3414484

03365034

1430 14-34 14-36 I4-38 14-40

:&* 14-68 14-70

14-78

1480

i 4 -88

1490

14.96 14.98

1


II.


e*K

X

e-*h(x)

e-'I^x)

1502 1504 1506 15-08 15-10

0-1038291 ** An 0-1037588 0-1036087 0-1036186 0-1035488

0-1003109 0-1002478 0-1001847 0-1001218 0-1000590

0-3201631 o-3i99543

15-12 15-14 15-10 15-18 15-20

0-1034791 0-1034095 0-1033400 0-1032707 0-1032016

0-0999964 0-0999338 0-0998714 0-0998090 0-0997468

1522 1524

0-1031325 0-1030636 0-1029949 0-1029203 0-1028578 0-1027895 0-1027213 0-1026532 0-1025853 0-1025175

0-0991921 009913 10

03170734

0-1024498 0-1023823 0-1023149 0-1022476 0-1021805

15-26

1528 15-30

1532 15-34 i5-3f I5-38 15-40 I5-42 J 5-44

15-46 I5-48

1550 I5-52 15-54

1556 1558 15-60 15-62

1564

5 -66 15-68 15-70 1

15-72 J5-74 15-76 I5-78

1580 15-82

15-86 15-88 15-90

1592 15-94

1596 15.98 16-00

713

*K

e*

X

1502

03303839

3335055'9 3402428-5 3471162-1 3541284-2 3612822-9

0-3I97459 0-3I95379 0-3193303 0-3191231 0-3189164

0-330I552 0-3299269 0*3296990 0-3294717 0-3292448

3685806-8 3760265-0 3836227-4 3913724-4 3992786-8

15-12 15*14 15-10 15*18 15-20

0-0996847 0-0996228 0-0995609 0-0994991 0-0994375

0-3187100 0-3185040 0-3182985 °'3 l8o ^23 0-3178885

0-3290184 0-3287924 0-3285670

4073446-5 4I55735-6 4239687-0 4325334-3 4412711-9

1522

0-0993760

0-3176841 03 1 74801 0-3172766

O3278933 C3276696

(x)

03207019 0-3205819

03203723

x

{x)

0-33I3037 0-33I073I 0-3308429 0-3306I32

O32834I9 O328I I74

15-04 15*06 15-08

1510

15-24

1526 15-28

1530 1532

0-3168705

0-3274464 0-3272237 0-3270014

4501854-6 4592798-I 4685578-8 4780233-7 4876800-9

0-0990701 0-0990092 0-0989485 0-0988879 0-0988274

0-3166681 0-3164661 0-3162644 0-3160632 0-3158623

0-3267796 0-3265582 0'3263372 0-326II68 0*3258967

4975318-8 5075826-9 5178365-4 5282975-3 5389698-5

0-1021135 0-1020466 0-1019799 0-1019133 0-1018468

0-0987670 0-0987067 0-0986465 0-0985064 0-0985265

0-3156618 0-3154617 0-3152619 0-3150626 0-3148636

0-325677I 0-3254580

5498577-6 5609656-2 5722978-8 5838590-7 5956538-o

15-54 I5-56 I5-58

0-1017805 0-1017143 0-1016482 1015822 0-1015164

0-0984666 0-0984069 0-0983472

00982877

0-3146650 0-3144668 0-3142689 0-3140714

0-3245857 0-3243687 0-3241522 0-323936I

0-0982283

03138743

O32372O4

6076868-1 6199628-9 6324869-8 6452640-6 6582992-6

15-62 15-64 15-66 15-68 15-70

0-0981690 0-0981097 0-0980506 0-0979916 0-0979328

0-3136776 0-3134812 •0-3128943

0-3235052 0-3232903 0-3230759 0-3228620 0-3226484

67I5977-9 6851649-6 6990002-1 7131270-7 7275332-o

15-72 15-74 15-76 15-78 i5-8o

0-3126994

O3224353

03125049

0-3222226 0-3220I03 0-3217085 0-3215870

7422303-4 7572243-9 7725213-4 7881273-0 8040485-3

15-82 I5 'ii 15-86 15-88 15-90

8202913-9 8368623-7 8537681-1 8710153-7 88861 10-5

15-94 15-96 15-98 16-00

0-1014507 0-1013851

01013197 0-1012544 0-1011892

00993146 00992533

0-1011241 0-1010592 0-1009944 0-1009297 0-1008651

0-0978740 0-0978153 0-0977567 0-0976983

0-1008007 0-1007363 0-1006722 o- 1 00608 0-1005441

0-0975816 0-0975235 0-0974654 0-0974075

00976399

00973496

03132852 03130896

0-3123107 0-3121169 0-3119235

03 1 1 7304 o-3ii5376 031 13453 031 1 1533 0-3109616

O3252392 0-3250210

O324303I

0-3213760

O32I I654 0-3209552 0-3207454 0-3205360

15-34 15-30 I5-38

1540 1542 15-44 15-40 I5-48 I5-50

1552

1560

1592


714

Table

X

III.

Functions of order one-third

*W*)

/1/3W

1

*£(*)

ZTgH 1/3 {x)

1

e*K ll3 (x)

-90

X

O-OOOOOOO

-

+ + + + +

0-2412455 0-3038819 0-3477275 0-3825227 0-4117819

- 3-8181573 - 2-9641628 - 2-5398832 - 2-2665744 - 2-0682566

3-8257712 2-9796989 2-5635758 2-2986264 2-1088503

-

86° 23' 4*72 84° 8' 47*63 82 12' 15*40 8o° 25' 14*29 78 44' 23*54

5-8973367 4-5650965 3-9129445 3-4996127 3-2048056

0-02 0-04 o-o6

+ + + + +

0-4372223 0-4598264 0-4802143 0-4988049 0-5158967

-

1-9140102 I -7884275 1-6828031 1-5917752 1-5118289

1-9633131 1-8465950 1-7499806 1-0080991 1-5974279

-

77^ 7; 75° 34 4 74 72 36 9' 71

57;2i 5i*i6 23*06 3*21 29*88

29795927

0-12 0-14 0-16 0-18 0-20

0-5317088 0-5464087 0-5601271 0*5729677 0-5850148

- 1-4405408 - i-37 6l 797 - 1-3174682 - 1-2634392 - I-2I33449

1-5355364 1 -4806867 1-4315952 1-3872889 1-3470145

-

69 44' 26*57 68° 20' 40*45

2-3231916 2-2397331 2-1651865 2-0980307 2-0370894

0-22 0-24 0-26 0-28

030

+ + + + +

0-32 o-34 0-36 0-38 0-40

+ + + + +

0*5963375 0-6069935 0-6170312 0-6264920 0-6354112

-

1-1665964 1-1227224 1-0813409 1-0421378 1-0048529

1-3101777 1-2763020 1-2450002 I,2 Q i5 537 1-1888973

-

62 55' 29*69 6i° 36' 8*57 6o° 17' 24*70

9814363 9303301 1-8831690 1-8394587 1-7987884

0-32 o-34 0-36 0-38 0-40

0-42 0-44 0-46 0-48

0-6438195 0-6517435 0-6592067 0-6662297 0-6728308

-

0-9692681 0-9351991 0-9024892 0-8710041 0-8406278

1-1636083 1-1398978 1-1176047 1-0965902 1-0767342

-

56 24' 23*72

1-7608136 1-7252429 1-6918274 1-6603536 1-6306366

0-42 0-44 0-46 0-48

050

+ + + + +

0-52 o-54 0-56 0-58 o-6o

+ + + + +

0-6790265 0-6848313 0-6902585 0-6953202 0-7000271

-

0-8112601 0-7828134 0-7552112 0-7283861 0-7022788

I-05793I9 1-0400917 1-0231329 1-0069839 0-9915813

" -

5 oo

M'M

1-6025156 I-575850I 1-5505163 1-5264049 1-5034188

0-52 o-54 0-56 0-58 o-6o

0-62 0-64

+ + + + +

0-7043893 0-7084159 0-7121152 0-7154951 0-7185627

-

0-6768367 0-6520129 0-6277661 0-6040589 0-5808580

0-9768685 0-9627948 0-9493146 0-9363868 0-9239742

-

43° 51' 25*98 42° 37' 33*23 41 23' 52*19 40 10' 22*19 38 57' 2T60

1-4814718 1-4604863 1 4403931 1-4211296

0-62 0-64 o-66 o-68 0-70

+ + + + +

0-7213248 0-7237876 0-7259570 0-7278387 0-7294377

-

0-5581337 0-5358591 0-5140100 0-4925646 0-4715032

0-9120431 0-9005629 0-8895054 0-8788453 0-8685590

-

37° 36° 35 34° 32

43' 52*84

1-3848710 1-3677782 1-3513186 1-3354533

+ + + + +

0-7307591 0-7318076 0-7325877 0-7331037 0-7333598

-

0-4508080 0-4304628 0-4104530 0-3907653 0-3713877

0-8586249 0-8490233 0-8397359 0-8307458 0-8220374

-

31 40' 14*12 3o° 27' 53*39 29° 15' 39*47

+ + + + +

0-7333600 0-7331080 0-7326077 0-7318627 0-7308764

-

03523093 03335201

0-8135962 0-8054086 0-7974623 o-7897454 0-7822472

-

o 0-02 0*04 o-oo

OO8 o-io 0-I2 0-14 O-IO

Ol8 0-20 0-22 0-24 O-20 0-28

066 o-68 0-70 0-72

074 0*76 0-78 o-8o

082 0-84 o-86 o-88 0-90 0-92 0-94 0-96

098 I-OO

00

0-3150111 0-2967741 0-2788016

To compute

00

00

66° 58' 1*30 65 36' 20*81 64 15' 32^17

58° 59'

i

4 -57

57 4i 35 T i3 55° 7' 3»"oi 53° 5i 15*93 52° 35' 15*65 5i 19 35*54 4'

48 49 10*14 47° 34' 22*35 46 19 49*69 * 45° 5' 3 1 20

30 52*38 18' 0*72 5' 17*40 52' 42*00

2-7996089 2-6511003 2-5256038 2-4175728

1 1

1

1

4026393

3201469

008 o-io

030

050

0-72 o-74 0-76 0-78 o-8o

082

28 y 32*02 26 51' 30*76

1-3053670 1-2910835 1-2772690 1-2638979 1-2509467

25° 39' 35*39 24 27 45*66 23 16' 1*32 22° 4' 22*12 20° 52' 47^84

1-2383936 1-2262184 1-2144022 1-2029275 1-1917780

0-92 o-94 0-96

functions of order - 1/3, increase the phase

by 6o c

0-84 o-86 o-88 0-90

098 I-OO

1


7i/sW 1-02

III.

Functions of order one- third

<(*)

Yu*(*)

+ 0-7296524 + 0-7281940

**K lla {x)

0-2610869 0-2436239 0-2264069 0-2094308 0-1926912

o-7749574 0-7678666 0-7609657 0-7542466 0-7477012

-

+ 0-7265045 + 0-7245872 + 0-7224452

-

+ + + + +

0-7200818 0-7175000 0-7147030 0-7116937 0-7084752

-

0-1761839 0-1599051 0-1438514 0-1280198 0-1124076

0-7413222 0-7351026 0-7290360 0-7231162 o-7i73372

-

+ + + + +

0-7050506 0-7014229 0-6975950 0-6935699 0-6893506

-

0-0970123 0-0818317 0-0668639 0-0521072

00375600

0-7116936 0-7061802 0-7007921 0-6955245 0-6903730

+ + + + +

0-6849400 0-6803413 0-6755573 0-6705909 0-6654453

- 0-0232209 - 0-0090889 + 0-0048372 + 0-0185581 + 0-0320747

0-6853336 0-6804020 -6755746 0-6708477 0-6662179

+ + + + +

0-6601234 0-6546281 0-6409626 0-6431297 0-6371326

+ + + + +

0-0453875 0-0584971 0-0714038 0-0841081 0-0906101

0-6616819 0-6572366 0-6528790 0-6486062 0-6444156

i-6o

+ + + + +

0-6309743 0-6246578 0-6181862 0-6115625 0-6047900

+ + + + +

0-1089100 0-1210079 0-1329039 0-1445980 0-1500900

0-6403046 0-6362706 0-6323113 0-6284245 0-6246079

9° 10 12° 13 14

1-62 1-64 1-66 1-68 1-70

+ + + + +

0-5978715 0-5908104 0-5836090 0-5762725 0-5688020

+ + + + +

0-1673799 0-1784675 0-1893528 0-2000354 0-2105152

0-6208594 0-6171771

06135590

72 74

+ + + + +

0-5612014 0-5534739 0-5456226 0-5376509 0-5295619

+ + + + +

+ + + + +

05213588

+ + + + +

I

-OA

I-OO I

08

I-IO I-I2 I-I

2 I-IO 1-18 1-20 1-22 *' 2

t 1-20 1-28 I

30

I

32

1-34 1

36

138 1-40

42 44 42 48

50 1 •52

.76

:g 82

% 88 90 92 1-96 1-98 2-00

715

-

-

-

-

19° 41' 18*29 18 29' 53*27 17 18' 32*58 7' 16*06 16 14° 56' 3'54

1-1809384 J 1703945 I-I60I329 I-I50I4II 1-1404073

13° 12° ii° io° 9°

1-1309205 1-1216703 1-1126469 1-1038412 1-0952444

44' 54-86

33 49*87 22' 48*44 11' 50*43 o' 55*70

7° 50' 4*15 6° 39' 15-66 5 28' 30*1

-

1-02 1-04 I-OO I

08

I-IO I-I2

\\l Il8 1-20

1-0868482 1-0786451 1-0706275 1-0627885 I-055I2I5

1-22 1-2A 1-26 1-28 I-30

0° 45' 55*39 o° 24' 36*88 i° 35' 6*76 2° 45' 34*33

I -0476204 1-0402790 1-0330918 1-0260535 I-OI9I588

1-32 i-34

1-38 1-40

3° 55' 59-6 7 6' 22*83 5 6° 16' 43*91 7° 27' 2*94 8° 37' 20*02

1-0124030 1-0057813 0-9992894 0-9929231 0-9866783

1-42 1-44 1-46 1-48 1-50 I

28' 17*91

0-9805512 0-974538I 0-9686354 0-9628399 0-9571482

1-54 1-56 1-58 i-6o

0-6100034 0-6065083

1 2 4'37 fo 3 §' 16 48 29*23 17° 5§' 32-56 19° § 34*38 20 18' 34*74

0-95I5574 0-9460644 0-9406663 0-9353606 0-9301444

1-62 1-64 1-66 1-68 1-70

0-2207919 0-2308653 0-2407351 0-2504011 0-2598629

0-6030722 0-5996933 0-5963702 0-5931013 0-5898852

21° 28' 33*69 22° 38' 31*26 23° 48' 27*48 24 58' 22*40 26° 8' 16*05

0-9250154 0-9199712 0-9150093 0-9101276

1-72

0-5130449 0-5046236 0-4960979 0-4874713

+ + + + +

0-2691204 0-2781733 0-2870212 0-2956640 0-3041014

0-5867204 0-5836056 0-5805395 0-5775209 o-5745485

27 28 27' 29° 37' 3o° 47

0-4787471 0-4699285 0-4610189 0-4520215 0-4429398

+ + + + +

03123332 03203591

0-5716212 0-5687379 0-5658974 0-5630987 0-5603409

7'

0-3281790 0-3357927 0-3432000

J-V* (2-00) =0-5603409 Y-Hi (200) =0-5603409

-

4°i7'47'4i

-

3

7

7*45

i° 56' 30*14 •

47; 35/18

57 48-50 8' 0-02 18' 9"8i

18'

31

33° 34° 35° 36 37°

x cos 97° 46' 9*65 x sin 97 46' 9*65

= =

57'

16 26' 36' 46'

8*47 59*68 49*72 38-62 26*41 13*12 58*77 43*39 27*01 9'6 5

- 0-0757500. +0-5551971.

09053239 0-9005961

08959423 0-8913605 0-8868489 0-8824057 0-8780291 0-8737176 0-8694694

08652832 0-8611573

136

52

1-70 1-78 i-8o 1-82 1-84

186 188 1-90 1-92 1-94 1-96 1-98 2-00


716

Table

X

III.


Vi/sW

/j/»W

1

*!>)

1

arg #1/3 (*)

e*K llz (x)

X

38° 55' 5**34

2-02 2-04 2-06 2-08 2-IO

2-02 2*04 2-06 2-08 2-IO

+ + + + +

0-4337771 0-4245367 0-4152219 0-4058363 0-3963830

+ + + + +

0-3504008 o-3573949 0-3641824 0-3707631 0-3771371

0-5576229 Q-5549437 0-5523025 0-5496984 0-5471306

41 42 24' 50*91 43° 34' 2 9 ? oo

0-8570902 0-8530808 0-8491275 0-8452290 0-8413842

2*12 2-XA 2-l6 2-l8 2-20

+ + + + +

0-3868655 0-3772872 0-3676514 0-3579615 0-3482210

+ + + + +

0-3833043 0-38926^7 0-3950165 0-4005657 0-4059065

0-5445981 0-5421002 0-5396362 0-5372051 0-5348065

44° 44' 6*25 45° 53 42-67 47° 3 '18*29 48 12' 53*12 49 22' 27*18

0-8375917 0-8338505 0-8301592 0-8265169 0-8229225

2-12 2-14 2-IO 2-l8 2-20

2-22 2'24 2-26 2-28

0-3384331 0-3286012 0-3187288 0-3088193 0-2988759

+ + + + +

0-4110411 0-4159696 0-4206923 0-4252096 0-4295216

o-5324394

o-5277974 0-5255212 0-5232740

50 32' 0*49 51 41' 33*06 52°5i' 4*91 o' 36*06 54 55 10' 6*52

0-8193748 0-8158730 0-8124159 0-8090028

230

+ + + + +

08056325

2-22 2-24 2-26 2-28 2-30

2-32 2-34 2'36 2-38 2-40

+ + + + +

0-2889021 0-2789012 0-2688766 0-2588316 0-2487696

+ 0-4336289 +~o-43753i8 + 0-4412307 + 0-4447262 + 0-4480187

0-52IP55 1 0-5188641 0-5167002 0-5145631 0-5124521

56 19' 36-32 5'46 2 5Z° 9; 5 S ° 38 33;95 59 48' 1*81 6o° 57' 29*06

0-8023043 0-7990173 0-7957706 0-7925634 0-7893949

2-32 2-34 2-36 2-38 2-40

2-42 2-44 2-46 2-48 2-50

+ 0-2386939 + 0-2286079 + 0-2185149

+ 0-1983209

+ 0-4511090 + 0-4539975 + 0-4566849 + 0-4591720 + 0-4614595

0-5103666 0-5083063 0-5062705 0-5042589 0-5022709

6' 55*70 62 63 1 6' 21*75 64° 25' 47*23 65° 35 12*14 66° 44' 36*50

0-7862643 0-7831710 0-7801140 0-7770928 0-7741066

2-42 2-44 2-46 2-48 2-50

2-52 2-54 2-56 2-58 2-60

+ + + + +

0-1882266 0-1781384 0-1680595 0-1579933 0-1479429

+ + + + +

0-4635482 0-4654389 0-4671325 0-4686300 0-4699324

0-5003061 0-4983640 0-4964442 0-4945462 0-4926698

67° 54'

0*31

3 23*59 7°° l2 ' 46"35 71 22' 8*59 72 31' 30*34

o-77ii547 0-7682366 o-76535i5 0-7624989 0-7596781

2-52 2-54 2-56 2-58 2-60

2-62 2-64 2-66 2-68 2-70

+ + + + +

0-1379115 0-1279023 0-1179186 0-1079633 0-0980398

+ + + + +

0-4710406 o-47i9557 0-4726788 0-4732111 0-4735538

0-4908144 0-4889797 0-4871653 0-4853708 o-4835959

73° 40' 51*60 74° 50 12*38 75° 59 32*68 52-52 7Z° 5 78 1 8' 11*92

0-7568886 0-7541297 0-7514009 0-7487017 0-7460315

2-62 2-64 2-66 2-68 2-70

2-72 2-74 2-76

0-0881509 0-0783000 0-0684899 0-0587238 0-0490046

+ + + + +

0-4737081 0-4736754 0-4734569 0-4730540 0-4724682

0-4818402 0-4801034 0-4783851 0-4766850 0-4750028

79° 80° 81° 82

27' 30*86 36' 49*37

6-7433898 0-7407762 0-7381900

2-72 2-74 2-76 2-78

2-80

+ + + + +

2-82 2-84 2-86 2-88 2-90

+ + + + +

0-0393353 0-0297189 0-0201583 0-0106564 0-0012161

+ + + + +

0-4717009 0-4707537 0-4696281 0-4683256 0-4668480

o-4733382 0-4716909 0-4700605 0-4684469 0-4668496

85° 86° 87° 88° 89

-13'

2*92 2-94 2 96 2-98

-

0-0081598 0-0174085 0-0267073 0-0358733 0-0449638

+ + + + +

0-465197° 0-463374 1 0-4613813 0-4592203 0-4568930

0-4652685 0-4637033 0-4621537 0-4606194 0-4591002

91 92

o' 17*63 9' 32*20 18' 46*41

278

300

4-

0-2084181

To compute

05301033

functions of order

- 1/3,

40

5' 32*10 15' 11*95

69

46 7*45 55 25*11 4' 42*36 84

07356309

59"2i

23' 15*66 32' 31*72 41' 47*39 5i' 2*69

93 94 28'

0*28 95° 37' I3 ? 8l

increase the phase

0-7330983

280

0-7305919 0-7281111

2-82 2-84 2-86 2-88 2-90

07256555 0-7232247 0-7208183

l

0-7184359 0-7160770 o-7i374i3 0-7114285 0-7091381

by 6o°

2-92 2-94 2-96 2-98

300

4


717

Table III. Functions of order one-third

X

AaW

Ym(*)

3-02

- 0-0539763 - 0*0629080 - 0-0717564 - 0-0805188 - 0-0891928

+ 0-4544013 + 0-4517471

320

-

0-0977759 0-1062656 01 1 46595 0-1229552 0-1311505

+ + + + +

0-4325221 0-4287877 0-4249076

3'22 3'24 3-26 3-28 3-30

-

0-1392429 0-1472303 0-1551105 0-1628813 0-1705405

+ + + + +

0-4208840 0-4167194 0-4124159 0-4079761 0-4034022

3'32

-

0-1780862 0-1855162 0-1928286 0-2000215 0-2070929

+ + + +

+

0-3986968 0-3938622 0-3889010 0-3838156 0-3786087

-

0-2140411 0-2208642 0-2275605 0-2341283 0-2405659

+ + + + +

0-3732827 0-3678404 0-3622843 0-3566170 0-3508413

360

-

0-2468718 0-2530444 0-2590821 0-2649836 0-2707474

+ + + + +

3*62 3-64 3-66 3-68 3'7°

-

0-2763722 0-2818568 0-2871997 0-2924000 0-2974564

+ + + + +

304 3'o6 3-08

310 312

3M 316 3-i8

334 3-36 3'38

340 3-42

344 3-46

3H8 350 352 3*54 3-56 3-58

hJJw

X

04517223

0-7068697 0-7046231 0-7023978 0-7001936 0-6980101

3-02

97° 55' 39*85 99° 4' 52*38 IOO 14 il59 101 o 23'/ 16*49

0-4502888 0-4488687 0-4474619 0-4460682 0-4446874

102° 32' 28*08 103 41' 39*37 104° 50 50*36 1*05 106 9' 11*46 107

0-6958470 0-6937040 0-6915807 0-6894769 0-6873922

0-4433192 0-4406202 0-4392890 o-4379697

108 18' 21*58 109 27' 31*42 110 36' 40*99 in 45' 50*29 112 54' 59*32

0-6853264 0-6832792 0-6812503 0-6792394 0-6772463

0-4366621 o-435366i 0-4340816 0-4328083 0-4315461

4' 8*09 1 1 115 13' 16*61 22' 116 24*87 117 31' 32*89 118 40' 40*66

0-6752708 0-6733124 0-6713711 0-6694465 0-6675385

0-4302948

0-4278244 0-4266049 o-4253958

119° 49' 48*19 120 58' 55*48 *22° 8' 2*54 123° 17' 9*37 124 26' 15*97

0-6656467 0-6637710 0-6619111 0-6600668

06582379

3-44 3-46 3-48 3-50

0-3449599 0-3389754 0-3328906 0-3267083 0-3204313

0-4241969 0-4230080 0-4218290 0-4206598 0-4195001

125° 35' 22*35 1 26 44' 28*51 127° 53' 34*46 2 40*19 129 130 ii' 45*72

0-6564241 0-6540254 0-6528413 0-6510719 0-6493168

3-52 3-54 3-56 3-58 3"6o

0-3140623 0-3076042 0-3010598 0-2944320 0-2877236

0-4183500 0-4172093 0-4160778 °-4 I 49554 0-4138420

131 132 133° 134° 135

20' 51*03 29' 56*15

362

1*06 5-78 57 10*30

0*6475758 0-6458489 0-6441357 0-6424361 0-6407500

+ 0-2809376 + 0-2740767

0-4127375 0-4116417 0-4105546 0-4094760 0-4084058

6' 14*63 137 138 15' 18*78 139° 24' 22*73 140 33' 26*51 141 42' 30*10

0-6390771 0-6374173 0-6357703 0-6341362 0-6325146

M2°

51' 33*52

36*76 9 39*83 146 18' 42*72 147° 27' 45*45

0-6309054 0-6293085 0-6277236 0-6261508 0-6245897

148 36' 48*01 149° 45' SO'4 1 1 50° 54' 52*65 J52 l 3 5 Jl?3 153 12' 56*65

0-6230403 0-6215023 0-6109758 0-6184605 0-6169562

+ 0-4489323 + 0-4459590 + 0-4428292 0-439545I

04361086

04419636

04290543

372

- 0-^023678

3-78 3'8o

-

0-3117518 0-3162224 0-3205442

3-82 3-§4 3 -86 3-88 3'90

-

0-3247164 0-3207383 0-3326092 0-3363283 0-3398952

+ + + +

0-2459438 0-2387529 0-2315048 0-2242025 + 0-2168489

0-4073440 0-4062904 0-4052440 0-4042073 0-4031776

392

- 0-3433091 - 0-3465698 - 0-3496766 - 0-3526292 - 0-3554274

+ + + +

0-4021558 0-4011416 0-4001351

370

3-94 3 96 3-98 4-00

4 0-2671440

+ 0-2601423 + 0-2530746

+-

0-2094471 0-2020000 0-1945106 0-1869819 0-1794168

[

o-4575959 0-4561061 0-4546308 0-4531696

3'74

03071333

M

«,/.(*)

1

03991361 0-3981444

J-vt (4°o) =0-3981444 Y_llz (4-00) =0-3981444

a*g #i/3

96 46' 26*99

39/ 48'

144° 145°

x cos 213 12' 56*65 = -0-3330932. x sin 213 12' 56*65 = -o-2i8ioo8.

304 3*o6

308 3'io 3'12 3-14

316 3-i8 3-20 3*22 3-24

326 3-28

330 3'32

334 3-36 3-38 3-40

342

3*64 3-66 3-68 3-70

372 374 3.76 3-78 3-8o 3-82

3§4

3-86 3-88 3-9o

392 3-94 3-96 3'98 4-00


718

Table

III.


YutW

/1/3W

1

tfSw

0-3971601 0-3961831 0-3952132

4-02 4-04 4-06 4-08 4-10

0-3580707 0-3605591 0-3628923 0-3650702 0-3670927

+ + + + +

0-1718183 0-1641895 0-1488527 0-1411506

03942503

4-12 4-14 4-16 4-18 4-20

0-3689599 0-3706718 0-3722285

+ + + + +

0-1334301 0-1256940 0-1179455 0-1101873 0-1024224

0-3923455 0-3914034 0-3904679 0-3895391 0-3886169

•©•3736302

0-3748770

01565333

0-3932945

0-3877012 0-3867919 0-3858890

arg

H m (x)

e*K ll3 (x) 4-02 4-04 4-06

54 21' 58*42 55° 3i °;°4 56 40 I ? SI 57°-49' 2*83 8' 4*00 58 5

0-6154630 0-6139805 0-6125087 0-6110475 0-6095967

60

5*03 5'9i 6*66 7*26 7*73

0-6081563 0-6067260 0-6053058 0-6038956 0-6024952

4-20

65° 1' 6Z 68° 10' 69° 19' 70° 28'

K

8*07 8*27 ? 34 § 8*28 8*09

0-6011045 o-5997234 0-5983518 0-5969897 0-5956368

4-22 4-24 4-26 4-28 4-3o

< K K

4-32 4'34 4-36 4'38 4-40

61 62 63° 64°

7,

16'

25 34 43

408 4-10

o-3759693 0-3769073 0-3776914 0-3783220 o-3787997

+ + + + +

0-0946539 0-0868845 0-0791172 0-0713550 0-0636006

0-3841018

0-3791248 0-3792981 0-3793201 0-3791916 0-3789I3 1

+ + + + +

0-0558570 0-0481270 0-0404134 0-0327191 0-0250469

0-3832175 0-3823392 0-3814669 0-3806006 0-3797400

71°

4-34 4'3» 4-38 4-40

7*78 7*34 6*78 6*10 5*29

0-5942930 0-5929584 0-5916327

4-42 4-44 4-46 4-48 4-50

0-3784856 0-3779098 0-3771866 0-3763170 0-37530I9

+ o-oi 73995 + 0-0097798 + 0-0021904

0-3788853 0-3780363 °-377 I 93° o-3763553 o-375523i

77° 22' 4-37 78° 31; 3-33 79 40' 2-18 8o° 49' o ? 9i 8i° 57' 59*52

0-5877086 0-5864178 0-5851356 0-5838617

05825961

4-42 4-44 4-46 4-48 4-50

4-52 4-54 4-56 4-58 4-60

0-3741423 0-3728394 0-3713941 0-3698078 0-3680815

-

0-0203684 0-0278093 0-0352065 0-0425573 0-0498592

0-3746963 o-373875o

83 84° 85° 86° 87

58-03 56-43 54*72 52*90 50*97

0-5813387 0-5800895 0-5788483 0-5776151 0-5763897

4-52 4-54 4-56 4-58 4-60

4-62 4-64 4-66 4-68 4-70

0-3662167 0-3642144 0-3620762

- 0-0571096 - 0-0643061 - 0-0714460 - 0-0785270 - 0-0855466

0-3706429 0-3698478 0*3690578 0-3682729 0-3674929

88° 51' 48*94 o' 46*81 9' 44-58 91 92 18' 42*24 93° 27' 39 ? 8i

0-5751721 0-5739622 0-5727599 0-5715652 0-5703779

4-62

4-72 4-74 4-76 4-78 4-80

0-3548595 0-3521915 o- 3493949 0-3464712

-

0-0925024 0-0993921 0-1062133 0-1129637 0-1196411

0-3667178 0-3659476 0-3651822 0-3644216 0-3636657

94° 36' 37*27 95° 45 34*64 96 54' 31*91

0-5691980 0-5680255

29*09

0-5657019 0-5645508

4-72 4'74 4-76 4-78 4-80

-

0-1262432 0-1327677 0-1392127 0-1455758 0-1518551

0-3629145 0-3621679 0-3614258 0-3606883 0-3599553

200° 2l' 23*16 201 ° 30' 20*06 202° 39' 16*87 20 3 ° 48' I3-58 204° 57' IO*2I

05634067

0-3592267 0-3585026 0-3577827 0-3570672 o-3563559

6'

6*75

0-5577890 0-5566856 o-5555886 o-5544982 o-5534i4i

4-22 4-24

426 4-28 4-30

432

4-82 4-84 4-86 4-88 4-90 4-92 4*94 4-96 4-98 5-00

03598033 0-3573972

03434221 •

0-3402493

03369544 03335393 0-3300057 0-3263554 -

-

-

03225903 0-3187124 0-3147234 0-3106254 0-3064205

- 0-0053659 - 0-0128864

-

0-1580485 0-1641539 0-1701695 0-1760932 0-1819232

To compute

03849923

03730591 03722485 0-3714431

functions of order

-

46

u

55,

75 u 4 7° 13

6'

15' 24' 33' 42'

90

98 99

206 207° 208 209 210

1/3, increase

3'

12' 26*17

15' 3*21 23' 59*58 32' 55*86

41' 52*06

the phase

05903159 05890079

05668601

0-5622696 0-5611393 0-5600158 0-5588991

by 6o°

J& 4-68 4-70

4-82 4-84 4-86 4-88

490 4-92 4-94 4-96 4-98 5-00


III.


*W*)

/1/3W 5'02

-

504

-

0-3021105 0-2976976

- 0-1876576 - 0-1932947

5-06 5-o8

-

02931838

-

-

510

-

0-2885714 0-2838623

-

512 5-M 516

-

-

5-18

-

520

-

0-2790589 0-2741632 0-2691776 0-2641042 0-2589454

522

-

0-2537034 0-2483807 0-2429794 0-2375020

-

5-24 5-26 5-28

530

-

-

"

•

532 5'34 5-3§

•

-

-

0-1988326 0-2042696 0-2096041 0-2148343 0-2199588 0-2249760 0-2298843 0-2346822

0-2393685 0-2439417 0-2484004

1

*!>>

**S

0-3556489

03549460 03542473 0-3535527 0-3528621

0-3521756 035 1 4930 0-3508144 0-3501397

03494688

02527435 0-2569696

02263285

-

0-2610776 0-2650664 0-2689349 0-2726820 0-2763068

O-3455230 0-3448782 0-3442370 0-3435994 0-3429653

o-3423346 0-34I7075 0-3410837

03392328

538

02090575

5*40

0-2031741

5-42 5 44

0-1972317 0-1912327 0-1851797

-

-

5-48 5'50

01790751 0-1729216

0-2798083 0-2831855 0-2864378 0-2895642 0-2925640

5-52 5'54 5-56 5-58

0-1667216 0-1604777 0-1541924 0-1478684 0-1415082

0-2954366 0-2981813 0-3007974 0-3032845 0-3056420

0-3386224 0-3380154 0'3374ii6 0-3368109

0-1351143 01 286894 0-1222361

5H6

560 5-62 5*64 5'66 5-68

03404634 03398464

05523364 05512650 .0-550I998

5-02 5-04 5-06

0-5491408 0-5480878

508 510

0-5470410 0-5460001 o-544965i o-543936o 0-5429127

5-12

5-i6 5-i8 5-20

5*OI 29' 0-27 37' 55 ;46

°-54 I 8952 0-5408834 0-5398773 0-5388767 0-5378818

5-24 5-26 5-28 5-30

40-62

0-5368923 o-5359o82 o-5349296 o-5339563 0-5329883

532 534 536 538 540

0-5320256 0-5310681 0-5301157 0-5291685 0-5282263

5-42 5*44 5-46 5-48 5-5o

217° 35' 27 T 56 218 44' 23*20 219 53' 18*76 221° 2' 14-25 222° II' 9*67 223° 20'

-

^l/3(*)

(*)

211° 50' 48*18 212° 59' 44*22 8' 40*17 214 215° 17' 36*05 216 26' 31*84

224 225° 226 227°

-

-

, /s

03481386

02319509 0-2206371 0-2148793

#

0-3488018 o-347479i 0-3468234 0-3461714

-

719

46 50*59 55' 45 T 6 4

229

4'

230° 231 232 233°

13' 35 ? 53

22 30*37 31' 25*14 40' 19*85

234° 49' 14*49 235 58 9*07 237° 7' 3-58 238 15' 58-02

239°24'52-40

5-22

240° 33' 46 : 72 241 42 40*97 242° 51' 35*16 o 29*30 244 245° 9' 23-37

0-5272892 0*5263570 0-5254298 0-5245074

05235899

5*54 5-56 5-58 5-60

0-3362135 o-3356i93 0-3350282 0-3344401 o-3338552

246 18' 17*38 247° 27' 11*33 248 36 5-22 249 44' 59*06 250 53' 52 ? 84

0-5226772 0-5217693 0-5208661 0-5199676 0-5190737

5-62 5-64 5-66 5-68 5-70

552

570

0-1092543

0-3078695 0-3099667 0-3119330 0-3137683 0-3154723

5'72 5-74

0-1027311 0-0961898 0-0896330 0-0830632 0-0764830

0-3170448 0-3184857 0-3197948 0-3209721 0-3220176

o-3332733 0-3326945 0-3321186 o-33i5457 0-3309758

252 253 254 255° 256

46*56 40-22 33*83 27*38 20*88

0-5181844 0-5172997 0-5164195 0-5155438 0-5146725

5-72 5-74 5-76

5-82 5-84 5-86 5-88 5*90

0-0698951 0-0633019 0-0567061 0-0501102

0-3229313

I4 "33 25 Zo 47 ^ 258° 56 7-72

0-5138056

05129432

5-82 5-84 5-86 5-88

00435167

03252714

0-3304088 0-3298446 0-3292834 0-3287250 0-3281694

592

0-0369283

0-3255288

5*94 5-96 5-98 6«oo

00303473

0-3256557-

0-0237764 0-0172181 0-0106747

0-3256526

576 5-78 5-8o

o-"57569

J-v*

03237134 0-3243639 0-3248832

03255199 03252580

2'

II' 20' 29' 38'

260

5'

1

"06

261° 13' 54-34 262 22 47*57

0-3276166 0-3270667 0-3265194 o-3259749 o-325433i

263 31' 264 40' 265 49' 266° 58' 268

7'

=

40*75 33*88 26-96 19*98 12*96

0-5120850 0-5112312 0-5103816 0-5095362 0-5086951 0-5078581 0-5070252 0-5061964

(6-oo)

=0-3254331 x cos 328

7'

12*96

y_i/s (6-°°)

=03254331 xsin 328

7'

12*96= -0-1718736.

+0-2763441.

V78 5-80

5

90

5*92 5-94

5-96 5-98 6-oo

TABLES OF BBSSEL FUNCTIONS

720

Table

/1/3W

III.


*!/•(*)

#1/3 (*)

I

argH, /3 (*)

e*K ll3 (x)

6-02 6-04 6-o6 6-o8 6-io

+ + + +

0-0041490 0-0023568 0-0080402 0-0152987 0-0217298

0-3248675 0-3243490 0-3237031 0-3229304 0-3220317

0-3248940 0-3243576 0-3238238 0-3232926 0-3227640

269 16' 5*89 270 24' 5877 27 l0 33' 5i ? 6o 272° 42' 44*39 273 °5i 37 1

0-50537I7 0-5045509 0-5037342 0-5029215 0-5021126

6-02 6-04 6-06 6-o8 6-io

6-12 6-14 6-16 6-i8 6-20

+ + + + +

0-0281313 0-0345007 0-0408350 0-0471337 0-0533927

0-3210077 0-3198593 0-3185872 0-3171924 0-3156758

0-3222380 0-3217146 0-32II936 0-3206753 0-320I594

6' 29*81 275 276 9' 22*46 277 18' 15*05 278 27' 7*60

O-II

0-5013077 0-5005066 0-4997094 0-4989160 0-4981263

6-12 6-14 6-16 6-i8 6-20

"

2 79° 36'

6-22 6*24 6-26 6-28 6-30

+ + + + +

0-0596103 0-0657842 0-0719122 0-0779920 0-0840213

0-3140384 0-3122813 0-3104053 0-3084118 0-3063018

0-3196460 0-3I9I350 0-3186265 0-3I8I204 0-3176167

2

%°l 44' 52*57 281° 53 44-99 2 2 37;36 284 11 29-69 285°2o' 21*98

0-4973404 0-4965582 o-4957796 0-4950048 0-4942335

6-22 6-24 6-26 6-28 6-30

6-32 6-34 6-36 6-38 6-40

+ + + + +

0-0899981 0-0959202 0-1017854 0-1075917 0-1133370

0-3040764 0-3017371 0-2992849 0-2967212 0-2940474

0-3I7II54 0-3166164

286 29' 14*22 287 38' 6*42 288 46' 58*58 289° 55' 50*70

o-4934659 0-4927018 0-4919413 0-4911843 0-4904308

6-32 6-34 6-30 6-38 6-40

6-42

0.-3l6lI90

0-3I56255 0-3I5I335

K

291

4 42*77

0-1190192 0-1246363 0-1301863 0-1356673 0-1410775

0-2912648 0-2883748 0-2853790 0-2822787 0-2790756

0-3146438 0-3141564 0-3136712 0-3131883 0-3127076

292 13' 34*80 293 22' 20*80 294° 3 1 ' l8 *75 295 40 10*67 296°49' 2*54

0-4896807 0-4889341 0-4881909 0-4874510 0-4867145

+ 0-1464147 + 0-1516774 + 0-1568635 + 0-1619714 + 0-1669992

0-2757711 0-2723669 0-2688647 0-2652659 0-2615725

0-3I2229I 0-3II7527 0-3112786 0-3108066 0-3103367

29 ?° 299° 300° 301 302

57' 54-38

0-4859814 0-4852510 0-4845250 0-4838017 0-4830817

6-52 6-54 6-56 6-58 6-6o

+ + + + +

0-1719453 0-1768080 0-1815856 0-1862766 0-1908793

0-2577860 0-2539082 0-2499409 0-2458860 0-2417452

0-3098690 0-3094033 0-3089398 0-3084783 0-3080189

303 304° 305° 307° 308

42' 12*98 51' 4-59

59 56*16 8 47*70 17' 39*20

0-4823648 0-4816511 0-4809406 0-4802333 0-4795290

6-62 6-64 6-66 6-68 6-70

*7j 67

+ + + + +

0-1953922 0-1998139 0-2041429 0-2083778 0-2125171

0-2375205 0-2332136 0-2288267 0-2243615 0-2198201

0-3075615 0-307I062 0-3066528 0-30620I5 0-3057522

309 26' 30*66 310 35' 22*08 3"° 44' 13*47 312° 53' 4*83 1' 56*15 314

0-4788279 0-4781298 o-4774347 0-4767427 0-4760537

6-72 6-74

6-82 6-84 6-86 6-88 6-90

+ + + + +

0-2165596 0-2205041 0-2243491 0-2280935 0-2317362

0-2152044 0-2105105 0-2057584 0-2009321 0-1960398

0-3053048 0-3048593 0-3044159 0-3039743 0-3035347

3 1 5*

1 °! 47*44 316 19 38*70 317° 28' 29*92 318 37' 21*11 319 46' 12*26

o-4753677 0-4746846 0-4740045 o-4733273 0-4726530

6-82 6-84 6-86 6-88 6-90

6-92 6-94 6-96 6-98 7-00

+ 0-2352760 + 0^2387118 + 0-2420^26 + 0-2452674 + 0-2483853

0-1910836 0-1860655 0-1809877 OI 75§524 0-1706616

0-3030969 0-30266IO 0-302227I 0-30I7949 0-3013646

320° 322° 323° 324° 325

0-4719815 0-4713130 0-4706472 0-4699843 0-4693242

6-92 6-94 6-96 6-98 7-00

6-42

tit 6-48 6-50 6-52

6-56 6-58 6-6o 6-62 6-64 6-66 6-68

670 672

+ + + + +

•

-

To compute

6 46*18 15' 37*94 24 29*66 33' 21*34

55'

3*38

3 54*47 12' 45*52

21 36*55 30' 27*54

functions of order -1/3, increase the phase

by

6o°.

t» 6-48 6-50

676 6-78 6-8o


X

III.


*W*)

/i/j(«)

721

1

ff Siw

1

afg

HZJ W (*) ,

*

l, 3

e?K*z(x)

X

0-4686668 0-4680122 0-4673604 0-4667113 0-4660648

7-04 7-06 7-08 7-10

7-02 7-°4 7-06 7 08 7-10

+ + + + +

0-2513952 0-2542964 0-2570881 0-2597694 0-2623395

-

0-1654177 0-1601228 0-1547701 01 493888 0-1439541

0-3000847 0-2996617 0-2992404

326 39' 18*50 3 2 Z°48' 9'42 328° 57' 0*32 330° 5'5i"i9 331 14 42*03

7-12 7-14 7-16 7-18 7-20

+ + + + +

0-2647979 0-2671439 0-2693769 0-2714962 0-2735015

-

0-1384774 0-1329609 0-1274068 0-1218174 0-1161950

0-2988209 0-2984032 0-2979872 0-2975730 0-2971604

332° 333° 334" 335° 336°

23' 32' 41' 50' 58'

32*83 23*61 14*36 5*07 55*76

0-4654211 0-4647800 0-4641415 0-4635057 0-4628725

7-12 7-14 7-16

7'22

+ + + + +

0-2753921 0-2771678 0-2788281 0-2803727 0-2818013

- 0-1105419 - 0-1048604 - 00991527 - 00934213 - 0-0876683

0-2967496

338° 339° 34°° 341° 342

46*42 16 37*05 25 27*65 34 l8'22 43' 8*77

0-4622419 0-4616138 0-4609884

7-22 7*24 7-26

04603654

f-28

+ + + + +

0-2831136 0-2843096 0-2853889 0-2863516 0-2871974

-

0-0818961 0-0761070

0-2947207

00703033

0-2939207

0-0644874

02935231

7-42 7'44 7-46 7-48 7'50

+ + + +

0-2879266 0-2885390 0-2890347 0-2894138

0-0528279 0-0469890 0-0411470

+ 02896766

-

7*52 7-54 7-56 7-58 7-60

+ + + + +

0-2898232 0-2898538 0-2897689 0-2895686 0-2892534

-

7-62 7

0-2888237 0-2882799 0-2876227

7-68 7-70

+ + + + +

7'72 7V74 7.76 7-78 7-80

+ + + + +

7-2 4

7-26 7'28 7-30

732 734

0-3009362

03005095

02963405 02959330 0-2955273 0-2951232

7'

7 02

7 -i8

7-20

0-459745°

730

0-4591271 0-4585116 0-4578986

7.32 7-34 7-36 7-38 7-40

0-2931272

343° 51' 59*28 345° 49*77 346° 9 40*23 8' 30*66 347 1 348 27' 21*07

0-4566801

349° 36' 11*45 3 5 o 45' 1*80 351° 53' 52*12 353° 2' 42*42 354 11 32*70

0-4560744 0-45547I2 0-4548703 0-4542718 o-4536757

7-42 7*44 7-46 7-48

0-0294630

0-2927328 0-2923400 0-2919488 0-2915592 0-2911711

0-0236256

0-2907845

02903995

0-0119713 0-0061589

00003594

0-2900160 0-2896341 0-2892536

20' 22*95 29' 13*17 38' 3*37

04530819

00177943

355 356 357° 358° 359°

7-52 7*54 7-56 7-58 7-60

0-2859699

+ + + + +

0-0054250 0-0111920 0-0169396 0-0226654 0-0283672

0-2888746 0-2884971 0-2881211 0-2877465 0-2873734

4' 33*81 361 362 13' 23*91 363 22' 13*98 364°3i' 4*03 365° 39' 54*05

0-2849756 0-2838702 0-2826545 0-2813292 0-2798952

+ + + + +

00340430 0-0396905 0-0453076 0-0508922 0-0564422

0-2870018 0-2866315 0-2862627

02858954 02855294

366 48' 44-05 367° 57' 34*o3 369 6' 23-98 370 15' 13*91 371 24' 3-82

04450067

7-82

+ 0-2783532

7-86 7-88

+ 0-2767042 + 0-2749490 + 0-2730886 + 0-2711241

+ + + + +

0-0619554 0-0674298 0-0728635 0-0782542 0-0836001

0-2851648 0-2848017 0-2844399 0-2840794 0-2837204

372° 373° 374° 375 377°

8'

13*01

o-4444463 0-4438880 o-44333i8 0-4427777 0-4422257

+ + + + +

+ + + + +

0-0888991 0-0941493 0-0993488 0-1044956 0-1095878

0-2833627 0-2830063 0-2826513 0-2822976

378°i7'

2*7.8

7'36 7*38 7-40

rU

790 7.92 7-94 7-96 7-98

800

02868525

0-2690564 0-2668867 0-2646159 0-2622454 0-2597762

- 0-0586615

00353042

J-ila (8-oo)

y_i/»( 8 °o)

02943199

02819453

46 53*54 55 43 ? 69

32' 53*70 4i 43*56 T 50, 33 40

59 23-22

T 3Z9° 2 5 52 53 380 34' 42-25 381 43' 31*96 382 52' 21*64

04572881

0-4524905 0-4519013 0451 3145 0-4507299

0-4501476 0-4495676

04489898 0-4484142 0-4478408

04472697 0-4467007

04461339 o-4455692

o-44i6757 0-4411278 0-4405819 0-4400381 o-4394962

=0-2819453 x cos 442 52' 21*64 = +0-0349823. =02819453 x sin 442 52' 21*64= +0-2797667.

750

7-62 7-64 7-66 7-68 7.70 7-72 7-74 7.76 7-78 7-80 7-82

7-86 7-88 7-90 7.92 7 94 7-96 7-98

800


722

Table

X

III.


Y V3 (x)

/1/3W

1

»!>)

1

ar g

H m (x)

«**i/s(*)

X 8-02 8-04 8-06 8-o8 8-io

8-02 8-04 8-o6 8-o8 8-io

+ + + + +

0-2572095 o- 2 545467 0-2517890 0-2489377 0-2459942

+ + + + +

0-1146236 0-1196012 0-1245187 01 293743 0-1341664

0-2815942 0-2812445 0-2808961 0-2805489 0-2802030

1' 11*30 384 385 10' 0*94 386 18' 50-56 387 27' 40*16 388° 36' 29*74

o-43 8 9564

8-12 8-14 8-i6 8-i8 8-20

+ + + + +

0-2429598 0-2398360 0-2366242 0-2333259 0-2299425

+ + + + +

0-1388931 0-1435528 0-1481438 0-1526645 0-1571133

0-2798584 0-2795151 0-2791730 0-2788322 0-2784927

389° 39o° 392 393° 394

45' I9 ? 29 5 4'

0-4362867 o-4357586

2'

04352324

8-22 8-24 8-26 8-28 8-30

+ + + + +

0-2264758 0-2229271 0-2192981 0-2155904 0-2118057

+ + + + +

0-1614886 0-1657888 0-1700125 0-1741581 0-1782243

0-2781543 0-2778172 0-2774813 0-2771467 0-2768132

395 29' 396 38' 39 7°47' 398° 55 400 4

26*78 16*21 5*63 55*03 44*41

0-4336652 0-4331466 0-4326298 0-4321148 0-4316017

8-22 8-24 8-26 8-28

8-32 8-34 l % 8.38 8-40

+ + + + +

0-2079456 0-2040118 0-2000061 0-1959301 0-1917857

+ + + + +

0-1822096 0-1861127 0-1899322 0-1936668 0-1973152

0-2764809 0-2761498 0-2758200 0-2754912 0-2751637

401° 13' 33*76 402 22 23*10 403°3i' 12*42 40 4 4o' 1*73 405 48' 51*01

04310905

8-32 8-34 8-36 8-38 8-40

8-42 8-44 8-46 8-48 8-50

+ + + + +

0-1875747 0-1832989 0-1789600 0-1745601 0-1701008

+ + + + +

0-2008763 0-2043488 0-2077315 0-2110234 0-2142234

0-2748373 0-2745121 0-2741880 0-2738651 o-2735433

4o6° 408 409 4IO' 411

0-4285611 0-4280605 0-4275617 0-4270646 0-4265693

8-42 8-44 8-46 8-48

8-52 8-54 ?" 5

0-1655842 0-1610121 01 563865 o- 15 17093 0-1469824

+ + + + +

0-2173304 0-2203434 0-2232615 0-2260837 0-2288092

0-2732227 0-2729031 0-2725847 0-2722674 0-2719512

412 41' 46*32 413° 5°' 35*48 414° 59 24*61 4 l6 o 8 ; I 3;73

0-4260757 0-4255838

I't 8-6o

+ + + + +

2*83

0-4246051 0-4241183

8-52 8-54 8- 6 5 8-58 8-6o

8-62 8-64 8-66 8-68 8-70

+ + + + +

0-1422079 0-1373878 0-1325240 0-1276185 0-1226734

+ + + + +

0-2314370 0-2339664 0-2363966 0-2387270 0-2409567

0-2716361 0-2713221 0-2710092 0-2706973 0-2703866

418 25' 51*92 419 34' 40*99 420 43' 30*04 421 52' 19*08 1' 8*10 423

0-4236331 0-4231496 0-4226678 0-4221876 0-4217090

8-62 8-64 8-66 8-68 8-70

8-72 8-74 8-76

0-1176907 0-1126725 0-1076208 0-1025377 0-0974252

+ + + + +

0-2430852 0-2451118 0-2470360 0-2488572 0-2505751

0-2700768 0-2697682 0-2694606 0-2691541 0-2688486

424° 9' 57*10 42 5 18' 46*08 426 27' 35*04 427° 36' 23*99 428 45' 12*93

0-4212321 0-4207568 0-4202831 0-4198109

8-8o

+ + + + +

8-72 8-74 8-76 8-78 8-8o

8-82 8-84 8-86 8-88 8-90

+ + + + +

0-0922855 0-0871206 0-0819327 0-0767237 0-0714958

+ + + + +

0-2521890 0-2536987

0-2685441 0-2682407 0-2679383 0-2676369

4 29°54' 1*85 2' 50*75 431

0-4188715 0-4184041 o-4i79383 0-4I74740

8-92 8-94 8-96 8-98 9-00

+ + + + +

0-0662512 0-0609918 0-0557199 0-0504375 0-0451467

+ + + + +

0-2586882 0-2596720

\

f

878

02551038 0-2564039 0-2575988

02605500 0-2613221 0-2619882

To compute

02673365 0-2670371 0-2667388 0-2664414 0-2661450 0-2658496

0-4384185 0-4378827 o-4373487 0-4368168

8*83 58*35 11' 47*84 20 37*32

0-4347081 0-4341857

0-4305810 0-4300733 0-4295675 0-4290634

57' 40*27 6'

29*52 15' 18*74 24' 7*95 32' 57*14

4 i7°i7

04250936

04193404

ii' 39*63 20' 28*50

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04170113

6*20 436 46' 55*03 437° 55 43*83 439 4 32-62 440 13' 21*40 435° 38'


0-4165501 0-4160905 0-4156323 o-4i5i757 0-4147206

by 6o°

8-12 8-14 8-i6 8-i8 8-20

830

850

8-82 8-84 8-86 8-88 8-90 8-92 8-94 8-96

898 9-00

TABLES Or BESSEL FUNCTIONS Table

X 9-02 9-04 9-06

9 08 9-10 9-12

914 9-16 9-18 9-20 9-22 9-24 9-26 9-28

930

/1/3M

+ 0-2637603 + 0-2635068 + 0-2632227 + 0-2628342

02635213

-

+ + + + +

0-2623417 0-2617456 0-2610463 0-2602443

0-2626639 0-2623799 0-2620968 0-2618147

02593402

+ 02583346

00130053 0-0182336

00234434 0-0286326

00337991

-

00885819 00933266

01118161

9-68 9-70

-

9.72 9-74 9-76 9-78 9-80

-

01 336999

9-82 9-84 9-86 9-88

-

9-48

95° 9-52 9*54

950 9-58 9-60 9-62

964 966

990 992 9*94

996 998 io-oo

1

+ 00133439 4- 0-0080532 + 0-0027709 - 0-0025010 - 0-0077604

0-0642137 0-0691672 0-0740826 0-0789581 0-0837918

944 946

H%{*)

+ + + + +

-

9-42

1

+ 0-0398497 + °°345485 + 0-0292452 + 0-0239420 + 0-0186408

00592239

934 936 938 940


î/sto

-

9.32

III.

0-0389411 0-0440566 0049 1 435 0-0541999

+•

723

0-2625482 0-2630023

02633504 02635927 02637293 0-2636861

0-2655552 0-2652618 0-2649693 0-2646778 0-2643873 0-2640977 0-2638090

44I 442 443° 444° 445

22' IO*I7 30' 58*91 39' 47*64

4» # 3&"3& 57 25*06 6' 13*75

447° 448° 449° 450° 451

15' 2*42 23' 51*08 32' 39*73 41' 28*36

02615334

452 453° 455 456 457°

50' 59' 7 16' 25'

0-2632346 0-2629487

«*#„,(*)

X

0-4142670 0-4138148

9-02

04133642 04129150

904 906 908

0-4124673

9-10

0-4120210 0-4115762 0-4111328 0-4106908

9-12

04102503

16*97 5*57 54*i6 42*74

0-4098112 0-4093735 0-4089372

04085023

9-22 9-24 9-26 9-28

3i*3<>

0-4080687

93O

+ 0-2560212 + 0-2547148 + 02533095

0-2612531 0-2609737 0-2606951 0-2604175 0-2601407

458° 34' 19*84 459° 43 8*37 460 51' 56*89 462° 0' 45*40 463° 9' 33*»9

0-4076366 0-4072058 0-4067764

9.32 9-34 9-36 9-38

0-4059216

940

+ 0-2518062

0-2598648

04054962

0-2502055

942

02595898

0-4050722

+ 02485083 + 0-2467156 + 0-2448282

0-2593157 o- 2 590424 0-2587700

464 18' 22*37 465 27' 10*83 466° 35' 59*28 4°Zl 44' 47*72 468 53' 36*15

9*44 9-46

0-4042281 0-4038080

948

2'

24*56 470 471 11 12*96 472° 20' 1*35 473° 28 49-72 474° 37' 38-o8

0-4033893 0-4029718 0-4025557 0-4021408 0-4017272

9-52 9*54

9-62

+.

•+

0-2572281

04063483

04046495

+ + + + +

0-2428471

02363512 02340052

0-2584984 0-2582277 0-2579579 0-2576889 0-2574207

+ + + + +

6-2315707 0-2290489 0-2264409 0-2237479 0-2209712

0-2571534 0-2568869 0-2566212 0-2563563 0-2560923

475° 46' 26*43 476° 55 14*77 478° 4 3*09 479 12' 51*41 480 21' 39*71

0-4013149 0-4009038 0-4004940 0-4000855 0-3996782

0-1378966 0-1420297 0-1460977 0-1500990

+ + + + +

0-2l8lI20 0-2151716 0-2121514 0-2090527 0-2058768

0-2558290 0-2555666

48 1 ° 482 4 83 484

30' 28*00 39' 16*27

0-3992721 0-3988673

4 8'

4*53

03984637

56 52*78 486 5 41*02

0-3980614 0-3976602

0-1540322 0-1578960 0-1616889 0-1654096 - 0-1690567

+ + + + +

0-2026253 0-1992996 0-1959010 0-1924311 0-1888915

0-2545250 0-2542665 0-2540089 0-2537520 0-2534959

-

+ + + + +

01852836

0-2532406 0-2529860

0-0980241 0-1026727 0-1072706

0-1163076 01 207433 0-1251217 0-1294411

0-1726290 0-1761252 0-1795441 0-1828845 0-1861452

914 9-16 9-18 9-20

02407732 0-2386075

0-1816090 0-1778693 0-1740662 0-1702011

02553050 0-2550442 0-2547842

02527323 0-2524792 0-2522270

J-\\i (1000) =02522270 x cos 557

y-i/a (io-oo) =0-2522270 xsin

0-3972603 0-3968616

9-50

956 958 9-60

964 9-66 9-68 -9-70 9.72 9-74 9-76 9-78

980 982

488° 23 17*47 489° 32 5'67 490°40 53*87 49 1 ° 49' 42*05

0-3960677 0-3956726

58' 30*22

03952786

9-92

i8"37 16' 6*52 24' 54*66 33' 42*78

0-3948858 0-394494 2 0-3941037 o-3937i44

9 94 9 96

4Q2° 494° 495 406° 497°

7'

33' 42*78 557 33' 42*78

= =

03964641

-0-2404711.

-00761059.

9-8 4

9-86 9-88

990

998 io-oo


724

Table

X

III.


*W*)

Jxnix)

1

hSw

1

arg

10-02 10-04 10-06 10-08 IO-IO

-

0-1893250 0-1924230 0-1954380 0-1983690 0-2012150

+ 0-1662759 + 0-1622921

IO-I2 10-14 IO-IO 10-18 10-20

-

0-2039750 0-2066482 0-2092336 0-2117305 0-2141379

+ + + + +

0-1458056 0-1415548 0-1372559 0-1329107 0-1285209

02497452

IO-22 IO-24 IO-26 IO-28

-

0-2164551 0-2186814 0-2208160 0-2228584 0-2248078

+ + + + +

0-1240885 0-1196151 0-1151028 0-1105533 0-1059685

0-2495010 0-2492576 0-2490148 0-2487728 0-2485314

-

0-2266637 0-2284255 0-2300928 0-2316649 0-2331416

+ 0-1013503 + 0-0967006 + 0-0920212 + 0-0873142 + 0-0825814

0-2482908 0-2480509 0-2478116 0-2475731 0-2473352

-

0-2345224 0-2358069 0-2369948 0-2380858 0-2390797

+ + + + +

0-0778247 0-0730460 0-0682473 0-0634306 0-0585977

0-2470980 0-2468615 0-2466257 0-2463905 0-2461560

52

- 0-2399763 - 0-2407753 - 0-2414768 - 0-2420805 - 0-2425864

+ + + + +

o-o5375o6 0-0488913 0-0440217 0-0391437

io-68 10-70

-

0-2435175 0-2436325 0-2436501

+ + + + +

10-72 10-74 10-76 10-78 io-8o

-

0-2435703 0-2433934 0-2431198 0-2427495 0-2422830

+ 0-0049268 + 0-0000518

0-2417207 0-2410629 0-2403100 0-2394626 0-2385212

IO3O IO-32

1034 1036 10-38 10-40 10-42 10-44 10-46 10-48

1050 1052 io-54 10-56 10-58 10-60 10-62 10-64

1066

10-82 10-84

1086 to-88 10-90 10-92

1094 10-96

1098 II-OO

-

0-2429945

02433049

- 0-2374863 - 0-2363584 - 0-2351382 - 0-2338264 - 02324236

+ 0-1582514 + 0-1541556 + 0-1500064

H m (x)

X

03933263

10-02 IO-04 10-06

0-2519755 0-2517247 0-2514747 0-2512254 0-2509769

498° 42' 30*90 499 51' 19*00 501 ° o' 7*09 8' 55*17 502 503° 17' 43-24

o-3929393 o-3925534 0-3921687 0-39I785 1

0-2507291 0-2504820 0-2502357 0-2499901

504 26' 31*30

0-3914026

IO-I2

505° 35' 19-35 506° 44' 7*39 507° 52' 55*4* 1' 43-43 509

03910213

ICI4

0-3906411 0-3902619 0-3898839

IO-IO 10-18 10-20

510 10' 31*43

0-3895070

511° 19' 19*42 5i2°28' 7*41

03891311

IO-22 IO-24 IO-26 IO-28 IO-30

IO08 IO-IO

514 45 43-35

0-3887564 0-3883827 0-3880101

515° 54' 3i"3° 3' J 9 : 2s 517 518 12' 7?i8 519° 20' 55'iii 520 29' 43^03

0-3876386 0-3872681 0-3868987 0-3865304 0-3861631

IO-32 10-34 10-36

38' 30*93 522 47' 18-82 56' 6*70 523° 525° 4 '54 r 58 526° 13' 42^44

03857968

10-42

0-3854316 0-3850674 0-3847043

1044

527° 22' 30*30 528° 31' 18*14 529° 40' 5*98

0383981

00342593

0-2459222 0-2456891 0-2454566 0-2452248 0-2449936

53o°48'53 ? 8i 53i°57'4i-62

0-3836210 0-3832620 0-3829039 0-3825468

0-0293704 0-0244790 0-0195870 0-0146963 0-0098090

02447631 0-2445332 0-2443040 0-2440754 0-2438474

533° 6' 29*43 534° 15' I7 T2 3 535 24' 5*02 536 32 52 ? 8o 537° 41' 40*57

0-3821908 0-3818357 0-3814816 0-3811285 0-3807764

10-62 10-64

538° 50' 28*33

03804253

539 59 16-08

0-3800751 o-3797259 o-3793777

03790304

10-72 IO-74 10-76 10-78 10-80

5 1 3° 36' 55-39

1°

- 0-0048141 - 0-0096692 — 00145113

0-2436201 0-2433935 0-2431674 0-2429420 0-2427172

541° 8' 3-83 542° 16' 51*56 543° 25' 39-29

- 0-0193387 - 0-0241495 - 0-0289417 - 0-0337136 - 00384633

0-2424930 0-2422695 0-2420466 0-2418242 0-2416025

544° 34' 27*00 545° 43 i4 r 7T 2" 4 I 546° 52 0' 50*10 548 549° 9' 37 ? 7«

-

0-2413814 0-2411610 0-2409411 0-2407218

550° 551° 552° 553 554

0-0431888 0-0478886 0-0525606 0-0572030 0-0618143

To compute

etKvzi*)

02405031

18'

03843422

25*46

27' 13*12 36' 0*78

44 48*43 53 36-07


10-40

10-46

1048 1050 1052 1054 1056 1058 10-60

1066 10-68 10-70

0-3786841

10-82

03783387

1084

o-3779942 0-3776507 o-3773o82

io-86 10-88 10-90

0-3769665 0-3766258 0-3762860 0-3759472

10-92

03756092 by

1038

6o°.

1094 10-96 10-98 II-OO

3

5


X

/1/3W

III.

725


î/s(*)

1

1

arg

H U3 {x)

e*K ll3 (x)

X

-

0-2309306 0-2293480 0-2276768 0-2259176 0-2240715

-

0-0663924 0-0709358 0-0754426 0-0799111 0-0843397

0-2402850 0-2400675 0-2398506 0-2396342 0-2394185

556

11*31 557° 558° 19' 58-93 28' 559° 46*54 560° 37' 34*13

0-3752722 °-37493 6 ° 0-3746008 0-3742665 0-3739330

II-02 II-04 ii-o6

0-222I392 0-220I2I7 0-2I80I99 0-2158348 0-2135675

-

0-0887266 0-0930702

02392033

11-18 II-20

-

561° 562° 564 565 56b

!2' 44*44 21' 32*00

0-3736005 0-3732688 0-3729380 0-3726081 0-3722791

11-12 11-14 ii-i6 ii-i8 II-20

11-22 11-24 11-26 11-28 II-30

- 0-2II2I89

-

0-2087902 0-2062824 0-2036967

-

0-1099790 0-1140819 0-1181319 0-1221276 0-1260674

567° 30' 19*55 568 39' 7*09 569° 47' 54*62 570 56' 42*15 5' 29*67 572

03719509 0-3716236 0-3712972 0-3709716 0-3706469

11-22 11-24 11-26 11-28 II-30

-

0-1982963 0-1954839 0-1925985 0-1896412 0-1866134

- OI 375373 - 0-1412393 - 0-1448785

573° 14' 17*18 574 23' 4*68 575 3i' 52*17 5?6° 40' 39 T 66 577° 49' 27*14

0-3703230 0-3700000 0-3696779 0-3693565 0-3690360

-

0-1835164 0-1803515 0-I77I202 0-1738238 0-1704636

-

0-1484535 0-1519029 01 554055 0-1587801 0-1620853

0-2360439 0-2358377 0-2356320 0-2354269

02352223

578° 580° 581 582° 583

-

0-1670413 0-1635582 0-1600158 0-1564157 0-1527593

-

0-1653202 0-1684833 0-1715738 0-1745904 0-1775320

-

0*1490482 0-1452839 0-1414681 0-1376024 0-1336883

-

-

0-1297276 0-1257217 0-I2I0725 0-1175816 OI I34507

-

OI935528

- 0-1092815 - 0-1050756 - 0-1008350 - 0-0965611 - 0-0922560

-

0-2046720 0-2066430 0-2085282 0-2103269 0-2120386

0-2320195

" -

-

0-2136627 0-2151988 0-2166464 0-2180050 0-2192744

02310452

II-02 II-04 1 1

-oo

11-08 II-IO

11-12 11-14

n-i6

11-32

"•34

V' 3 * 1138 11-40 11-42 ?

1

'44

11-46 11-48

1150 11-52

"54 11-56 , 1158 n-6o

1162 1164 n-66 n-68 11-70 11-72

u-74 11-76 11-78 11-80 11-82

1184 n-86 n-88 11-90

1192 "•94 11-96 11-98 I2-00

'-

O2OIO343

0-0879212 0-0835585 0-0791698 0-0747568 0-0703214

00973689 0-1016210 0-1058249

- o- 1 299499 - 0-1337736

0-2389887 0-2387747 0-2385613 0-2383484 0-2381361 0-2379244

02377132 0-2375026 0-2372926

02370831 0-2368741 0-2366658 0-2364579 0-2362506

2*23*69

n

#

46' 21*72 55' 9*31

3-56*88

58' 14*61

1 1

08

II-IO

11-32

"•34 II36 II- 3 8

11-40

2*08

0-3687164 0-3683975

15' 49*53

03680795

24' 36*98 33' 24*42

0-3677623

11-46 11-48

03674460

n-50

0-2350182 0-2348147 0-2346117 0-2344092 0-2342072

584 42' 11*86 5 8 5° 5°' 59-29 586 59' 46*70 8' 34*11 588 589 17' 21*52

0-3671304 0-3668157 0-3665018 0-3661886 0-3658763

1152 "•54 1156

01803977

0-2340058

0-1831865 0-1858973 0-1885292 - 0-1910814

02338049

590° 591° 592 593° 595

26' 8*92 34' 56"3i 43' 43*69 52' 1 "°7

18*44

0-3655648 0-3652541 0-3649441 0-3646350 0:3643266

596°io' 5*81 18' 53-16

0-3640190 0-3637122

27' 40*51 36' 27*85 45' 15*18

03634062 03631009

11-72 11-74 11-76 11-78

0-3627964

1 1

6oi° 54' 2*51 2' 49*84 603 604° "' 37 -i 605 20' 24*46 606° 29' 11*76

0-3624927 0-3621897 0-3618875 0-3615861 0-3612854

11-82 11-84

607° 37' 59*05 608 46'- 46*34 609° 55' 33*62

0-3609854 0-3606862 0-3603878 0-3600901 0-359793I

11-92 11-94 11-96

0-1959428

01982505 0-2004750 0-2026158

0-2336044

02334046 0-2332052 0-2330063 0-2328070 0-2326101 0-2324127

02322159 02318237 0-2316283 0-2314335

02312391 0-2308518 0-2306589 0-2304664 0-2302745

J-va (12-00) =0-2302745 x cos 672 ^-i/3( r2 '°°)=o 23 02 745 xs»n 672 -

597° 598 599° 6oo°

7'

1'

'

61 1°

612

4*20*90 13'

=

8*17

13' 8*17 +0-1547365. 13' 8*17= -0-1705373.

II-42

n-44

1 1

-58

n-6o 11-62 11-64

n-66 n-68 11-70

-So

n-86 n-88 1 1

-90

1198 12-00


726

Table

Vi/sW

/1/3W

12-02 12-04 I2-06 12-08 I2-IO

-

12-12 12-14 I2-IO

12-20

- 00433371 - 0-0387947 - 00342443 - 0-0296877 - 0-0251267

-

12-22 12-24 12-26 12-28 I2-30

-

0-0205632 0-0159989 0-0114356 0-0068753 0-0023196

12-32 12-34 I2-36

+ + + + + + + + + +

I2- 3 8

12-40 I2-42

1244 12-46 12-48 12-50

Functions of order one-third (

X

I2l8

III.

-

1

H, >)

*°Kv»{*)

X

613° 21' 55*43 42*69 29*93 17*17 4*41

o-3594968

12-02 12-04 I2-O0 12-08 I2-IO

b^M

I

0-2300830 0-2298920 0-2297015 0-2295114 0-2293219

614 30 615 39' 6i6°48' 6i7°57'

0-2291327 0-2289441 0-2287559 0-2285682 0-2283809

619° 5' 620 14' 38*86 621 23 25*08 622 32' 13*29 623°4i' 0*50

03580265

0-2256333 0-2261782 0-2266320 0-2269945

03568631

12-20

-

0-2272658 0-2274458 0-2275347 0-2275326 0-2274397

0-2281941 0-2280078 0-2278219 0-2276365 0-2274515

624 49' 47*70 625° 58' 34-89 7' 22*08 627 628 16' 9*26 629 -24' 56*43

o-356574i 0-3562857 o-355998o 0-3557110 o-3554 2 48

12-22 I2-24 12-26 12-28 I2-30

0-0022296 0-0067705 0-0113014 0-0158205 0-0203259

-

0-2272560 0-2269819 0-2266176 0-2261634 0-2256196

0-2272670 0-2270829 0-2268993 0-2267161 0-2265333

6 30° 33' 43"6o 631 42' 30*76 632°5i' 17*92 6 3< 0' 5*07 8' 52*21 635

o-355i392 o-3548543 0-3545700 0-3542865

12-32

03540036

I2-36 12-38 12-40

0-0248159 0-0292889 0-0337429 0-0381763 0-0425874

-

0-2249866 0-2242647

0-2263510 0-2261692 0-2259877 0-2258067 0-2256262

63°° 637° 638 639°

640

17' 39*35 26' 26*48 35' J 3- 61 44' 0-73 52' 47*84

0-35372I5 0-3534400 o-353i59i 0-3528790 o-3525995

I2-42 12-44 12-46 12-48 12-50

-

0-2204979

642° 643 644 645 646

i'34 ? 95 10 22*05 19' 9*15 27' 56*24 36' 43*33

0-3523206 0-3520424 0-3517639 0-3514881 0-3512118

1252

0-2180943 0-2167645 0-2153502

0-2254460 0-2252664 0-2250871 0-2249082 0-2247298

12-62 12-64 12-66 12-68 12-70

0-0658652 0-0613901 0-0568980 0-0523905 0-0478696

0-2204540

02215437 0-2225430 0-2234519 0-2242700

- 0-2249971

02234544 0-2225562 0-2215705

'

03592013 03589065 03586125 0-3583191

o-3577346 o-3574434 o-357i529

12-12 12-14 12-16

1218

1234

12-52 12-54 12-56 12-58 12-60

+ + + + +

0-0469744

12-62 12-64 12-66 12-68 12-70

+ + + + +

0-0684894 0-0726967 0-0768684 0-0810028 0-0850983

-

0-2138521 0-2122710 0-2106077 0-2088628 - 0-2070373

0-2245518 0-2243743 0-2241971 0-2240204 0-2238441

6 4Z° 45' 3o-4i

03509363

648° 54 I7'49 650° 3 4'56 651 11 51*62 652 20' 38*68

0-3506614 0-3503871 0-3501135 0-3498405

12-72 12-74 12-76 12-78 I2-80

+ 0-0891534 + 0-0931665

0-2051320 0-2031477 0-2010854 0-1989460

0-2236682 0-2234927

0-2231429 0-2229687

653 29' 25*73 6 54 ° 38' I2'7§ 655° 46' 59 ? 82 656 55 46-86 658 4' 33-89

03495681 03492964

+ 0-0971361 + 0-1010607 + 0-1049388

-

12-82 I2-84 12-86 12-88 12-90

+ + + + +

0-1087869 0-1125496 0-1162795 0-1199571 0-1235811

-

o- 1 944399

0-1920752 0-1896373 0-1871278 0-1845472

0-2227948 0-2226214 0-2224484 0-2222757 0-2221035

659° 66o° 661 ° 662 663

13' 20*92 22' 7*94 30' 54*96

0-3482159 o-3479473 -3476794 0-3474120 0-347I453

1282 1284

12-92

+ + + + +

0-1271502 0-1306629 0-1341180 0-1375142 0-1408503

-

0-1818970 0-1791782 0-1763920 0-1735397 - 0-1706224

0-2219317 0-2217602 0-2215892 0-2214186 0-2212483

664° 666° 667 668° 669

57' I5 T97

0-3468792 0-3466137 0-3463488 0-3460846 0-3458209

12-92

1294 12-96 12-98 13-00

00513356 0-0556694 0-0599741 0-0642479

02193390

01967305

To compute

02233176

functions of order

- 1/3,

39'4 I,r 97 48' 28*97

6'

2*96

14' 49*95 23' 36*94 32' 23*92

increase the phase

0-3490254 0-3487549 0-3484851

by 6o°

12-54 12-56 12-58 12-60

12-72 12-74 12-76 12-78 12-80

12-86 12-88 12-90

1294 12-96

1298 1300


7i/ 3

W

III.


î/.W

"!>)

1

argH 1/3 (*)

e*K ll3 (x) o-3455578 o-3452954 0-3450335 0-3447722 o-3445i 15

[3-02 13-04 13-06 [3-08

13-06 13-08 13-10

+ + + + +

0-1441250 °- I 47337i 0-1504854 0-1535688 0-1565861

- 0-1676415 - 0-1645982 - 0-1614938 - 0-1583297 - 0-1551071

0-2210785 0-2209090 0-2207399 0-2205712 0-2204029

670 41' 10*89 671° 49' 57-S6 672 58' 44*82 674° 7' 3i*78 675 16' 1874

13-12 13-14 13-16 13-18 13-20

+ + + + +

o-i595363 0-1624183 0-1652310 0-1679735 0-1706447

-

0-1518276 0-1484923 0-1451029 0-1416606 0-1381670

0-2202350 0-2200675 0-2199003 0-2197335 0-2195671

676 25' 5*69 677° 33' 52*63 678° 42 39*57 679 51 26*50 68i°

o' 13*43

13-22 13-24 13-26 13-28

+ + + + +

0-1732437 0-1757696 0-1782214 0-1805984 0-1828996

-

0-1346234 0-1310315 0-1273926 0-1237083 0-1199802

0-2194011 0-2192355 0-2190702 0-2189053 0-2187408

682 683 684 685 686°

17' 47*27 26' 34*18 35' 21*09 44' 8*00

+ + + + +

0-1851244 0-1872718 0-1893413 0-1913320

0-1162097 0-1123985 0-1085481 0-1046601 0-1007361

0-2185766 0-2184128 0-2182494 0-2180864 0-2179237

687 689 690 691° 692

52' 54*90

01932433

-

+ + + + +

0-1950745 0-1968252 0-1984946 0-2000822 0-2015875

-

0-0967777 0-0927866 0-0887643 0-0847125 0-0806329

0-2177613 0-2175994 0-2174378 0-2172765 0-2171157

693° 694 695 697° 698

36' 49*33 45' 36*20 54' 23*06

+ + + + +

0-2030101 0-2043496 0-2056054 0-2067772 0-2078647

-

0-0765272 0-0723969 0-0682438 0-0640696 - 0-0598759

0-2169551 0-2167949 0-2166351 0-2164757 0-2163160

699 20' 43*63 700 29' 30*48

- 00556645 - 0-0514370 - 0-0471951 - 0-0429407 - 00386752

13-02

1304

1330 13-32

1334 1336 13-38

1340 13-42

1344 13-46 13-48 13-50

1352 1354 1356 13-58 13-60 13-62 13-64 13-66 13-68

1370 13-72

1374 1376 13-78 13-80 13-82 I3-8 13-86 13-88

4

1390 1392 13-94 13-96 13-98 14-00

+ 0-2088676 + 0-2097855 + 0-2106183 -f 0-2113658 + 0-2120278 + + + + +

0-2126040 0-2130946 0-2134993 0-2138181 0-2140510

-

+ + + + +

-

+ + + + +

727

9'

0*35

41*79 io' 28*68 19' 15*57 28' 2*45 i'

[310

03442514

[3-12

o-34399i9 0-3437330 o-3434747 0-3432169

[3-16 t3'i8 [3-20

o-3429597 0-3427031 o-342447i 0-3421917 0-3419368

[3-22 [3-24 13-26 13-28 [3-30

0-3416825 0-3414288 0-3411756 0-3409230 0-3406709

[332 ^3-34 C3-36 [3-38 [3-40

0-3404194 0-3401685 o-3399i8i 0-3396683 0-3394I90

'3-46 [3-48 C3-50

701° 38' 17*33 702° 47 4*17 703 55 5i*oo

0-3391702 0-3389221 0-3386744 0-3384273 0-3381808

[3-52 t3\54 [3-56 ^3-58 [3-60

0-2161578 0-2159994 0-2158413 0-2156836 0-2155262

705°

4' 37*83 706 13 24*65 22' 11*47 707 708 30' 58*29 45*11 709 39'

o-3379347 0-3376892 0-3374443 o-337i999 0-3369560

[3-62 13-64 [3-66 [3-68

3'

9*92

11' 56*78

[3-42 [

3H4

E370

0-0344006 0-0301184 0-0258305 0-0215384 0-0172440

0-2153692 0-2152125 0-2150561 0-2149001

710 48' 31*92 711° 57' 18*72 713° 6' 5*52

02147445

7X5

23' 39*10

0-3367126 0-3364698 0-3362275 o-3359857 o-3357444

0-2141981 0-2142594 0-2142350 0-2141251 0-2139298

0-0129489 0-0086548 0-0043635 0-0000766 + 0-0042041

0-2145891 0-2144341 0-2142795 0-2141251 0-2139712

716° 717° 7i8° 719 721

32' 25*89

0-3355037

[3-82

41 12*67 49' 59-45 58' 46*22

03352635 03350237

7'

32*99

13-84 [3-86 [3-88 13-90

0-2136494 0-2132840 0-2128339 0-2122994 0-2116809

+ + + + +

0-2138175 0-2136642 0-2135112 0-2133585 0-2132061

722 723 724° 725° 726

16' 25' 33' 42' 51'

19*76 6*52 53*28 40*03 26*78

0-0084770 0-0127404 0-0169926 0-0212319 0-0254567

7M°

M

^-i/s (14-00) =0-2132061 x cos 786 51' 26*78 xsin 786 51' 26*78

y-i/8( I 4'°°) =0-2132061

= =

52*31

o-3347845 o-3345459 0-3343077 0-3340700

03338329 0-3335962 o-33336oo

+00837943. +0-1960494.

[3-72 '3-74 [3-76 C3-78

[380

[3-92 [3-94 [3-96 t3-98 [4-00


728

Table


Y lla (x)

Ju»( x)

X

III.

1

»!>)

1

ars

Hm (*)

728°

o' 13*52

729

9

e*K ll3 (x)

X

0-3331244 0-3328892 0-3326546 0-3324204 0-3321867

14-02 14-04 14-06 1 4 08 14-10

0-33I9536

14-12 I4-I4 14-16 14-18 14-20

14-02 14-04 14-06 14-08 14-10

+ + + + +

0-2109787 0-2101933 0-2093250 0-2083743 0-2073417

+ + + + +

0-0296652 0-0338559 0-0380272 0-0421773 0-0463046

0-2130541 0-2129024 0-2127510 0-2126000 0-2124493

731° 26 33*73 732 35 20*46

14-12 14-14 14-16 14-18 14-20

+ + + + +

0-2062277 0-2050330

+ + + + +

0-0504076 0-0544847 0-0585342 0-0665445

0-2122989 0-2121488 0-2119990 0-2118495 0-2117004

733° 44' 7 ? i8 734° 52 53-90 1 40*62 736 737° 10' 27*33 19' 14*04 738

14-22 14-24 14-26 14-28

+ 0-1994581

0-0705021 0-0744259 0-0783145 0-0821664 0-0859801

0-2115516 0-2114031 0-21 12549 0-2III070 0-2I09594

739° 740° 741° 742° 744

0*75 36 47 ? 45 45 34 T i4 54 20*83 ? 3 7 52

0-3307950 0-3305648 0-330335°

02037580 0-2024034 0-2009699

00625547

730°i7'47"oo

28'

03317209 0-3314887 0-3312570 0-3310257

03301057

14-22 14-24 14-26 14-28

0-3298769

1430

11' 54-21 20' 40*89 29' 27*57

0-3296485 0-3294206

1432

38 I4 24 749 47' 0*91

0-3289663 0-3287398

14-34 14-36 I4-38 14-40

0-2100804 0-2099349 0-2097898 0-2096449 0-2095004

75°°55'47 T 58 752° 4 34 r 24 753 13 20*90 754° 22' 7*55 755 30' 54 -20

0-3285138 0-3282883 0-3280632 0-3278386 0-3276145

14-42 14-44 14-46 14-48 I4-50

0-1345743 0-1376501

0-2093562 0-2092I22 0-2090686 0-2089252 0-2087822

756 ° 39' 40*85 757° 48 27*49 750° 57, I4 ? i3 6 0*77 760 761° 14' 47 ? 40

0-3273908 0-3271676 0-3269448 0-3267225 0-3265006

I4-52 14-54 14-50 I4-58 14-60

+ + + + +

0-1406666 0-1436229 0-1465178 0-1493501 0-1521190

0-2086394 0-2084969 0-2083548 0-2082I29 0-2080713

762° 763 764° 765

23' 34*03 32' 20*65 41' 7*27 49 53" 8 9 58' 40*51

0-3262792 0-3260583 0-3258378 0-3256177 0-3253981

14-62 14-64 14-66 14-68 14-70

0-1387970 0-1355800 0-1323131 0-1289979 0-1256355

+ + + + +

0-1548233 0-1574621 0-1600344 0-1625393 0-1649758

0-2079300 0-2077889 0-2076482 0-2075077 0-2073676

768 769

770° 25'

27*12 13*73 0*33 771° 33 46 ?93 772° 42' 33*53

0-3251789 0-3249602 0-3247419 0-3245240 0-3243006

14-72 14-74 14-76 14-78 14-80

+ + + + +

0-1222275 0-1187753 0-1152803 0-1117439 0-1081677

+ + + + +

0-1673432 0-1696405 0-1718669 0-1740217 0-1761041

0-2072277 0-2070881 0-2069488 0-2068097 0-2066710

773 51' 20*12 775° 0' °-7i 776° 8 53*30. 777° 17,39*88 778 26 26*46

0-3240896 0-3238731 0-323657° 0-3234413 0-3232261

14-82 14-84 14-86 14-88

+ + + + +

0-1045530 0-1009014 0-0972143 0-0934933 0-0897400

+ + + + +

0-1781133 0-1800487 0-1819095 0-1836952 0-1854051

02065325

779° 35' 13-04

0-2063943 0-2062564 0-2061187 0-2059813

780 43' 59*61 781 52' 46*18 783° i'32"75 784 io' 19*31

0-3230112 0-3227969 0-3225829 0-3223694 0-3221562

14-92 14-94 14-96 14-98 15-00

1430

+ + + +

0-1978687 0-1962026 0-1944604 0-1926429

+ + + + +

14-32 14-34 14-36 14-38 14-40

+ + + + +

0-1907510 0-1887856 0-1867475 0-1846376 0-1824569

+ + + + +

0-0897541 0-0934869 0-0971773 0-1008236 0-1044246

0-2I08I2I 0-2106652 O-2IO5I05 0-2I0372I 0-2I02261

745° 746 747° 748°

14-42 14-44 14-46 14-48 14-50

+ + + + +

0-1802063 0-1778868

0-1730454 0-1705256

+ + + + +

0-1079789 0-1114852 0-1149420 0-1183482 0-1217023

14-52 14-54 14-56 14-58 14-60

+ + + + +

0-1679410 0-1652930 0-1625825 0-1598109 0-1569792

+ + + +

+ 0-1250032 0-1282497 0-1314404

14-62 14-64 14-66 14-68 14-70

+ + + + +

0-1540886 0-1511404 0-1481359 0-1450763 0-1419629

14-72

14-76 14-78 14-80

+ + + + +

14-82 14-84 14-86 14-88 14-90 14-92 14-94 14-96 14-98 15-00

M-74

0*26

0-175.4995

To compute

766

r

1

7' 6'


03291932

by

6o°

1490

1


X

III.


>W*)

/1/3W

729

!

arg 1

H m {x)

X

)

0-0859558 0-0821423 0-0783610 0-0744336 0-0705416

+ 0-1870386 + 0-1885953 + 0-1900745 + 0-1914758 + 0-1927988

0-2058442 0-2057074 0-2055709 0-2054346 0-2052986

7 8 5 °i9'

786 787 788° 789

5"8 7 27 52*42 36' 38-97 45' 25*52 54 12*07

03219435

15-10

+ + + + +

0-3217313 0-3215194 0-3213080 0-3210970

15-02 15-04 15-06 15-08 15-10

15-12 15-14 15-16 15-18 15-20

+ + + + +

0-0666265 0-0626899 0-0587336 0-0547589 0-0507677

+ + + + +

0-1940430 0-1952080 0-1962935 0-1972992 0-1982247

0-2051628 0-2050273 0-2048921 0-2047572 0-2046225

791 792 793 794 795°

58*61

0-3208864 0-3206762 0-3204664 0-3202570 0-3200481

15-12 15-14 15 8 I5-I8 I5-20

15-22

4 + + + +

0-0467614 0-0427416 0-0387101 0-0346684 0-0306181

+ 0-1990697 + 0-1998342

0-2044881

796 46' 51*28 797° 55' 37"8o 799° 4 24-32 8oo° 13 10*84 8oi° 21' 57*35

03198395

02043540

15-22 I5-24 I5-26 15-28 I5-30

+ + + + +

0-0265609 0-0224983 0-0184321 0-0143638 0-0102950

+ 0-0062274 + 0-0021626

1502 1504 1506 1508

1524

+ 0-2005178 + 0-2011203 + 0-2016418

0-2042201 0-2040865

+ + + + +

0-2020820 0-2024408 0-2027184 0-2029145 0-2030294

0-2038200 0-2036872

- 0-0018978 - 0-0059522 - 0-0099990

+ + + + +

-

0-0140366 0-0180634 0-0220777 0-0260781 0-0300630

15-70

-

0-0340307 0-0379798 0-0419086 0-0458157 0-0496995

15-72 15-74 15-76 15-78 15-80

-

15-82 15-84

-

15-26

1528 1530

I53 2 1534 I5-36 15-38 15-4°

1542 15-44 I 5 - 4°

1548 15-50

1552 1554 1556 15-58 15-60

1562 15-64 15-66

1568

1586 15-88 15-00

1592 15-94

1596 1598 16-00

-

02039531

29' 18*22 38' 4*75

0-3196314

03194237 0-3192164

03190094

43*86 30;37 16*87 3*37 49*87

0-3188029 0-3185968 0-3183911 0-3181857 0-3179808

1532

0-2034223 0-2032902

802 30' 803° 39' 804 48' ^°5°57; 807 5

0-2030630 0-2030154 0-2028867 0-2026772 0-2023869

0-2031584 0-2030269 0-2028956 0-2027646 0-2026338

808 14' 36*36 809 23' 22*85 8io° 32' 9*34 81 1° 40' 55*83 812 49 42*31

0-3177763 0-3175721 0-3173684 0-3171650 0-3169621

15-42 15-44 I5-46 I5-48 I5-50

+ + + + +

0-2020162 0-2015652

0-2025032 0-2023730 0-2022429 0-2021132 0-2019836

813 815° 816 817 818

58' 28*79 7'

0-3167595 0-3165573

16'

03163555

15-52 15-54 15-56 I5-58 15-60

+ + + + +

0-1989651 0-1981178 0-1971924 0-1961894 0-1951093

0-2018544 0-2017253 0-2015965 0-2014680

819 42' 21-15 820 51' 7*61 82i°59 , 54 t07 8' 40*52 823 824 17' 26-98

0-0535585

+

01939526

0057391

0-1927199 0-1914118 0-1900289 0-1885717

0-20I2II7 0-20I0839

0-0611960 0-0649716 0-0687165

+ + + +

0-0724292 0-0761082 0-0797522 0-0833597 0-0869294

+ + + + +

0-1870411 0-1854377 0-1837622 0-1820154 0-1801980

0-0904599

+ + + + +

00939498 0-0973977 0-1008025 £-1041627

02010343 0-2004237 0-1997338

0-1783110 01 763550 0-1743311 0-1722400 0-1700828

02035546

02013397

B. F.

15*27 i?74 24' 48*21 33' 34 r 68

0-3161541 o-3i5953i

0-3I57524

03155522 o-3i535 2 3 0-3151528 03 ! 49537

0-3M7549

0-2008290 0-20070I9

825° 26' 13-43 826 34' 59*88 827° 43' 46*32 828 52' 32*76 1' 19*20 830

0-200575I 0-2004485 0-200322i 0-200I960 0-200070I

83i°io' 832 18' 833° 27' 834 36 835 45'

0-3137668

o- 1999445 0-1998191 0-1996939 0-1995690 0-1994442

836° 53' 2' 44*18 838 839 11' 30*60 840 20' 17*01 841 29' 3*42

O2OO9563

J_ ll3 (16-00) =0-1994442 (16-00) =01994442

Y-va W.

2'

ii' 45*15 20' 31*69

x cos 901 x sin 901

5*63 52*06 38-49 24*92 ii"34 57*7<">

29' 3*42 29' 3*42

= =

0-3I45565 o-3i43586 0-3141609 0-3139637 0-31357.03

o-3!3374i 0-3131784 031 29830

0-3127879 0-3125932

03123989 0-3122050 0-3120114

15-34 15-36 I5-38 I5 40

15-62 15-64 15-66 15-68 15-70 15-72 15-74 15-76 I5-78 15-80 15-82 15-84 15-86 15-88

1590 15-92 15-94 15-96 15-98 16-00

- 0-1993773. -00051662. 24


730

Table IV. Values of

X

O-I 0-2 o-3 o- 4

o-5

o-6

11 09 I-O i-i 1-2

i-3 i-4 i-5

i-6

XI 1-9 2-0 2-1

2-2

23 2-4 2-5 2-6

u 2-9

30 3-i

3-2

33 34 3-5

3-6

37 3-8 3'9 4-o 4'i 4-2

4'3 4'4 4*5

4-6

n

4-9 5-o

;.w

/.w

Jn (x)

/«(*)

/.(*)

X

+ + + + +

0-0012490 0-0049034 0-0111659 0-0197347 0-0306040

+ + + + +

0-0000208 0-0001663 0-0005593 0-0013201 0-0025637

+ + + + +

0-0000003 0-0000042 0-00002IO 0-000066I 0-OOOI607

+ + + +

o-ooooooi 0-0000006 0-0000026 0-0000081

0-2

+ + + + +

0-0436651 0-0587869 0-0758178 0-69^5863 0-1149035

+ + + + +

0-00^3997 0-0009297 0-0102460 0-0144340 0-0195634

+ + + + +

0-0003315 0-0006101 0-0010330 0-0016406 0-0024766

+ + + + +

0-0000199 0-0000429 0-0000031 0-0001487 0-0002498

o-6

+ + + + +

0-1365642 0-1593490 0-1030267 0-2073559 0-2320877

+ + + + +

0-0256945 0-0328743 0-0411358 0-0504977 0-0609640

+ + + + +

0-0035878 0-0050227 0-0068310 0-0090629 0-0117681

+ + + + +

0-0003987 o-ooooioi 0-0009008 0-0012901 0-0017994

1-2 i'3 !'4 i-5

+ + + + +

0-2569678 0-2817389 0-3061435 0-3299257 0-3528340

+ + + + +

0-0725234 0-0851499 0-0908020 0-1134234 0-1209432

+ + + + +

0-0149952 0-0187902 0-0231965 0-0282535 0-0339957

+ + + + +

0-0024524 0-0032746 0-0042936 0-0055385 0-0070396

i-6 1-7 I-O 1-9 2-0

+ + + + +

0-3746236 0-3950587 0-4139146 0-4309800 0-4460591

+ + + + +

0-1452767 0-1623255 0-1799789 0-1981148 0-2166004

+ + + + +

0-0404526 0-0476471 0-0555957 0-0643070 0-0737819

+ + + + +

0-0088284 0-0109369 0-0133973 0-0162417 0-0195016

21

+ + + + +

0-4589729 0-4695615 0-4776855 0-4832271 0-4860913

+ + + + +

0-2352938 0-2540453 0-2726906 0-2910926 0-3090627

+ + + + +

0-0840129 0-0949836 0-1066687 0-1190335 0-1320342

+ + + + +

0-0232073 0-0273876 0-0320690 0-0372756 0-0430204

2-6

+ + + + +

0-4862070 0-4835277 0-4780317 0-4697226 0-4586292

+ 0-3264428 + + + +

0-3430664 0-3587689 0-3733889 0-3867701

+ + + + +

0-1456177 0-1597218 0-1742754 0-1891991 0-2044053

+ + + + +

0-0493448 0-0562380 0-0637169 0-0717854 0-0804420

+ + + + +

0-4448054 0-4283297 0-4093043 0-3878547 0-3641281

+ + + + +

0-3987627 0-4092251 0-4180256 0-4250437 0-4301715

+ + + + +

0-2197990 0-2352786 0-2507362 0-2660587 0-2811291

+ + + + +

0-0896796 0-0994854 0-1098400 0-1207178 0-1320807

0-3382925

0-2810592 0-2500861 0-2178490

+ + + + +

0-4333147 o-4343943 o-433347o 0-4301265 0-4247040

+ + + + +

0-2958266 0-3100286 0-3236110 0-3364501 0-3484230

+ + + + +

0-1439079 0-1561363 0-1687200 0-1816009 0-1947147

44 45

0-1845931 0-1505730 0-1160504 0-0812915 0-0465651

+ + + + +

0-4170686 0-4072280 0-3952085 0-3810551 0-3648312

+ + + + +

0-3594094 0-3692925 0-3779603 0-3853066 0-3912324

+ + + + +

0-2079912

4-6

02213550

U 49

+ + + + + + + + + +

03105347

o-i

0-2347252 0-2400168 0-2611405

03 0-4

05

11 09 I-O I-I

2-2

23 24 2-5

n 2-9

30 31 32 33

34 35 3-6

37

3-8 3'9 4-0 4*1

42 4*3

50

TABLES OF BESSEL FUNCTIONS Table IV. Values of

X

I

2

'/«,(*)

7iW

+ 0-765198 + 0-223891

+ 0-440051 + 0-576725 + 0339059

6

- 0-260052 - 0397150 - 0-177597 + 0-150645

2

+ 0-300079 + 0-171651

3 4 5

- 0-066043 - 0-327579 - 0-276684

/.(*)

+ + + + +

+ 0-157798

/.(*)

7ioW

7nW

—

—

12

+ 0-047689

- 0-176785 - 0223447

X

JM

;,w

,/bW

I

2

3 4 5 b

2 9 10

0-000021 0-OOI202 0-011394 0-049088 + 0-131049

+ + + +

+ 0245837

+ 0-339197 + 0-337576

+ 0-245312 + 0043473

+ + + + + +

0-000002 0-000175 0-002547 0-015176 0-053376 0-129587

+ + + + +

0233584

0-320589 0-327461 0-216711 0-018376 - 0-170254

0-105357 0-265471 0-219603 0-015040 + 0-182499

+ 0-347896 + 0-185775 - 0-055039 - 0-234062 - 0238286 - 0-073471

+ + + + + +

0-127971 0-223455 0-305067 0-317854 0-224972 0-045095

+ + + + + +

0-058921 0-I2632I 0-214881 0-291856 0-308856 0-230381

+ + + + + +

0-023539 0-060767 0-124694 0-207486 0-280428 0-300476

+ + + + + +

0-008335 0-025597 0-0622I7 0-I23II7 0-20IOI4 0-270412

+ 0-000006 + 0-000076 + 0-000545

+ o-oooooi + 0-000015 + 0-000133

+ 0-000003 + 0-000030

+ + + + + +

+ + + + + +

10

-

0-000250 0-007040 0-043028 0-132087 0-261141 + 0-362087

0-000002 0-000037 0-00035I 0-002048

/»(*)

11 12

0391232 0357642

+ + + + +

+ + + +

/»(*)

9

0-132034 0-201129

0-000013 0-000195 0-001468 0-006964

/«(*)

0-000770 0-003275 0-010830 0-028972 0-064295 0-120148

0033996

+ + + +

X

0-002656 0-009624 0-027393 0-063370 0-121600 0-195280

0-002477

0-000002 0-000084 0-000939 0-005520 0-02II65

0243725

7 8

+ + + + + +

+ + + + +

12

6

0-128943 0-309063 0-430171 0-364831 0-114768

0-000022 0-000493 0-004029 0-018405 0-056532

11

5

0019563

+ + + + +

+ 0-204317 - 0-014459 - 0-201584

4

JM

/#(*)

- 0-167556 - 0-291132 - 0-180935 + 0-058379 + 0-227348 + 0-195137

ii

9 IO

/.(*)

+ + + + + +

- 0-301417 - 0-112992 + 0-144847 + 0-254630 + 0-139048 - 0084930

- 0-004683

Jn (x)

0-114903 0-3^2834 0-486091 0-364128 0-046565 - 0-242873

- 009033^ - 0-245936 - 0-171190

+ 0-234636

731

_ + 0-000205 + 0-OOIOI9 + + + +

0-003895

0011957 0-030369 0-065040

/»(*)

/«(*)

/»(*)

— —

—

—

+ 0-000006

+ o-oooooi

+ 0-000051 + 0-000293 + 0-001286

+ + + + + +

+ 0-004508 + 0-013009 + 0-031613

0-OOOOI2 0-000078 0-000393 0-001567 0-005110 0-013991

+ + + + + +

0-000002 0-000019 0-OOOII2 0-000506 0-001856 0-005698

X

I

2 H

4 •>

6 I 9 IO II

12

X

I

2 3

4 S

6

I Q IO 11

12

X

4 5

6 2 8

9 IO II

12


732

Table IV. Values of

X

I 9 IO ii

12

+ + + + + +

o-oooooi 0-000005 0-000030 0-000152 0-000628 0-002152

ViW

- 0-2881947 - 00259497

- 0-1750103 - 0-3026672 - 0-1580605

_ + + + + +

o-oooooi 0-000007 0-000043 0-000199 0-000759

Y

+ 0-2298579

+ 0-0556712

12

- 0-1688473 - 0-2252373

+ 0-1637055 - 0-0570992

+ 0-1986120 + 0-2157208

X

VfW

+ 0-2499367

IO II

6

9 IO II

12

- 0-1970609 0-0637022 0-2564011 0-2851178

+ + + +

0-1354030

- 0-0892528 - 0-2298179

I 9 10 11

12

e

Y

(x)

- 0-4268259 - 0-1993068 + 0-0375581 + 0-2267718 + 0-2803526

+ 0-1673728 - 0-0402973

Y

7

I 9 IO

0-000002 0-OOOOI2 0-000059 0-000251

II

12

+ 0-3282489 + 0-2680006 + 0-0265422 - 0-2050949 - 0-2513627 - 0-0914830

+ 0-1290061

+ + + +

0-0983910 0-2903100 0-2829432 0-0900258 - 0-1449495

- 0-2485118 - 0-1512177

Y

V.to

(x)

- 0-6565908 - 0-4053710 - 0-2000639

+ 0-0172446 + 0-2010200 + 0-2718414 + 0-1895207

X

Y*{*)

9

12

-

+ 0-1786071

- 0-0120492

11

+ 0-2614047

+ 0-1590189

12

2-2906609

6

09921953 0-5752760 0-3727057 o- 1992993

03598142

- 56-3168097 - 12-3614737 - 3-7448595 - 1-5064942 - 0-7849097

- 208-3353554 - 37-83I7507 - 9-543ioi8 - V1778801 - i-3634543

I 9 10

- 0-1983240 - 0-0228763

- 0-3485399 - 0-1971461

- 0-4987558 - 0-3385583

- 0-7396546 - 0-4799704

11 12

5-7667633 1-9399240 0-9067010 0-5454645

II

1-1052194 0-6114352 0-3876699 0-1999469 + 0-0010755

- 16-9318836 - 4-5504447 - 1-6914765 - 0-8394376 - 0-5203290

-

I

9 IO

-

Y13 (x)

lt (x)

6

X

(x)

Y lt (x)

Y 10 (x)

X

6

Y

+ + + +

nw

+ 0-1043146 + 02490154

9

_ —

2 (x)

- 0-0605266 - 0-2630366 - 0-2267557 - 0-0058681

+ 0-2235215

X

/20M

/»(*)

nw

X

6

/«(*)

Jn (x) and Yn (x)

X 6

I 9 10

7

1


Y

X

OI 0-2

04 o-5

o-6

11 09 I-O i-i 1-2

i-5

1-6

U 2-0 2-1 2*2

23 2-4 2-5

2-6

n 2-9 3.0

31 32 33

45 4-6

n

4-9 5'0

- 5099-3323786 - 639-8190662 - 190-7748150 - 81-2024845 - 42-0594943

o-i 0-2

- 24-6915728 - 15-8194791 - 10-8146466 - 77753605 - 5-8215176

o-6

- 0-3085099 - 0-1906649 - 0-0868023

1-2603913 1-1032499 0-9781442 0-8731266 0-7812128

-

3- 8 927946

+ 0-0056283 + 0-0882570

-

+ + + + +

0-1621632 0-2280835 0-2865354 0-3378951 0-3824489

-

0-6981196 0-6211364 0-5485197 0-4791470 0-4123086

-

I-43I47I5 1-2633108

+ + + + +

0-4204269 0-4520270 o-47743i7 0-4968200 0-5103757

-

0-3475780 0-2847262 0-2236649 0-1644058 0-1070324

-

+ + + + +

0-5182937 0-5207843 0-5180754 0-5104147 0-4980704

+ + + + +

0-4813306 0-4605035 0-4359160 0-4079118 0-3768500

2-9614776 2-3585582 1-9459096 1-6506826

i-o i-i

1-0223908 0-9321938

1-2 i-3 1-4 i-5

0-8548994 0-7869991 0-7259482 0-6698787 0-6174081

-

1-7896705 1-5670362

3895534

U

1-2458651 1-1277838

1-9 2-0

- 0-0516786 + 0-0014878 + 0-0522773 + 0-1004889 + 0-1459181

- 0-5675115 - 0-5194317 - 0-4726169

-

1-0292956 0-9459092 0-8742197 0-8116122 0-7560555

2-1 2-2

+ + + + +

-

0-2918869 0-2476693 0-2038152 - 0-1604004

-

0-7059567 0-6600575 0-6173586

2-6 2-7 2-8 2-9

0-5012882 0-4649097 0-4291009

0-1883635 0-2276324 0-2635454 0-2959401 0-3246744

11304119

- 0-4266740 - 0-3813358

03364356

- 01175355 - 0-0753587 - 0-0340296 + 0-0002760 + 0-0453714

0-1477100 0-1060743 0-0645032 0-0233759 - 0-0169407

+ + + + +

0-4153918 0-4166744 0-4141147 0-4078200 0-3979257

+ 0-0830632 + 0-1191551 + 0-1534519 + 0-1857626 + 0-2159036

-

+ + + + +

0-3845940 0-3680128 0-3483938 0-3259707 0-3009973

+ + + + +

- 0-2234600 - 0-2493876 - 0-2723038 - 0-2920546 - 0-3085176

0-9

3-5898996 2-9296706 2-4419696

0-3496295 0-3707113 0-3878529 0-4010153 0-4101884

0-0560946 0-0937512 o- 1 295959 0-1633365 0-1947050

SI

45072313

+ + + + +

+ 0-3431029

+ 0-3070533 + 0-2690920

0-3 0-4 o-5

-

-

+ + + +

4-3 4-4

- 127-6447832 - 32i57 x 446 - 14-4800940 - 8-2983357 - 5-4413708

6-458951 3-3238250 2-2931051 1-7808720 I-47I4724

3-6

4-1 4-2

X

-

+ 0-2296153 + 0-1890219

3 9 4-o

Y*(*)

- I-5342387 - 1-0811053 - 0-8072736 - 0-6060246 - 0-4445187

34 35

n

Yn (x)

Y2 (*)

YAx)

(-x)

733

+ 0-2737452 + 0-2445013

20735414

1

05770644 05385416

i-6

23 2-4 2-5

30 31 32

33

0393631 03583353

35

-

03230993

3-6

0-2878581 0-2525864 0-2172943 0-1820221

13 39 40

0-2437015 0-2689954 0-2916395 031 15049 0-3284816

-

0-1468365 0-1118267 0-0771012 0-0427844 0-0090137

+ 0-3424796 + 0-3534308

+ + + + +

0-0240631 0-0562908 0-0875092 0-1175556 0-1462672

+ 0-2135652

+-

+ 0-1812467 + 0-1478631

+ 0-3660328 + 0-3676629

0-3612893

3*4

4-i 4-2

43 4-4 4-5 4-6

4-9

50


734

Table IV. Values of X

o-i

0-2 o-3 o- 4

°'5

o-6

%l o-g I'O i-i

1-2

i*5

i-6

n i-9

2-0 2-1

2-2

23 24 2-5

2-6

n 2-9 3 -o

31

Y<(x)

- 305832-2979 - 19162-4148 - 3801-0162 - 1209-7389 - 499-2726

- 243-02293 - 13263406 - 78-75129 - 49-88983 - 3327842 - 23-153427 - 16-686187 - 12-391145 - 9-443I93 - 7-361972 -

5856365 4-7437I7 3-905897

3264432 2-765943

Y>{x)

Y*(*)

- 24461484-502 - 765856-775 - 101169-657 - 24113-576 - 7946-301

- 2445842617-9 - 38273676-3 - 33685209 - 601629-7 - 158426-8

- 3215-6142 - I499-9983 - 776-6983 - 435-6898 - 2604059 - 163-88133 - 107-65135 - 7332353 - 5I-5I9I3 - 37 I 903i

- 27-492153 - 20-756338 - 15*969987 - 12-499113 - 9-935989

-

2-3733331 2-0603205 1-8079562 1-6023566 I-433I973

-

-

1-2926953 1-1749076 1-0752421 0-9901112 0-9166828

- 3-271567 - 2-821150 - 2-454762 - 2-154277 - 1-905946

0-8526997 0-7963470 0-7461539 0-7009203 0-6596606

-

1-6992271 I-5259577 I-379757I

- 0-6215621 - 0-5859520

-

Yn (z)

8-011973 6-546165

5-4M324 4-529576 3-830176

- 53350-547 - 21295-913 - 9629-977 - 4791-108 - 2570-780 - 14666768 - 8804084 - 55 1 -6360 - 358-5506 - 240-5734 - 165-96960 - ii7'35239 - 84-81625 - 62-52037 - 46-91400 -

35-77892 27-69498 21-73258 17-27008 13-88751

- 11-290256 - 9-273796 - 7-691764 - 6-438430 - 5436470

Y

7

(x)

- 293476652667 - 2295654722 - 134639666 - 18024776 - 3794296 - 1063795-33 - 363572-8o - 143672-96 - 63445-75 - 3058896 - 15836-229 - 8696-433 - 5018-701 - 3021-772 - 1887-397 - 1217-2798 - 807-6135 - 549-47I7 - 382-366 - 271-5480

X o-i

0-2

03 o-4 o-5 o-6

Zl 09 i-o

11 1-2 i-3 1-4 i-5

i-6

n 19 2-0

- 196-43901 - I44-5I734 - 107-97306 - 81-82481 - 6282986

2-1

48-8373I 38-39572 30-50994 24-48750 I9-83994

2-6

-

- 16-218237 - 13370058 - 11-110890 " °'3 34 3 - 7-848866

2-2

23 2-4

25

U 2-9

30

1-1494603

-

4-628678 3-972271 3-434928 2-991999 2-624512

4-o

- 05522725 - 0-5200615 - 0-4889368

-

1-0581497 0-9790651 0-9100926 0-8494985 0-7958514

-

2-3177427 2-0601699 1-8427079 1-6581398 1-5006918

-

6-667659 5-702567 4-908985 4-252470 3-706224

4-1 4-2 4*3 4-4 4-5

-

0-7479619 0-7048359 0-6656385 0-6296652

- 1-365713° - 1-2494327 - 1-1487739

4-1 4-2

- 1-0612100 - 0-9846543

-

3-249247 2-864972 2-540242 2-264544

03404998

-

029425

4-5

4-6

-

0-3110929 0-2815701 0-2519027 0-2220872 0-1921423

- 05650943 - 0-5355591 - 05073471 - 0-4801469 - 0-4536948

-

- 1-8280526 - 1-6548682 - 1 5053290 - 1-3757009 - 1 2628988

32 33 3'4

35 3-6

n 39

XI 4-9 5-o

0-4585842 0-4287478 0-3992226 0-3698472

1

2555924

05963194

0-9173730 0-8579174 0-8050705 0-7578045 0-7152474

2

31 32 33 3-4

35 3-6

37

3-8

39 4-o

43 44 4-6 4-7 4-8 4-9

50


X

Y a (x)

Y9 (x)

OI

- 410842855308 2 - 1606575569 2 - 62798159 '2 - 6302655 '2 - 1060819 [2]

- 65731922082781:3

0-2 o- 4

o-5

o-6

09 i-o i-i

1-2 i-3 i-5

i-6

n 1-9 2-0

- 200085-33 - 100577-97 - 53495'9i - 29859-17 - I7375-I3 - 10485-229 - 6533582 - 4188-852 - 2754-916 - 1853922

23 24 2*5

2-6 2-7 2-8

29 3-o

- 1273-8144 - 891-9608 - 635-4947 - 460-0405 - 3379597

- 251-67985 - 109-81514 - I44-85794 - 111-77710 - 87-14989 -

31 3-2

33 3-4

35

n

3*9 4-o

4-2

43 4'4 4-5

4-6

49 5-o

in

- n83i335i320 4 5[5; - 11563671430IV - 200810527 5 - H330366[5; - I2i9636[5

O-I 0-2

- 197581500 2 - 42447194 2 - 11213538 '2] - 3469552 '2 - 1216180 [2]

- 659430617 - 165354373 - 49949263 - 17396869 - 6780205 - 2894495-9 - 13323432 - 653392-5 - 338225-9 - 183447-3 - 103635-01 - 60684-92 - 36684-77 - 22816-93 - I4559-83 -

9508-814 6342-471 4312-860 2985-112 2IOO-II2

- 1499-9618 - 1086-4347 - 797-2497 - 592-2137 - 444-9595 -

33792355 259-23861 200-77848 156-90853

12367548

- 47J64393 - 19884570 - 8993478 - 4318759 - 2183993 -

1554086

1

-

636012-7 362658-9 213405-5 129184-5

-

80230-30 51000-98 33117-32 21928-30 14782-85

- 10132-671 - 7053-083 - 4980-319 - 3564032 - 2582-607 - 1893-5218 - 1403-6955 - 1051-4532 - 795-3720 - 607-2744

- 5-329II4

34-71866 28-75588 23-95941 20-07785 16-91853

- 142-69412 - II4-93902 - 93-17343 - 75-99248 " 62-34502

- 4-646265 - 4-07I477 - 3-585472 - 3-172769 - 2-820869

- 14-332870 - 12-205480 - 10-446246 - 8-984362 - 7-763883

[

^144157

]

467-7617

3633271 284-4645 224-4160 178-3306

-

o-6 0-7 o-8 0-9

10 i-i

1-2 i-3 i-4 i-5

XI 19

.

-

9-729277 8-300474 7-121782

03 04 o-5

i-6

-

51-43888 42-67291 35-58795 29-83101 25-12911

20 2-1 2-2

23 2-4 2-5 2-6

n 2-9

30 3-i 3-2

33 34 35 3-6

n

39 40 41 4-2

43 4-4

45 4-6

Si

49 50

are the numbers of digits between the last digits given and the decimal integral part of F10 (o-i) is a number containing 19 digits of which the

For example, the

14 are given.

X

98-27476 78-69575 63-48271 51-57168 42-17814

23-612043 19-517110 16-243027 13-607138 11-471092 -

41

first

- 3347888 75 [3. - 25I92597[3, - 3390825 [3

*\o(*)

-

-

3-6

The numbers

68-61497 54-52I73 43-70218 35-32025 28-77095

Yn (x)

1 2850308895 [3

- 24768540-5 - 7250160-1 - 2504646-8 - 982142-7 - 425674-6

2-1 2-2

points.

-

735

1

I


736

Table TV. Values of e~x In {x) X

e~ x h(x)

e~*I z {x)

e-x h{x)

e-*I & {x)

X

o-i 0-2

0-0011320 0-0041073 0-0083969 0-0135860 0-0193521

0-0000189 0-0001368 0-0004191 0-0009027 0-0016043

0-0000002 0-0000034 0-OOOOI57 0-0000450 o-oooiooo

__ o-ooooooi 0-0000005 o-oooooio 0-0000050

o-i

00254458

0-0025257 0-0036585 0-0049877 0-0064938

00081553

0-0001886 0-0003182 0-0004948 0-0007233 0-0010069

0-0000113 0-0000222 0-0000394 0-0000047 0-0000999

00013479 0-0017471 0-0022045 0-0027189 0-0032885

0-OOOI468 0-0002072 0-0002826 0-0003746 0-0004843

03 o- 4

05

o-6 o-7 o-8 I-O

0-0316770 0-0379022 0-0440151 0-0499388

i-i

00556193

1-2 i-3 i-4 i-5

0-0610206 0-0661209 0-0709088

0075381

0-0099497 0-0118547 0-0138406 0-0159110 0-0180231

i-6 1-7 1-8 i-9

0-0795406 0-0833947

0-0201679 0-0223299

0-0039110 0-0045834

00869539

00244955

00053023

0-0902306

2-0

00932390

0-0266527 0-0287912

2-1

00959939 00985103

0-0309022 0-0329781

0-1008034 0-1028881 0-1047787

00350127

09

2-2 2-3 2-4

25 2-6

n 29 30 31 32 33

34 3'5

36 3 "Z 3-8

39 40

0-1064892 0-1080327 0-1094217 o- 1 106680 0-1117825

0-1127758 0-1136572 01 144358 0-1151197 0-1157167

0-1162339 0-1166776 0-1170540 0-1173686 0-1176265

!'3 i-4 i-5

0-0122829

0-0030293

2-6

00132534

OOO33813

0-0142344 0-0152228 0-0162159

0-0037511 0-0041380 0-0045409

U 29

0-OI72II2 0-0182063 O-OigigiO 0-0201876 O-02II70O

0-0049590 0-0053913 0-0058369 0-0062947 0'Oo6763o

0-0576999 0-0588928 0-0600338 0-0611243

0-022I447 0-023II02 0-0240654 0-0250090 0-0259400

0-0072431 0-0077318 0-0082288 0-0087333

n

00092443

40

0-0268576 0-0277610 0-0286495 0-0295227 0-0303800

0-0097611 0-0102826 0-0108082 0-0113371 0-0118685

41

0-03I22I2 0-0320458 0-0328538

46

OO336449

0-0124017 0-0129361 0-0134711 0-0140060

0-0344190

00145403

50

0-0370010

00389387 0-0408227 0-0426507 0-0444207 0-0461318

00477833

00493750 0-0509071 0-0523802 0-0537949

00551523 00564535

0-0666994 0-0674822 0-0682274 0-0689364 06961 07

49 50

i-i 1-2

2-1 2-2 2-3 2-4 2-5

0-1182204 0-1181933 0-1181380 0-1180568 O- 1 1 795 T 9

4-8

I-O

0-0015615 0-OOl8l42 0-0020879 0-00238I9 0-0026960

0-1179905 0-1181048 0-1181791 0-1182166

47

11 09

0-0077019 0-0085701 0-0094659 0-0103857 O-OI 13259

01178323

4-6

o-6

1-9 2-0

4-2

45

05

0-0060642 0-0068654

41 4'4

03 0-4

0-0006I29 0-00076I 0-0009298 0-OOIII92 0-0013298

0-0621661 0-0631607 0-0641096 0-0650147 0-0658774

43

0-2

i-6

n

30 3-i

32 33 34 35

36 3-9

4-2

43 4*4 4-5

XI 4-9


X

O'l 0-2

o-3 o- 4

o-5

o-6

n 0-9 I-O i-i

1-2 1-3 1*4 i'5

i-6

:i 1-9 2-0

9-8538448 4-7759725 3-0559920 2-1843544 1-6564411

199-503964° 49-5124293 2i-74574°3 12-0363013 7-5501836

7990-0124305 995-0245583 292-9991958 122-5473670 62-0579095

0-7775221 0-6605199 0-565347 1 0-4867303 0-4210244

1-3028349 1-0502835 0-8617816 0-7165336 0-6019072

5-1203052 3-6613300 2-7198012 2-0790271 1-6248389

35-4382031 21-9721690 14-4607876 9-9566542 7-1012628

0-3656024 0-3185082 0-2782476 0-2436551 0-2138056

0-5097600 0-4345924 0-3725475 0-3208359 0-2773878

1-2924388 1-0428289 0-8513976 0-7019921 0-5836560

5-2095375 3-9106886 2-9922325 2-3265275 1-8338037 1-4625018 1-1783157 0-9578363 0-7847324 0-6473854

2-4270690 I-7527039 1-3724601 1-1145291 0-9244191

o-6

SI 0-9 I-O i-i 1-2

i-3 i-4 i-5

i-6

0-1459314 0-1288460

0-1826231 0-1596602 0-1398659

0-1227464 0-1078968 0-0949824 0-0837248 0-0738908

0-2176851 0-1873570 0-1617334 0-1399880 0-1214602

o-5373847

2-1

04485459

2-2

o-3762579 0-3170382 0-2682271

23

0-0652840 0-0577384 0-0511127 0-0452864 0-0401564

0-1056168 0-0920246 0-0803290 0-0702383 0-0615105

0-2277714 0-1940711 0-1658685 0-1421668 0-1221704

2-6 2 .y 2"8

00356341

0-0539444 0-0473718 0-0416512 0-0366633

0-1052398 0-0908577 0-0786032 0-0681323 0-0591618

31 32

01138939

2-6

0-0553983

n

00492554 0-0438200 0-0390062 0-0347395 0-0309547 0-0275950 0-0246106 0-0219580

3*5

00195989

0-0316429 0-0281169 0-0249990 0-0222394

3-6

0-0174996 0-0156307

0-0197950 0-0176280

0-0284968 0-0222321 0-0196614 0-0174014

P 39

05

0-4887471 0-4118051 0-3488460 0-2969093 0-2537598

0-0702173 0-0623476

33 34

0-4

02093625

2'4 2*5

3*2

O-I 0-2 0-3

0-2406339

00791399

3'i

X

0-1879548

23

3 -o

3 (x)

01654963

0-1007837 0-0892690

2-9

K

K*(x)

(*)

2-2

2-1

K n (x)

«iW

Ko

737

00323071

00251593

0-0514581 0-0448273 0-0391079 0-0341649 0-0298849

XI 19 2-0

2-4 2-5

29 30

33 3*4 3'5

3-6 3 '1 3-8

00139659

00157057

0-0124823 001 1 1597

0-0139993

45

0-0099800 0-0089275 0-0079880 0-0071491 0-0063999

0-0111363 0-0099382 0-0088722 0-0079233 0-0070781

4-6

00057304

0-0063250

0-0084804

00136993

4-6

&

0-0051321 0-0045972 0-0041189 0-0036911

00056538

00075380

0-0050552 0-0045212 0-0040446

0-0067036 0-0059643 0-0053089

0-0120691 0-0106415 0-0093900 0-0082918

rl

40

4*i 4-2

43 4*4

4*9 5-o

00124835

0-0154123 0-0136599 0-012,1146

0-0107506 0-0095457

0-0261727 0-0229477 0-0201416 0-0176965 0-0155631

39 4-0 4-1

42 43 4'4 4'5

4-8 4.9 5-o


738

Table IV. Values of

X

O-I 0-2

K

t (x)

o- 4

479600-2498 29900-2492 5881-7297 1850-2468

o-5

7522451

03

o-6 0-7 o-8 o-9 i-o I-I 1-2

i-3 !-4 i-5

359-502336 191-994207 111-175708 68-456722 44-232416 29-708098 20-596272 14-661702 10-672824 7-918871

K,{x)

38376010-00 1197004-99 157139-12 37127-48 12097-98 4828-8027 2216-1917 1126-2179 618-4609 360-9606 221-26843 141-21917 93-21809 63-31409 44-06778

K

6

K n (x)

(x)

3838080599-8 59880149-8 5243852-5 930037-3 242711-8 80839-547 3l8 ^'§75 14188-899 6940-244 3653-832 2041-2393 1197-4227 73I-7239 462-9164 301 7041

K,(x)

X

460608047990 3594005995

O-I 0-2

20991 1239 27938248

03 °4

5837^2

o-5

1621619-74 548248-34 2I3959-70

93I5505

o-6 0-7 o-8 0-9

44207-02

i-o

22489-333 12115-446 6847-593 4031-169 2457-700

i-i 1-2

3-54I634 2-775011 2-195916

31-328146 22-686864 16-698431 12-468991 9-431049

2-4 2-5

1-7530699 1-4106641 1-1432756 0-9325836 0-7652054

7-215746 5-578234 4-352869 3-425650 2-716884

36-113765 26-766271 20-068791 15-206127 11-632743

213-58012 151-57608 109-05961 79-45628 58'55405

2-6 2 'Z 2-8 2-g 3*o

0-6312432 0-5232937 o-43576i5 0-3643764 0-3058512

2-1700581 1-7445711 1-4109012

8-977621 6-984668 5-474694

1 1473430 o-9377736

4320732

43-60523 32 8 75$ 7 24-87388 19-02623 14-66483

31

0-2576343 0-2177299 0-1843662 0-1568967 o-i337274

0-7701024 0-6351824 0-5260364 0-4373011 0-3648244

27418356 2-2026750 1-7786158 1 -4430764 1-1760828

11-383660 8-895214 6-993730 5-530512 4-397108

0-1142604 0-0978523 0-0839814 0-0722228 0-0622288

0-30537 01 0-2563998 0-2159108 0-1823141 0-1543425

0-9625106 0-7908246 0-6521676 o-5396949 0-4480852

3-513739 2-821236 2-275387 1-842914 1-498598

0-0537I39 0-0464423 0-0402191 0-0348822 0-0302965

0-1309802 0-1114092 0-0949678 0-0811187 0-0694236

o-373i778 031 17023 0-2610745 0-2192429 0-1845713

1-2232080 1-0019872

4-6

0-0263491

u 49

00229453

0-0595239 0-0511250 0-0439839 0-0378998 0-0327063

i-6 1-7 i-8 i-9

2-0 2-1

2-2

23

3-2

33 34 35 3-6

U 39

40 4-i 4-2

43 4'4 4'5

50

5 -973 ! 29 4'5 7 567

0-0200054 0-0174623 0-0152591

201-77404 138-02271 96-31069 68-40128 49-35ii6

3-43I763

o-i55749o

01317219 0-1116385 0-0948088 0-0806716

I

544'6334 996-9648 658-7697 444*4771 305-5380

08235478 0-6790539 0-5616138 0-4658258 0-3874361 0-3230800 2700847 0-2263181

I# 3

1-4 i-5

i-6

U 1-9 2-0 2-1

2-2 2-3 2-4 2'5

2-6

n 2-9

30 31 3-2

33 3-4

35 3-6

37 3-8 3-9 4-0 4-1

42 43 44 45 4-6 4*7 4-8 4*9

50

21

1

1


X

«iW

o-i 0-2

644889647992 [2]

03

98ouoi7[2]

o- 4

9787687[2] i6 3 68 3 8[2]

25i6402998[2]

o*5

0-9 i-o

379I8633-59 10996818-60 3758483-72 1456018-75 622552*12

i-i

288269-12

1-2

14254429

I "5

74475'03 40774-60 23240-24

o-6

SI

i-6 I -7

1-8

13717-316 8348-321 5220-075

1-9 2-0

3343496

2*1

1

2-2

23 2-4 2*5

2-6

3 29 30 3-i

32 33 3-4

35

36

U 39 40 4'i

42 43 44 4*5

2188-117

459981 99I-34I3 683-9099 478-7011

3395354 243*7750i 1 76994 A 129-84408 96-17151 71-86762 54-I5I9I 41-11923 31-44899 24-21577 18-76452

14-627050 II-465773

9035174 7-155283 5-693179

4-549986 3-651659 2-942393 2-379869 1-931814

46

1-5734796 1-2857868

49 50

0-8664794 0-7143624

I0539552

first

Kn (x)

9 (x)

*

10 (*)

io3i869497592o[3] 20i348i7990[3J 522935335C3] 39i78686[3]

18574295846304(5]

5243719^] 1012785182 25 1 904 1 04 75383634 25977933 1 000504 42I5494-70 1912706-01

92346336 470026-62

25035361 138717-80 79569-40 47059-44 28600-23 17810-48 11337-247 7361-331 4866 694 3270-797 2231-581 1543-7592 1081-6417 766-8400 549-6277 397-9588

1 81 23852594^]

X

o-i 0-2

3i38 5 92i2[5]

03

1 7630197(5]

i889 376[5]

0-4 o\5

3042i474i[2]

o-6

64885309O] 16998902J2] 52ioi47[2] i8o7i33[2]

69269092 28833134 1 286089 6083974 3027484

S3 0-9 i-o i-i

1-2 i-3 i-4 i-5

1574292-56 850847-84 475814-46 274293-04 162482-40

1-6 1-7 i-8

98636-38 61220-41 38771-08 25009-68 16406-92

2-1 2-2

10931-338 7387*939

2-6 2 '1 2-8 2-9

5059530 3507-654 2459-620

19 2-0

23 2-4

25

30

1743-1174

31

12476333

3-2

119-48709 90-17775

901 3053 656-7945 482-5358

33 34 35

68-52285 52-40296 40-31822 31-19792 24-27131

357-24I3 266-3991 200-0162 I5I-I457 114-9141

3-6

290-87739 214-49139

15947366

n 39 40

87-87352 67-56482 52-22047 40-56082 31-65296

41 42

5-938798 4-764583 3-836264

24-812255 I9-533I25

4-6

3099405

12-252049 9-758563

18-979250 14-913071 11-771986

9333122 7-430286

2-512278

15 439946

43 44 4-5

u

4-9

50

in [ ] are the numbers of digits between the last digits given and the decimal For example, the integral part of K10 (o-i) is a number containing 19 digits of which the

The numbers points.

K

739

14 are given.


740

Table V. Values of

X

I

2

3 4 5

6 7 8

9 IO ii 12

/*<*>

J-l(*)

0-671397 + 0-513016 + 0-065008 - 0-301921 - 0-342168

+ 0-431099 - 0-234786 - 0-456049 — 0-260766 + 0-101218

- 0-091016 + 0-198129 + 0-279093 + 0-109608 0-137264

+ 0-312761 + 0227356 - 0-041045 - 0-242326 - 0-211709

-1

-

-

0240569 + 0-001064

13 14 15

- 0-123589 + 0-092980 + 0-211241 + 0-133968

+ 0-194364 + 0-200012 + 0-029158 - 0-156506

16 17 IO 19 20

- 0-057428 - 0-186045 - 0-141233 + 0-027435 + 0-162881

- 0-I9I025 - 0'053240

21

+ 0-145672 - 0-001506 - 0-140786 - 0-147489 - 0-02II20

22 23 24 25

26

+ 0-119324

3

+ 0-146854

29 30

+ 0-040849 - 0-098326 - 0-143930

+ 0-124181 + 0-180980 + 0-072807 - 0-095367 - 0-170103 - 0-088648

+ 0-069085 + 0-158173

+ 0-IOI229 - 0-044859

36 - 0-131887 - 0-084414 3Z 38 + 0-038360 39 + 0-123138 40 + 0-094001

- 0-017017 + 0-I00400 + 0-123619 + 0-034067 - 0-084139

- 0-019766 - 0-112839 - 0-101207

- 0-123031 - 0-049245

4i 42

43 44 45

46

% 49 50

+ 0-002129 + 0-101208 + o- 1 06088 + 0-014382 - 0-088476 - 0-108712 - 0-029606

+ 0-067544 + 0-I20267 + 0-062483 - 0-050842 - 0-115491 - 0-073722 + 0-034263 + OI08885

/-*

w

X

+ 2-876388 + 0-828221 + 0-369041

- 0-327930

+ 0-038889

- o- 199052 + 0-075931 + 0-254504 + 0-197983

+

- 0-072950 - 0283437 - 0-250619 - 0-024773

- 0332205 - 0-128524 + 0-143781

-

0-022934 0-204663 0-193660 0-014070 + 0-165437

+ 0-240472

+ 0-234314

0-107392 - 0-108427 - 0213323 - OI23534

+ 0-072423 - 0-137671 - 0-2I42S6 - o- 100880

- 0-066647 11 - 0-22I2I2 12 - 0-175790 13 + 0-016554 14 + 0-l8l2I2 15

+ 0-187436 + 0-042305 - 0-132027 - 0-179536

+ 0-069367 + 0-189178 + OI34334

+ 0-092573 + 0-I935II + 0-II9229

+ 0-178019 16 + 0-019864 17 - 0-146570 18

- 0-036960 - 0-166521

- 0-055782 - 0-172580

- 0-175144 - 0-047829

- 0-141131

- 0-131058

0-009238 + 0-144640 + 0-144611

+ + + +

+ 0-115528 21 + 0-168843 22 + 0-069702 23 - 0-087161 24 - 0-159948 25

- 0-064663 -!-

0-102303

+ 0-I70034 + 0-082527 - 0-075230 - 0-159018 - 0-096639

+ 0-013353

- 0-057748

W

0-049497 0-223925 0-412710 0-440885 0-240377

+ 0-006053 + 0-118244 + 0-I20228

35

J%

(*)

+ + + + +

- 0-132954 - 0-115235

- 0-001844 - 0-II6II5 - 0-121878

n + i) (x)

+ 0-477718 + 0-185286 - 0-169651

+ 0-131087 + 0-117665

- 0-057900

/

- 1-102496 - 0-395623 + 0-087008 + 0-367112 + 0-321925

+ 0-240298 + 0-491294

+ 0-022470

- 0-145147 - OI I0835

J -I

(*>

+ 0-050298 + 0-146606 + 0-107444 - 0-027268

3i 32 33

+ 0-077777 + 0-138882 34 + 0-072398

/|

J±

- 0-102602 - 0-122610 - 0-030910

0-230608 0-273962 0-082683 0-158435

H-

-f

+ OOI4793

+ 0-196659

0-024692 0-151550 0-138086 0-002038

- 0-123217 - 0-145193 - OO35665 + 0-102148 + 0-143181

+ 0-109441 + 0-I4I203

+ 0-053672

+ 0-045034

- 0-081454 - 0-138826 - 0-068982 + 0-06I230

- 0-088581 - 0-138331 - 0-061964 + 0-068053

- 0-014568 - 0-294372

I

2 3

4 5

6 I

+ 0269886 9 + 0-164179 10

19

20

- 0-130474 - 0-087011 - 0-I4I260 + 0-060991 - 0-025141 '+ 0-148969

26 27 28 + 0-100268 29 - 0-036788 30 - 0-136281 - 0-110029 + 0-014465 + 0-122202 + 0-II6630

3i

32 33 34 35

+ 0-132360 + 0-I33000 + 0-005987 36 + 0-081700 + 0-076088 - 0-107025 - 0-041613 - 0-I240I2 - 0-091898

- 0-048040 - 0-I255I6 - 0-087514

- 0-I20334 11 - O-O24520 39 + 0-091031 40

+ 0-122549 + 0-022766 + 0-028733 + 0-046558 + 0-II40II + 0'Il6l64 - 0-009898 + 0-099636 + 0-096331

+ 0-121365 41 + 0-04IIOI 42

+ 0-086489

- 0-120218 - 0-060234

- 0-004863 - 0-102596

- 0-OI0326 - 0-I05223

+ 0-053148 - 0-104983 - 0-I02622 + 0-115797 - 0-011925 - 0-00699I + 0-071879 + 0-0900I2 + 0-092968 - 0-036481 + 0-108013 + 0-106479 - 0-109477 + 0-027428 + 0-023037

- 0-074495 - 0-119958 - 0-055643 4-

43 44 45

0-057689 46

+ 0-II6253 + 0-068096

8

- 0-040876 49 - 01 10530 50

TABLES OF BESSEL FUNCTIONS Table V. Values of

AW

X

w

741

J± n + ^ {x) (

X

/.(*)

/-#(*>

+ 0-000807 + 0-015887

+ 90-079718 + 5-034028 + 1-269137 + 0-625147 + 0-332945

4 5

+ 0-054598

6

- 0-193887 - 0-304868 - 0-217577 + 0-004188

I 9 10

+ 0-200397 + 0-230091 + 0-081000

11 12 13

+ 0-183031 + OOO7984

- 0-120260 - 0-210673

14 15

- 0-123323

16

+ 0-083050 + 0-178478

- 0-l6l920 - 0-187495 - OO55OO5 + 0-II65I9 o-i8oiii

+ 0-060438 + 0-182978 + 0-144546

\l 19

- 0-014639

20

+ 0-113625

+ 0-086555

- 0-047611 - O-T 598IO - 0-126452 + 0-017196

- 0-077008 - 0-166640 - 0-107753 + 0-042601

- 0153403 - 01 53694 - 0-021144 + 01 24043

21 22 23

24 25

- 0-133507 - 0-164423 - 0-049581 + 0-103998 + 0-159426

+ 0-155133

26

+ 0-071548

+ 0-139950 + 0-133898 + 0-009064

+ 0149737 + 0-121443

+ 0049333 - 0-095706

26

- 0-012033 - 0-130821 - 0-129349

- 0-151235 - 0-071438 + 0-068766

11 29 30

+ 0-088450

32 33 34 35

2 3

+ 0-007186 + 0-068518 + 0-210132

4 5

+ 0-365820 + 0-410029

6

+ 0-267139

7 8

9 IO

- 0-003403 - 0-232568 - 0-268267 - 0099653

ii 12 13

+ 0-129440 + 0-234840 + 0-140709

15

- 0-062450 - 0-199063

I

M

16 \l 19

20 21 22 23

- 0-158507 0-014610 0-165146 0-164856 0-021518

+ + + +

29 30

- 0-076458 - 0-151096 - 0-088575 + 0-050802

3

J -I

- 13-279444 - 1-674928 - 0-702076 - 0-348902 - 0-027552

- 0-067254 - 0-240524

- 0-183879 - 0-266416

- 0-210178 - 0-015220 + 0-176039 + 0-207411 + 0-063130 - 0-124998 - 0-195020 - 0-093620

- 0-119436 - 0-137049

- 0-133192 - 0-067238 + 0-057447 + 0-127156 + 0-080519

- 0-132005 - 0-054717 + 0-069461 4- 0-128176 + 0-070464

+ 0-019912 + o-i 19745 + 0-109751 + 0-001705

- 0-119045 - 0-032729 + 0-081099 + 01 19045 + 0-048542

- 0-037567 - 0-118904 - 0-090974 + 0-018494 + 0-108779

-

- 0-121619 - 0-083128 + 0-029265 + 0-112774

- 0-114951 - 0-021284

- 0-064303 - 0-116541 - 0-062195 + 0-047346 + 0-111781

+ 0-098712

+ 0-092836

- 0-000443

-

36

+ + + +

49 5°

-i

0-005119 0-112964 0-116289 0-014818 - 0-097428

- 0-031691

+ 0-098646 + OI 36634 + 0-05IOI2

8

- 0-151943

+ 0-064567 + 0213437

- 0-077891

0-140218

+ 0-101394

46

+ 0-047I22

- 0013372 + 0-110760 + 0-132602 + 0-035744 - 0090154

4-

42 43 44 45

+ O-I9930O +' O333663

+ 0-384612 + 0-280034

3i

4i

+ 0-077598

+ 0-237949 + 0-322411 + 0-184099

32 33 34 35

11 39 40

2

- 0-027012 - 0-127357 - 0-110507

- 0-082918 - 0-103842 - OOI6375

0049057

0-010367 0-102038 0-099715 0-007388

I

2

3

24 25

3i

- 0-043448 - 0-132704 - 0-101062

- 0-105122

+ 0089305 + 0-117016 + 0-038744 - 0-072710 - 0-116187 - 0-056004 + 0-055710 + 01 12833

36

11 39 40 4i

42 43 44 45

46

% 49 50


742

Table V. Values X

I

2 3 4 5

6 7 8

9 IO

7-V

(*)

+ 0-000074 + 0-002973 + 0-022661 + 0-082606 + 0-190564 + + + +

0-309779 0-363446 0-285580 0-084388 - 0-140121

ii 12

- 0-253757 - 0-186414

13 14 15

+ 0-007055 + 0-180113 + 0-203854

16 17 io 19

+ 0-067428

20

+ 0-059532

21 22

+ 0-170603 + 0-132919 - 0-015626

23 24

25

26

3

- 0-113872 - 0-192649 - 0-109663

- 0-144405 - 0-144089

- 0-019716 + 0-116939 + 0-147035

29 30

+ 0-047975

31

- 0-144100 - 0-070242

32 33 34 35

36 ii 39 40 4i

42 43 44 45

46

% 49 50

- 0-089606

+ 0-063176 + 0-136818 + 0-087324

•/-V

oiJ ±(n + ^(x)

W

hi-

(*)

- 797-438019 - 20-978200 - 3-10533^ - 1-057678 - 0-571750

+ + + + +

0-000006 0-000467 0-005493 0-027066

- 0-319846 - 0-073127

+ + + + +

0-183316 0-291096 0-345551 0-287020 0-112283

+ 0-531787 + 0-308802 + 0-086412 - 01 30550

-

0-101814 0-235447 0-207468 0-041513 + 0-141509

- 0-246614 - 0-085855 + 0-115406 + 0-222482 + 0-164272

+ 0-208276 + 0-113813

-

+ 0-158877 + 0-284832 + 0236755 + 0-046217 - 0-157348 - 0-232116 - 0-130102 + 0-063274

+ o-i94373 + 0-163023 + 0-002131 - 0-151520 - 0-171891 - 0-047880 + 0-110486 + 0-168084 + 0-079936 - 0-073044

0085579

1

2

3 4 5

6 I 9 10

- 0-264618

0-010308

11 .

12 J3

14 15 16

0165924 20

+ 0-002808 + 0-143468 + 0-159167

+ 0-178483 + 0-098451

21 22

- 0-059244 - 0-160681 - 0-122994

23 24 25

26

+ 0041567 -

o- 106000

11 19

- 0-157027 - 0-101996 + 0-039548 + 0-141606 + 0-116419

- 0-158079 - 0-073801 + 0-070397 + 0-149019 + 0-096493

+ + + +

-

- 0-037760 - 0-134906 - 0-111543 + 0-008521

- 0-139905 - 0-045989 + 0-085043 + 0-137843 + 0-068414

3i

- 0-059088 - 0-131075 - 0085598 + 0-034271 + 0-120760

36

0-009951 0-123523 0-124785 0-015884 + 0-103879

+ on 17599

- 0-083440 - 0-127550 - 0-056866

- 0-098362 - 0-124012 - 0-039312

+ 0-108276 + 0-006668

+ 0-062800 + 0-120424 + 0-072282

+ 0-078107 + 0-123365 + 0-057931 - 0057530

- 0-065662 - 0113110

+ 8681-738496 + 110-346069 + 10-117087 + 2-283448 + 0-924903

0-184280 0-056824 + 0-109179

+ 0-128214 + 0-038110

+ 0-082467 + 0-114556 + 0-043062

X

- 0-062725 - o- 1 80008 - 0-147369

- 0-038120 - 0-126274 - 0-099837 + 0-014761 + 0-113283

- 0-098498 - 0-113059 - 0-025987

/-vW

- 0-042429 - 0-116528

+.

0-120357

+ 0-017176

- 0-119127

- 0-084486

- 0-073116

+ 0-022691 + 0-093419 + 0-093609 - 0003933

+ 0-037178 + 0-111907 + 0-084974 - 0-017490

0-017102 0-137260 0-135698 0-017726 - 0-111453

+ 0-098102 -

0-010256 0-107796 0-106408 0-010259

+ 0-092913 + 0-110876 + 0-034596 - 0-076725 - 0-111958

11 29 30 32 33 34 35

11 39 40 41 42 43

44 45

46

% 49 50

TABLES OF BESSEL FUNCTIONS Table V. Values of

X

2 3 4 5

7¥

.

+ 0-000063 + + 0-001140 + + 0-007957 + + 0-031941 +

0-000008 0-000207 0-001974 0-010243

13 14 15

- 0-068653 - 0-214523 - 0-218661 - 0-081213

17 IO 19

+ 0-101797 + 0-200906 + 0-147348 - 0-013500

20

-

16

0155322

- 0-112842

0-063457 0-185515 0*169350 0-030877

7?f

6

+ + + + +

+ + + + +

11

+ 0-044768

12 13 14

+ + + +

•*5

0-089695 0-152818 0-221379 0-269315

16 + 0-265267 + 0-189882 \l + 0-054141 19 - 0-094968 20 - 0-191398

0-012324 0-037852 0-089213 0-167162 0-252556

- 0-221691 - 0-137449

+ 0-027861 + 0-165024 + 0181568

3

4 5

6 + 0-000272 + 0-001446 + 0-005680 I + 0-017442 9 + 0-043438 10

0159277 + 0-089780 11 0235095 + 0-155899 12 + 0-227859 13 0273186 + 0-276957 14 0-179412 + 0-269236 15

- 0-209985 - 0-169805 - 0-032875

- 0-20I228

/v <*>

J* (*)

/v w

—

—

— — —

W — 4-

+ 0-020106 + + 0-045914 + + 0-089532 + + 0-149989 + + 0-215531 + + + + +

+ + + + +

2

+ 0-185304 16 + 0-040241 17

0-000003 0-000024 0-000152 0-000718 0-002683

0-262335 0-261336 o-i934 J 9 0-066273 - 0-078969

0-003827 0-014205 0-040045 0-089590 0-163007

X

+ 0-024269 - 01 30704

+ + + + +

0-000014 0-000103 0-000551 0-002261 0-007421

(*)

- 0-150416 - 0-217076 - 0-156106 - 0-004326 + 0-141612

/¥

(*)

+ o-oooooi

9 10

+ + + + +

0-285372

+ + + +

/v

—

+ 0-283766 + 0-305116 + 0243253 + + 0-149630 + 0-280630 + 0-294700 + - 0-040059 + 0-162139 + 277030 + - 0-192766 - 0-015412 + 0-171849 + - 0-222722 - 0-171205 + 0-005862 +

+ 0-000007

7 8

W

0-001069 0-004763 0-015904 0-041882 0-089759

5

0-000063 0-000402 0-001846 0-006568 0-018837

7V

+ + + + +

/y M

X

w

+ 0-000005 + o-oooooi + 0-000434 + 0-000086 + 0-000015 + 0-000002 + 0-002887 + 0-000727 + 0000165 + 0-000034

+ 0-087406 + 0-035199 + + 0-177161 + 0-088535 + I + 0-275940 + 0-171837 + 9 + 0-330196 + 0263308 + IO + 0-286089 + 0-316850 +

+ 0-133432

(z)

+ o-oooooi + 0-000034

6

ii 12

;v

/j/to

/*<*>

(*)

Jn + ^

743

+ + + +

O-OOOOOI 0-000005 0-000039 0-000212 0-000898

+ + + +

o-oooooi 0-000009 0-000058 O-OOO20O

0-008237 0-021263 0-046907 0-089312 0-147378

+ + + + +

0-003108 0-0090I7 0-022324 0-047774 0-089050

+ + + + +

0-001086

0-210215 0-255927 0-257478 0-196122 0-076893

+ + + + +

0-144957

0003532 0-009760 0-023297

0048533

-

0-II2207

18 19

0-179418 20

X

5

6 7

+ 0-000002 8 + 0-OOOOI5 9 + 0-000082 10

+ + + + +

+ 0-088758 + 0205354 + 0-142701 + 0-2500I6 + 0-200884 + 0-253715 + °* 2 4454 I + 0-198153 + 0-250059 +

0-000355 0-001288

11 12

0003955 13 0-010469 14 0-024193 15 0-049201 16 0-088443 17 o- 140592 18 0-196755 19 0-239451 20


744

Table V. Fresnel's integrals

X o-5 I-O i\5

2-0 2-5 3-o 3'5

4 -o 4'5 5-o

\\]j^{t)dt

*/V*w*

X

+ + + + +

0-550247 0-721706 0-779084 0-753302 0-670986

+ + + + +

0-092366 0-247558 0-415^48 0-5628^9 0-665787

255

+ + + + +

0-56I020 0-452047

+ + + + +

0-711685 0-700180 0-642119 0-556489 0-465942

28-0 28-5 29-0 29*5

0-391834 0-349852 0-347099 0-381195 0-441485

30-5 31-0 3i-5

33-o

O368193 0-325249 0-328457

26-0 26-5 27-0 27-5

300

*/V 4

(/)<*<

+ + + +

+ + + + +

0-425797 0-448300 0-482927 0-521054 0-553369

+ + + + +

0-521695 0-484566 0-451832 0-431358 0-427908

+ + + + +

0-572142 0-573060 0-556212 0-525995 0-489969

+ + + + +

0-442034 0-470019 0-504844 o-537944 0-561307

+ + + + +

0-456974 0-434973 0-429129 0-440605 0-466343

+ + + + +

0-569407 0-560508 0-537026 0-504881 0-472012

+ + + + +

0-55749© 0-567709 0-561313

+ + + + +

0-446415 0-434212 0-438182 0-457140 0-486272

+ + + + +

0-540094 0-509417 0-476871 0-450396 0-436345

+ + + + +

0-518359 0-545560 0-561321 0-561957 0-547503

+ + + + +

0437971

+ 0-521665 + 0-490870 + 0-462670

0-558799 0-561608

+ 0-443897 + 0-439006

+ + + + +

+ + + + +

+ + + + +

0-466829 0-446755 0-439878 0-427720 0-468209

+ + + + +

0-496215

+ + + + +

0-537309 0-511657

6-o 6-5 7-o 7-5

0-372439 0-443274 0-522202

O59OI 16 0-631845

+ + + + +

8-o 8-5 9-o 9-5 io-o

+ + + + +

0-63930I 0-6I2868 0-560804 0-496895 0-436964

+ + + + +

0-512010 o-575457 0-617214 0-628573 0-608436

io- 5

0-395087

12-0 12-5

+ + + + +

+ + + + +

0-563176 0-504784 0-447809 0-405810 0-388217

I30

+ 0-542511

I3'5 14-0 14-5

+ 0-584583 + 0-604721 + 0-598871 + 0-569335

+ + + + +

0-398268 0-432489 0-481770 0-533736 0-575803

+ + + + +

0-524009 °-4743io 0-432343 0-407985 0-406589

+ + + + +

0-598183 0-596126 0-570890 0-529259 0-481750

405

18-0 18-5 19-0 19*5 20-0

+ 0-427837

+ + + + +

0-439989 0-413893 0-409336 0-426853 0-461646

43-o 43'5 44-0 44'5

45o

20-5 2 I-O

+ 0-587849 + 0573842

21-5 22-0 22-5

+ 0-542266 + 0-501167 + 0-460707

+ + + + +

0-504875 0-545885 0-574811 0-584939 0-574246

45'5 46-0 46-5 47-0 47-5

+ 0-559004 + o-5533oi

23O

+ + + + +

+ + + + +

0'545782 0-506824 0-467029 0-436051 0-421217

48-0 48-5 49-o 49-5 50-0

+ 0-456160 + 0-443930 + 0-445486

ii-o 1 1 -5

I50 15-5 16-0

165 17-0 17-5

23-5 24-0 24-5

250

O38039O 0-395149 °-434557 0-488146

+ 0-465971 + 0-511332 + ° 552774 + 0-580389 -

0-430662 0-418080 0-425635 0-451078 0-487880

320 32-5

335 34-o 34-5 35-o

355 36-0 365^ 37-o

375 38-0 38-5 39-o

395 40-0

41-0 4i'5 42-0 42*5

w*

0-526896 0-558628 0-575524 0-573766 + 0-554127

+ + + + +

5S

ij'ji

0-449025 0-471341 0-500382 0-529002

0550239

+ 0-534676 + 0-507802 + 0-479313

0-400311 + 0-484658

.+

0-499873

0532930

0-454670 0-482187

0513690 0-541464

0549384 0525282 0-495309

0524837 0-547099

0557650 0554044

0483428 0459523 0-445722


745

Table V. Fresnel's integrals

*/>-i«>

dt

»/>i<"

dt

Maxima and minima 0-02 0-04 o-oo 0-08 o-io

+ + + + +

0-1128334 0-1595514 0-1953707 0-2255314 0-2520611

+ + + + +

0-12 0-14 O-IO 0-18 0-20

+ + + + +

0-2759976 0-2979565 0-3183378 °'3374 I 86 0-3554002

+ 0-0110444 + 0-0139124 + 0-0169904 0-0202639 + 0-0237204

7-853982 IO-995574 14-137167

0-22 0-24 O-20 0-28 0-30

+ + + + +

o-3724338 0-3886365 0*4041012 0-4189028 0-4331026

+ + + + +

0-0273496 0-0311421 0-0350898 0-0391853 0-0434218

17*278760 20*420352 23-56I945

0-32 o-34 0-36 0-38 0-40

+ + + + +

0-4467517 0-4598932 0-4847941 0-4966121

+ + + + +

0-0477932 0-0522937 0-0569181 0-0616612 0-0665185

0-42 0-44 0-46

048 050

+ + + + +

0-5080410 0-5191018 0-5298125 0-5401895 0-5502472

+ 0-0817309

0-52 o-54 0-56 0-58 o-6o

+ + + + +

0-5599985 0-5694551 0-5786275 0-5875253 0-5961571

+ + + +

0-62 0-64 o-66 o-68 0-70

+ + + + +

06045308 0-6126537 0-6205324 0-6281731 0-6355815

+ + + + +

0-1263176 0-1322398 o- 1382290 0-1442820

0-72 0-74 0-76

+ 0*6427627 + 0-6497217 + 0-6564631 + 0-6629910 + 0-6693095

+ + + + +

0-1565683 0-1627958 0-1690757 o -1 754054 0-1817820

0-82 0-84 o-86 o-88 0-90

+ + + + +

0-6754224 0-6813330 0-6870448 0-6925609 0-6978843

+ + + + +

0-1882030 0-1946656 0-2011673 0-2077055 0-2142775

0-92 o-94 0-96 0-98 I-OO

+ + + + +

0-7030179

+ + + + +

0-2208809 0-2275131 0-2341717 0*2408543 0*2475583

04725635

07079643 0*7127261

07173059 0*7217059

0-0007522 0-0021274 0-0039078 0-0060153 0-0084044

4-

+ 0-0714853 + 0*0765575

of Fresnel's integrals

#=(»-£)»-

//-i«><"

1-570796

+ 0-779893

4712389

+ + + +

0-321056 0-640807 0-380389 0-605721

29-845130

+ + + + +

0-404260 0*588128 0-417922 0-577121 0-427036

32-986723 36-128316 39-269908 42-411501 45 553093

+ + + + +

48-694686

+ 0-442848

x = n*

*/>!«><•<

26703538

0-569413

0433666 0-563631 0-438767 0-559088

+ 0-0870016 + 0*0923658

4-

0-0978198 0-1033602 0*1089835 0-1146863 0-1204654

OT503961

3-I4I593 6-283185 9-424778 12-566371 15*707963

18-849556 21*991149 25-132741

.

+ + + + +

0-713972 0-3434I5

O62894O 0-387969 0-60036l

+ 0-408301 4-

0-584942

31*415927

+ 0-420516 + o-574957 + 0-428877

34-557519 37*699112 40-840704 43*982297 47-123890

+ + + + +

50*265482

+ 0-443747

28274334

0-567822 0-435059 0-562398 0-439868

0558096

1

'

'


746

Table VI. Functions of equal order and argument

»* /»

Jn(n)

0-4400506 0-3528340 0-3090627 0-2811291 0-2611405

0-4400506 0-444543° 0-4457456 0-4462646 0-4465441

n

/»' (n)

(«)

0-3251471 0-2238908 0-1770285

Jn' (n)

0-3251471

03554045

0-149042,1

0-3682342 0-3755633

0-1300918

03803908 0-3838497 0-3864704 0-3885364 0-3902143 0-3916089

0-2234550 0-2148806 0-2074861

0-4467152 0-4468293 0-4469100 0-4469696 0-4470153

0-1162502 0-1056130 0-0971341 0-0901865 0-0843696

0-2010140 0-1952802 0-1901489 0-1855174 0-1813063

0-4470512 0-4470800 0-4471036 0-4471233 0-4471399

0-0794142 0-0751323 0-0713880 0-0680806

0-3927897 0-3938047 0-3946882

00651336

0-3961557

0-1774532 0-1739079 0-1706299 0-1675857 0-1647478

°-447 I 540

0-0624879 0-0600969 0-0579234 0-0559374

0-3967734 0*3973300 0-3978347 0-3982948

16

00541 141

03987163

20

0-1620927 0-1596009 °' I 572555 0-1550422 o- 1529484

0-4472015 0-4472080 0-4472138 0-4472191 0-4472239

0-0524332 0-0508777 0-0494332 0-0480874 0-0468301

0-3991041

03994624

21 22

0-3997946 0-4001035 0-4003917

23 24 25

0-1509633 0-1490774 0-1472823 0-1455706 °-i439359

0-4472282 0-4472321 o-4472358 0-4472391 0-4472422

0-0456522 0-0445460 0-0435048 0-0425226

0-4006614 0-4009143 0-4011521 0-4013762 0-4015877

26 27 28 29 30

0-4472450 0-4472476 0-4472500 0-4472523 o-4472544

0-0407151 0-0398812 0-0390889 0-0383350

0-4017879

3i

04019775

34 35

0-1423721 0-1408742 Q-I394373 0-1380567 0-1367305

32 33 34 35

36 3Z 38 39 40

o-i35453i 0-1342222 0-1330349 0-1318885 0-1307805

0-4472564 0-4472583 0-4472600 0-4472616 0-4472632

0-0369309 0-0362758 0-0356491 0-0350489 0-0344734

0-4026472 0-4027956

36

04029374

38 39 40

41

42 43 44 45

0-1297089 0-1286716 0-1276667 0-1266925 0-1257473

0-4472646 0-4472660 0-4472673 0-4472685 0-4472697

0-0339210 0-0333904 0-0328800 0-0323888

0-4033281 0-4034479

46

0-1248297

42 48 49 50

0-1230719 0-1222291 0-1214090

0-4472708 0-4472718 0-4472728 0-4472738 0-4472747

0-0314594 0-0310192 0-0305941 0-0301833 0-0297861

0-2458369

02335836 9 10 ii 12

13 14 15

16 II 19

20 21

22 23 24 25

26

3 29 30 31 32 33

01239383

0-4471662 0-4471768 0-4471861 0-4471943

03954655

00415942

0-4021576 0-4023288 0-4024918

00376165

0-4030732

04032033

0403563 0-4036738 0-4037805

00319156

04038833 0-4039824 0-4040781 0-4041705 0-4042599

9 10 11

12 13 14 15

\l 19

4i

42 43 44 45

46 47 48 49 50

For values of n exceeding 50, the following approximations may be used with seven-figure accuracy: ~\ 1 000586 92885 r 1213 -| 0-44730 73184 F

J,,' (w)

J

225» 2J

J

23 1 3i5onaJ

L 0-41085 01939 r L

"'

»*

"

L

0-08946 14637

Hi

r

i4625n 2 J 947

"I

L "69300«aJ


747

Table VI. Functions of equal order and argument

-V-W

n i

07812128

2 3 4 5

0-6174081 0-5385416 0-4889368 '453&948

6

0-4268259 0-4053710 0-3876699

Y»'

(n)

(»)

*f

YH

'

n

(«)

0-7812128 0-7778855 0-7767114 0-7761387 0-7758072

02615525

0-8694698 0-8101709 0-7865654 o-7733765 0-7647843

0-775594 1 0-7754409 o-7753399 o-775259o 0-7751961

0-2297650 0-2060642 o- 1 876060 0-1727588 0-1605149

0-7586672 0-7540520 0-7504241 0-7474840 0-7450441

0-7751458 0-7751049 0-7750711 0-7750426 0-7750184

01502159 0-1414121 0-1337852 0-1271029 0-1211915

0-7429809 0-7412092

0-7749976 0-7749796 o-7749638 o-7749499 0-7749374

0-1159184 0-1111803 0-1068955 0-1029987 0-0994367

02650095

0-7749266 0-7749168 0-7749079 o-7748999 0-7748925

0-0961658 0-0931499 0-0903586 0-0877663 0-0853514

0-2615652 0-2582933 o-255i79i 0-2522100 0-2493744

0-7748859 0-7748798 o-7748742 0-7748690 0-7748642

0-0830953 0-0809819 0-0789973 0-0771295 0-0753678

0-2466622 0-2440643

0-0737029 0-0721267 0-0706318 0-0692116 0-0678605

0-7273206

31

0-2391794 0-2368784

0-7748598 o-7748557 0-7748519 0-7748483 0-7748450

07269914 07266790 07263820 07260991

32 33 34 35

0-7748419 0-7748389 0-7748362 o-7748336 0-7748312

36

39 40

0-2346635 0-2325292 0-2304705 0-2284828 0-2265620

4i 42 43 44 45

0-2247042 0-2229059 0-2211637 0-2194748 0-2178364

0-7748289 0-7748267 0-7748246 0-7748227 0-7748208

46

0-2162458 0-2147007 0-2131988

0-7748191 o-7748i74 0-7748158 0-7748143 0-7748128

I 9

03727057 03598 142

IO II

°*34 8 5399

12 13

03385583

M 15

0-3296303 0'32i5755 0-3142546

16

0-3075580

03013982 !S 19

0-2957040 0-2904173 0-2854894

20

0-2808800 0-2765546 0-2724839 0-2686456

21

22 23 24 25

26

3 29 30 31

32 33 34 35

36

U

» 49 50 For values of accuracy.

02415724

02117381 0-2103166

n exceeding

^

YM '(n). The

-n*Yn

0-8694698 0-5103757 0-3781412 0-3069147

coefficients are numerically equal to

1

I 9 10 II 12 I3

!4 15

0-7360358 0-7350670

16

07341890

11

0-7333887 0-7326559

20

19

0-7319817

21

07313591 07307820

07297446

22 23 24 25

0-7292763

26

07288371 07284242 07280352 07276680

11 29 30

0-7302453

0-7258295

07255720

0-0641718 0-0630496 0-0619751

0-7253259 0-7250904

$ 39

07248647

40

0-0609450 0-0599565 0-0590071 0-0580942

0-7246483 0-7244405

41 42 43 44 45

0-0563695 0-0555539 0-0547671 0-0540074

00532735

225«*J

5

6

0-0665732

00572157

~[

0-7371112

4

00653451

50, the following approximations

077475 90021 [~ L ni 071 161 34100 r

07396683 07383135

1

2 3

07242407 07240486 07238636 0-7236853 o-7235i34

07233475 0-7231873 0-7230324

may

H 49 50

be used with seven-figure

001016 59059 f_ + ,

_*

46

L

1213

"1

l4625^»_r

+

23

V3

times the corresponding coefficients in

0-15495 18004 [

rt

947 "| 6930on8J

Jn

(n)

and


748

Table VII. Zeros, jQt

n

Jo,n

n

,y

,

n

,

h,

«

,

yo,n

Vi, n,

of

J

0*0,

JUH

Y

(x),

J1

(x),

yi,n

Y x (x) n

133610975

3-83I7060 7-OI55867 IO-I73468I I3-32369I9 I6-470630I

2-1971413 5-4296811 8-5960059 11-7491548 14-8974421

30-6346065

16-5009224 19-6413097 22-7820280 25-9229577 29-0640303

I9-6I58S85 22-7600844 25-903672I 29-0468285 32-I896799

18-0434023 21-1880689 24-3319426 27-4752950 30-6182865

33-7758202 36-9170984 40-0584258 43-I9979I7 46-3411884

32-2052041 35-3464523 38-4877567 41-6291045 44-7704866

35-3323076 38-4747662 4I-6I70942

33-7610178 36-9035553 40-0459446 43-1882181 46-3303993

49-4826099 52-6240518 55-7655 Io8 58-9069839 62-0484692

47-9118963 51-0533286 54-1947794 57-3362457 60-4777252

5I-0435352 54-I855536

60-4694578 63-6II3567

55-7565449 58-8984962 62-0404111

\l 19 20

65-1899648 68-3314693 71-4729816 74-6145006 77-7560256

63-6192158 66-7607160 69-9022246 73-0437403 76-1852624

66-7532267 69-89507I8 73 0368952 76-I786996 79-3204872

65-1822951 68-3241522 71-4659861 74-6077996 77-7495953

21 22

11 29 3°

80-8975559 84-0390908 87-1806298 90-3221726 93-4637188

79-3267901 82-4683228 85-6098598 80-7514008 91-8929453

82-4622599 85-6040I94 88-745767I 9I-8875042 95-02923I8

80-8913753 84-0331412 87-1748947 90-3166370

3?

96-6052680

32 33 34 35

997468199 102-8883743 106-0299309 109-1714896

95-0344930 98-1760436 101-3175968 104-4591523 107-6007100

98-I709507 IOI-3I266l8 I04-4543658 I07-5960633 IIO-7377548

96-6000923 99-7418072 102-8835147 106-0252153 109-1669097

112-3130503 115-4546127 118-5961766 121-7377421 124-8793089

110-7422697 113-8838313 117-0253944 120-1669592 123-3085253

II3-8794408 II7-02II2I9 I20-I627983

112-3085985 115-4503820 118-5919607 121-7336349 124-8753051

2-4048256 5-5200781 8-6537279 II-79I5344 14-9309177

0-8935770 3-9576784 7-0860511 10-2223450

2 8

18-0710640 2I-2II6366 24'35247 I 5

9

27H93479I

IO ii 12

i

2

3 4 5

6

13 J4 15

16 \l 19 20 21

22 23 24 25

26

36

11 39 40

447593I9O 47-90I4609

573275254

I233O447O5 I26-446I387

49-4725057

1

2

3 4 5

6 I 9 10 II 12

13 14 15

16

526145508

934583692

23 24 25 26 11 29 30 3i 32

33 34 35

36

11 39 4°

TABLES OF BESSEL FUNCTIONS Table VII. Zeros, j lt n ,y2 n j3 n ,yz n ,oi ,

n

Ji.n

i

VI356223

2

8-4I7244I 11-6198412

3 4 5

6

ii

12 13 14 15

16 11 19 20 21

22 23 24 25

26 11 29 30

,

,

y»,«

J 2 (x), Yt (x), Jz (x), Ys (x)

Ja n

y*,n

n

17-9598195

3-3842418 6-7938074 10-0234780 13-2099868 16-3789666

6-3801619 9-7610231 13-0152007 16-2234640 19-4094148

4-5270247 8-0975538 1 1 3964667 14-6230726 17-8184543

21-1169971 24-2701123 27-4205736 30-5692045 33-7165195

19-5390400 22-6939559 25-8456137 28-9950804 32-1430023

22-5827295 25-7481667 28-9082508 32-0648524 35-2186707

20-9972845 24-1662357 27-3287998 30-4869896 33-6420494

1 9 10

36-8628565 40-0084467 43-I534538 46-2979967 49-4421641

35-2897939 38-4357335 41-5810149 44-7257771 47-8701227

38-3704724 41-5207197 44-6697431 47-8177857 50-9650299

36-7947910 39-9457672 43-0953675

11 12 13

462438744 49-39I4980

J4 15

52-5860235 55-7296271 58-8730158 62-0162224 65-1592732

51-0141287 54-1578545 57-3013461 60-4446401 63-5877658

54-1116156

525383976

16

572576516 60-4032241 63-5484022 66-6932417

55*6846964 58-8304911 6I-9758587 65-I2086l2

11 19

20

68-3021898 71-4449899 74-5^76882 77-7302971 80*8728269

66-7307471 69-8736034 73-0163509 76- 1 59003 79\30i57 I 3

69-8377884 72-9820804 76-1261492 79-2700214 82-4137195

68-2655491 71-4099642 74-554I409 77-6981084 80-8418910

2r 22 23 24 25

84-0152867 87-1576839 90-3000252 93-4423160 96-5845614

82-4440651 85-5864927 88-7288612 91-8711766 95-OI3444I

85-5572629 88-7006678 91-8439487 94-9871177 98-1301857

83-9855095 87-1289817 90-2723230 93-4I55465 96-5586637

26 27 28 29 30

98-1556685 101-2978536 104-4400031 107-5821201 110-7242073

101-2731621 104-4160552 107-5588722 110-7016197

99-7016848 102-8446186

IO9I302542

1

138443033

II2-272969I

3i 32 33 34 35

113-8662672 117-0083021 120-1503138 123-2923041 126-4342746

116-9869284 120-1294994 123-2720205 126-4144954 129-5569276

115-4156229 118-5582204 I2I-7007659 124-8432635 I27-9857I67

36 3Z 38 39 40

I479595I8

i 9 IO

,

749

31

997267657

32 33 34 35

102-8689327 106-0110655 109-1531673 112-2952406

36

1

11 39 40

12^-8632917 128-0052530

15-4372877 118-5793107 I2I-72I3II<;

105-98^74728

1

2

3

4 5 6


750

Table VII. Zeros, j4

J4.H

,

w

y* n j5 ,

,

,

n,

y*,*

y6, „ of ,

J4 (*), F4 (*),

H,n

877M838

J,

(a;),

F5 (*)

.Vs,»

6-7471838 10-5971767 14-0338041 17-3470864 20-6028990

7-5883427 11-0647095 I4-3725367 17-6159660 20-8269330

5-6451479 9-3616206 12-7301445 15-9996271 19-2244290

24-0190195 27-1990878 30-3710077 33-537I377 36-6990011

22-4248106 25-0102671

39-8576273 43'Oi37377 46-1678535

41-3263833 44-4893191 47-6493998 50-8071652

52-4715514

38-2786681 41-4359606 44-5910182 47-7442881 50-8961052

55-6216509 58-7708357 61-9192462 65-0669953 68-2141749

54-0467255 57-1963482 60-3451302 63-4931972 66-6406512

22 23 24 25

71-3608607 74-507II55 77-6529918 80-7985341 83*9437799

697875753 729340384

26 27 28 29 30

87-0887615 90-2335065 93-3780390 96-5223797 99-6665468

85-5163019 80-6611620 91-8057980 94-9502321 98-0944839

31 32 33

102-8105563 105-9544223 109-0981571 112-2417718 115-3852762

101-2385704 104-3825064 107-5263053 110-6699788 113-8135372

I043393353 107-4843998 IIO-6292667 II3-7739523 II6-9I847I3

102-7667232 105-9118934 109-0568569 112-2016312 115-3462317

31 32

118-5286792 121-6719886 124-8152114 127-9583541 131-1014225

116-9569899 120-1003451 123-2436104 126-3867924

120-0628368 123-2070606 I26-35II534 129-4951246 132-6389830

118-4906725 121-6349657 124-7791228 127-9231536 131-0670674

36

9 10 ii

12 13 14 15

16 17 ie 19

20 21

34 35

36 3Z

38 39 40

493203607

287858937 31-9546867 35-1185295

76-0800980 79-2258022 82-3711919

1295298972

12-3386042 15-7001741 18-9801339 22-2177999

2V43034II 28-6266f83 3I-8lI7l67 3^-98878l3 38-I598686

23-8265360 27-0301349 30-2203357 33-4011056 36-5749725

9 10

397436277

11

42-9082482 46-0696791 49-2285437 52-3853121

12 13 14 15

57-II73028 60-2702451 63-4220540 66-5728919 69-7228912

55-5403458

16

61-8462803 64-9975855 68-1479890

\l

72-8721613 76-0207934 79-1688641

21 22

85-4635703

71-2976113 74-4465520 77-5948946 80-7427095 83-8900562

88-6I03082 91-7566925

901835423

539630266

82316438O

949027585 98-0485369 101-1940546

586939271

87-0369859 93-3297633 96-4756819 99-0213268

19

20

23 24 25 26 27 20 29 30

33 34 35

37.

3« 39 40

TABLES OF BESSEL FUNCTIONS Table VII. Zeros, jv 3 8n ,dn , of [Note. The v'3-

last

,

n

J_ ll3

,

y ly 3 n of

{x)

,

,

+ J vz

two functions are equal

Jv 3

(z),

(x),

J_ 1/3

751

Y v 3 (x) with - «/ 1/3 x ;

(a;)

(

zeros,

)

to

{JWa;)cos30 - ri/3 (a;)sin3o

},

^3.

;J 1/8 (a;)cos 120 -

ri/3 (x)sin

120

respectively.]

J 11*,'

2-9025862 6-0327471 9-1705067 12-3101938

ym 13530196

I 5*45 o6 490

4-4657883 7-6012412 10-7402128 13-8803575

9 10

18-5914863 21-7325412 24-87373I4 28-0150117 31-1563549

17-0210330 20-1619929 23-3031228 26-4443623 29-5856767

11

34'2977437

12 13 14 15

374391666

32-7270444 35-8684514 39-0098884

40-5806158 43-7220857 46-8635719

16 \l 19 20 21 22 23

31

36

\

337741762 36-9I5594I 40-0570394 43-1985061 46-3399899

65-1891127 68-3306564 7I-47220^4 74-6I37562

90-8451519 93-9867191

79-8496829 82-9912426 86-1328048 89-2743691 92-4I59353

97-1282878 100-2698581 103-4114297 106-5530025 109-6945765

95-5575032 98-6990728 101-8406437 104-9822160 108-1237894

112-8361516

111-2653639 114-4069394 117-5485159 120-6900931 123-8316712

"5-9777275 11 39 40

27-4914601

30632794I

64-1419325 67-2834747 70-4250213 73-5665718 76-7081259

877035867

32 33 34 35

24350I925

65-7127030 68-8542475

81-4204625 84-5620234

29 30

18-0679953 2I-20902IO

48-4343202 51-5758256 54-7I734 10 57-8588648 61-0003956

-9957961 75-I373484 78-2789040

26

2-3834466 5-5IOI956 8-6473577 11-7868429 14-9272068

50-0050715 53-1465821 56-2881019 59-4296294 62-5711634

71

24 25

421513485 45-2928269

SH

119-1193044 122-2608821 125-4024605

49-4814874

526229964 55-7645I47 58-9060410 62-0475740

d„

0-8477186

39441020 7-0782997 10-2169407 J3-3569532

16-4975630 19-0384856 22-7795923 25-9208165 29-0621201 32-2034801 35-3448813 38-4863138 41-6277704 44-7692461 47-9107371 51-0522406 54-1937545 57-3352769 60-4768067

9 10 11

12 13 14 15

16

\l 19 20 21 22

77-7553U2

63-6183427 66-7598840 69-9014299 73-0429798 76-1845333

80-8968692 84-0384298 87- 1 799926 90-3215576 93-463I244

79-3260899 82-4676492 85-6092109 88-7507749 91 8923408

26

96-6046929 99-7462629

95-0339085 98-1754777 101-3170485 104-4586205 107-6001938

3i 32 33

110-7417681 II3-8833435 117-0249197 120-1664969 123-3080748

36

IO28878343 106-0294070 109-1709808 112-3125557 II 5454i3i5

118-5957082 121-7372858 124-8788641

23 24 25

II 29 30

34 35

11 39 40

}


752

Table VIII. Integrals of functions of order zero

Maxima and minima X

ij" Jo (*)dt

lf'Y

(t)dt

lj o

0-02 0-04 O'OO 0-08 O-IO

+ + + + +

0-0099997 0-0199973 0-0299910 0-0399787 0-0499583

- 0-0320078 - 0-0551846 - 0-0750205 - 0-0926801 - 0-1087153

0-12 0-14 O-IO 0-18 0-20

+ + + + +

0-0599280

-

0-22 0-24 O-20 0-28

+ 0-1095571 + 0-1194252 + 0-1292695 + 0-1390880 + 0-1488788

- 01833430 - 0-1932224 - 02025397 - 0-2113348

+ + + + +

0-1586399 0-1683694 0-1780654 0-1877260 o- 1973493

- 0-2274894

-h

02069333

- 0-2606471 - 0-2661901

030 032 o-34 0-36 0-38 0-40 0-42 0-44 0-46 0-48

00698858 0-0798295

00897573 0-0996672

050

0-2164763 0-2259764 + 0-2354316 + 0-2448403

0-52 o-54 0-56 0-58 o-6o

+ + + + +

0-2542004

0-62 0*64 o-66 o-68 0-70 0-72 0-74 0-76

4

-r

0-1370979 0-1498103 0-1617001 0-1728544

-

0-2349040

- 0-2419080 - 0-2485215 - 0-2547624

+ + + + +

0-3002II7 0-3092437 0-3182150 0-3271238 0-3359684

-

0-3025960 0-3053111 0-3077877 0-3100319 0-3120498

o-8o

+ + + + +

0-3447472 0-3534586 0-362I0I0 0-3706727 0-379I722

-

0-3138471 0-3154290 0-3168010 0-3179678 0-3189344

0-82 0-84 o-86 o-88 0-90

+ 0-3875979 + 0-3959484 + 0-4042220 + 0-4124174 + 0-4205330

- 03197054 - 0-3202852 - 0-3206781 - 03208884

0-92

+ 0-4285674 + C4365I92 + 0-4443870

- 0-3207771 - 0-3204634 - 0-3199827 - 03193386 - 0-3185347

0-96 0-98 I-OO

+ 0-452I694 + 0-4598652

i/VoO

at

+ °-735 22 °8 + + + +

0-3344230 0-6340842 0-3845594 0-6028269

I8-07I0640 2I-2II6366 24-35247I5 27-493479I 30-6346065

+ + + + +

0-4064156 0-5864441 0-4192836 0-575991 0-4279931

33-7758202 36-9I70984 40-0584258 43-I9979I7 46-34II884

+ 0-5685888 + 04343856

49-4826099

+ O4433085

% = yro,n

!

05629957

+ 0-4393331 + 0-5585784

*/l

y° W

(It

0893577O

- 0-320929I

3-9576784 7-0860511 IO-2223450 I3-36I0975

+ 0-I920I49

16-5009224 19-6413097 22-7820280 25-9229577 29-0640303

+ 0-0978827 - 0-0898033 + 0-0834339 - 0-0782474 + 0-0739188

32-2052041 35-3464523 38-4877567 41-6291045 44-7704866

- 0-0702357 + 0-0670523 - 0-0642652 + 0-0617985 - 0-0595953

47-9118963

+ 0-0576118

- 0-1474447 -;-

0-1237411

-

01 085949

- 0-3209201

16, the integrals

may

be calculated with the help of Table I

X

J (t)dt=lwx\J {x)Ua '(c)+Jx(x)iI (xy( j

H

(t)dt

2-4048256 5-5200781 8-6537279 11-7915344 I4-9309I77

02996358

For values of x between o and from the formulae (cf. § 10-74) i

Jo.

hjj

and

- 0-2714049 - 0-2763037 - 0-2808977

0-2727682 0-28I9722 0-29II206

094

(t)dt

- 0-2196416

0-2851974 0-2892124 0-2929516 0-2964234

078

J

X —

01234500

-

O2635IO3

of

X

J

,

if

Y

(t)dt

= inx{Y (x)U

'(x)+

Y^H,^)}.

—

BIBLIOGRAPHY* ADAMOFF, A. the asymptotic representation of the cylinder functions Jv (2) and «/„' (2) for large values of the modulus of z. Petersburg, Ann. Inst, polyt. 1906, pp. 239—265. [JahrbuZh itber die Fortschritte der Math. 1907, pp. 492—493.] On

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AICHI, K. the Solution of the Modified Bessel's Equation.

AIREY, J. R. Bessel Functions (Dec. 29, 1910). Proc. Phys. Soc.

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The Vibrations of Circular Plates and their Relation to Bessel Functions (Feb. 15, 1911). Proc. Phys. Soc. xxm. (1911), pp. 225—232. The Oscillations of Chains and their Relation to Bessel and Neumann Functions. Phil. Mag. (6) xxi. (1911), pp. 736—742. Tables of Neumann Functions G n (x) and n (x). Phil. Mag. (6) xxu. (1911), pp. 658 663. The Asymptotic expansions of Bessel and other functions. Archiv der Math, und Phys. (3) xx. (1913), pp. 240—244. The Vibrations of Cylinders and Cylindrical Shells. Archiv der Math, und Phys. (3) xx. (1913), pp. 289—294. Tables of the Neumann functions or Bessel functions of the second kind. Archiv der Math, und Phys. (3) xxn. (1914), pp. 30—43. Bessel and Neumann Functions of Equal Order and Argument. Phil. Mag. (6) xxxi. (1916), pp. 520—528. The Roots of Bessel and Neumann Functions of High Order. Phil. Mag. (6) xxxil.

Y

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J

'

A

AIRY, SIR

On

GEORGE

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Trans.

Soc. v. (1835), pp. 283—291. On the Intensity of Light in the neighbourhood of a Caustic (May 2, 1836 March 26, 1838). Trans. Camb. Phil. Soc. vi. (1838), pp. 379—402. On the Diffraction of an Annular Aperture (Dec. 4, 1840). Phil. Mag. (3) xvur. (1841),

Camb. Phil.

;

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pp.

AKIMOFF,

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Transcendantes de Fourier-Bessel a plusieurs variables (July 10, 1916 June 25, 1917 Dec. 24, 1917). Comptes Rendus, clxiii. (1916), pp. 26—29; clxv. (1917), pp. 23—25, 1100—1103. :

*

iiber

;

In the case of a few inaccessible memoirs, references are given to abstracts in the Juhrbuch die Fortschritte der Math, or elsewhere.


754

ALDIS, W.

S.

Tables for the Solution of the Equation

d2 u (June

16, 1898).

On

dy

1

n2 \

/,

Proc. Royal Soc. lxiv. (1899), pp.

203—223.

the numerical computation of the functions G^^x), Gi(x) and

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Proc. Royal Soc. lxvi. (1900), pp.

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(June

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ALEXANDER,

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Expansion of Functions in terms of Linear, Cylindric, Spherical and Allied Functions (Dec. 20, 1886). Trans. Edinburgh Royal Soc. xxxin. (1888), pp. 313—320.

ANDING, Sechsstellige Tafeln der Besselscken

E.

Funktionen imaginaren Arguments (Leipzig, 1911).

ANGER,

T.*

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Function Ikh mit Anwendungen auf das Kepler'sche Problem. Neueste Schriften der Naturforschenden der Ges. in Danzig, v. (1855), pp. 1 29.

Untersuchungen

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—

ANISIMOV, The pp.

1

V. A. Proceedings of

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—33.

de

APPELL, P. E. l'inversion approchee de certaines integrates reelles et sur 1'extension de l'equation Kepler et des fonctions de Bessel (April 6, 1915). Comptes Rendus, clx. (1915),

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Sur

AUTONNE, Sur

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d2 y dx*

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BAEHR, Sur

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I

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47

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(.r

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'

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L.

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—68.

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cos(#cos«)sin2 G>rfo>=0

—358.

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Ableitung einiger Formeln aus der Theorie der Bessel'schen Functionen (June 6, 1891). Astr. Nach. cxxvin. (1891), col. 1 4.

—

BARNES, E. W. the homogeneous linear difference equation of the second order with linear coeffi71. cients. Messenger, xxxiv. (1905), pp. 52 On Functions denned by simple types of Hypergeometric Series (March 12, 1906). Trans. Camb. Phil. Soc. xx. (1908), pp. 253—279. The asymptotic Expansion of Integral Functions denned by generalised Hypergeometric Series (Dec. 3, 1906). Proc. London Math. Soc. (2) v. (1907), pp. 59—116. On

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BASSET, A. B. of finding the potentials of circular discs by means of Bessel's functions (May 10, 1886). Proc. Camb. Phil. Soc. v. (1886), pp. 425—443. On the Potentials of the surfaces formed by the revolution of Limacons and Cardioids about their axes (Oct. 25, 1886). Proc. Camb. Phil. Soc. vi. (1889), pp. 2—19. A Treatise on Hydrodynamics (2 vols.) (Cambridge, 1888). On the Radial Vibrations of a Cylindrical Elastic Shell (Dec. 12, 1889). Proc. London Math. Soc. xxi. (1891), pp. 53—58. On a Class of Definite Integrals connected with Bessel's Functions (Nov. 13, 1893). Proc. Camb. Phil. Soc. vin. (1895), pp. 122—128. On a method

* See also under Bourget and Cauchy.

BIBLIOGRAPHY BATEMAN,

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Certain definite integrals connected with the Legendre and Bessel functions. Messenger xxxiii. (1904), pp. 182—188. generalisation of the Legendre polynomial (Jan. 1, 1905). Proc. London Math. Soc (2) in. (1905), pp. 111—123. The inversion of a definite integral (Nov. 8, 1906). Proc. London Math. Soc. (2) iv. (1906), pp. 461—498. On an expansion of an arbitrary function into a series of Bessel functions. Messenger, xxxvi. (1907), pp. 31—37. The Solution of Linear Differential Equations by means of Definite Integrals (Jan. 25, 1909). Trans. Camb. Phil. Soc. xxi. (1912), pp. 171—196. The History and Present State of the Theory of Integral Equations. British Association Report, 1910, pp. 345—424. Notes on integral equations. Messenger, xli. (1912), pp. 94 101, 180 184. Some equations of mixed differences occurring in the Theory of Probability and the related expansions in series of Bessel's functions. Proc. Int. Congress of Math. i. (Cambridge, 1912), pp. 291—294. Electrical and Optical Wave-motions (Cambridge, 1915).

A

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BAUER, Von den

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Reihen von Kugelfunctionen einer Variablen. Journal fur Math. lvi. (1859), pp. 101—121. Bemerkungen iiber Reihen nach Kugelfunctionen und insbesondere auch iiber Reihen, welche nach Producten oder Quadraten von Kugelfunctionen fortschreiten mit Anwendun<* auf Cylinderfunctionen (July 3, 1875). Miinchener Sitzungsberichte, v. (1875), pp. 247 272° Coefificienten der

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BECKER,

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Die Riccatische Differential-Gleichung. Programm Karlsbad, 1908 (25 der Math. 1908, p. 395.]

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[Jahrbuch


BELTRAMI,

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BERNOULLI, DANIEL. Correspondence with Leibniz 1697—1704. [Published in Leibnizens Oes. Werke, Dritte Folge (Mathematik), in. (Halle, 1855).] Notata in praecedens schediasma 111. Co. Jacobi Riccati, Actorum Eruditorum quae Lipsiae publicantur Supplementa, vm. (1724), pp. 73 75. Solutio problematis Riccatiani propositi in Act. Lips. Suppl. Tom. vm. p. 73. Acta Eruditorum publicata Lipsiae, 1725, pp. 473 475. Exercitaticm.es quaedam mathematicae (Venice, 1 724), pp. 77 80. Theoremata de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae. Comm. Acad. Sci. Imp. Petrop. vi. (1732 33) [1738], pp. 108 122. Demonstrationes Theorematum suorum de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae. Comm. Acad. Sci. Imp. Petrop. vn. (1734 35)

—

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[1740], pp.

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BERNOULLI, JOHN. Methodus generalis construendi omnes aequationes differentiales primi gradus. Acta Eruditorum publicata Lipsiae, 1694, pp. 435—437. \0pera, I. Lausanne aud Geneva, 1742, p. 124.]

BERNOULLI, NICHOLAS

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—

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Ueber die Entwickelung der Functionen zweier Winkel u und u' in Reihen welche nach den Cosinussen und Sinussen der Vielfachen von u und u' fortgehen (June 21, 1821). Berliner Abh. 1820—21 [1822], pp. 56—60. [Abhandlungen, n. (1876), pp. 362—364.]


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BRAJTZEW, J. R. Fourier-Besselschen Funktionen und deren Anwendung zur Auffindung asymptotischer Darstellungen von Integralen der linearen Differentialgleichungen mit rationalen Koeffizienten. Warschau Polyt. Inst. Each. 1902, nos. 1, 2. [Jahrbuch uber die Fortschritte der Math. 1903, pp. 575 577.] Uber

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—

BRASSINNE, Sur diverses equations

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—

Mem. de

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—

BRIDGEMAN, On

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* See also ibid. p. 136. t This paper, which might have been mentioned in § 6*4, contains the formula §6*4 (1) the author does not give the formula § 6-4 (2).

;

but

—

BIBLIOGRAPHY

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—

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UNFERDINGER, F. C a tiber die beiden Integrate fe*n * ? (njc-cosx)dx (April 16, 1868). berichte, lvii. (1868), pp.

611

—

Wiener Sitzungs-

620.

VALEWINK, Over asymptotische ontwikkelingen

G. C. A.

{Dissertation,

Haarlem, pp.

138).

[Jahrbuch

iiber

die Fortschritte der Math. 1905, p. 328.]

VAN VLECK, On

the Roots' of Bessel- and P-functions.

—

pp. 75

E. B.

American Journal of Math. xix. (1897),

85.

,

VERDET,

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I.

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E.

E.

differentieltes ordinaires

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du second

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Annates de la Fac.

6.

VOLTERRA, V. Sopra alcune questioni di inversione di integrali definiti. Ann. di Mat. pp. 139—178. VORONOI,

(2)

xxv. (1897),

G.

Sur une fonction transcendante et ses applications a la sommation de quelques senes. Ann. sci. de VtkoU norm. sup. (3) xxi. (1904), pp. 207—268, 459—534. Sur le developpemeut, a l'aide des fonctions cylindriques, des sorames doubles 2 2 If (pm +2qmn + rn )

oh pmt + Zqmn+rn*

est

une forme

positive a coefficients entiers.

Kongresses in Heidelberg (1904), pp. 241—245.

Verh. des dritten Int.


786

WAGNER,

C.

Beitrage zur Entwicklung der Bessel'schen Function

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—

—

WALKER,

G. T. Some formulae for transforming the origin of reference of Bessel's functions. Messenger, xxv. (1896), pp. 76 80. WALKER, G. W. The scattering of electromagnetic waves by a sphere. Quarterly Jburnal, xxxi. (1900), 49. pp. 36

—

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WALKER,

J.

The Analytical Theory of Light (Cambridge, 1904).

WALLENBURG,

G.

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der willkiirlicheu Constanten cxxxvii. (1903),

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—

On

—

—

WATSON, G. N. a certain difference equation of the second order.

Quarterly Jotirnal, xli. (1910),

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—

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On pp.

(3) iv. (1878), pp.

infinite integrals involving

177—186.

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—

Quarterly Journal, xliii. (1911), pp. 161 177. series of Bessel functions (Dec. 6, 1917). Proc.

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the integration of Riccati's equation (Jan. 27, 1890). Proc. Phys. Math. Section, Naturalist^ Soc. Imp. Univ. Kazan, vm. (1890), pp. 337 342.

—

INDEX OF SYMBOLS [The numbers refer

Aw

An

,

243, 244, 263 6;

,

pages on which

pFq («i, «2,

• • •

584

A n>y (0,283 «#,(«,*); 598 «#,,„(*), 529 «#«,,(*). 571 a,

to the

6

Bn 6 ,

ay

5

Pi

>

P*>

...,/>,;*),

P F

(a;), k

(a),

#,(*), 597

529

«#,,«(*),

ber

bei (#), 81

(#),

ber„

(a;),

bei„ (x), 81

554 C„(*), 658 C/,, (a), 320 Cn "(*), 50 »r<«), 82 (•*[?), 129 c, 508 C„,

^

•

• • >

100

639 646

§A*.P\

J,

308

(*),

3.(*),61 j,

507

jm, 576

1>l'\ £>^)>37i

/«(*), 55 /„(*), 483

j**j,',j.", j,,»,

559

if,

/(*)*, 536

^n CO,

K

232 Gn (*),64; 65 (a, a), 640 0„(a:, o), 648 G, 211

;

G

78 [see also

K„tli (z), 46 iT(20n 288 ,

K„(s), 78

ker

(e),

(*),

81

£„(*), 71 I- r

(*),329 111

I, l r ,

H, 577 fr,» (*),#.»(#), 73

M, 6

H,(*),328

if (a;), 661

herj», hei(*), 81

iW,

;

505

471, 475, 514

her„(s), hei„(.z), 81 ,i,

JV»,

598 £»(*), 74

h\

#,6

J",(5), 10,

,

kei

£» (0,276 £»,„(0 284 5rm ,,(z),303

13

324 661 ,

h(z)> 77

0,197

E

n (z),

56

Jn (z),Q,

Ein (a), 320

./-«(*),

E„(*), 308

^,«, 10

e,

er ,

111

P(0), 258 F(a, 0, 90 P(0, a), 253 P„(
0„(0, 271, 272

16 14,

19

16

J„8 (*),30 ./,(*), 38 */"(/!., a?), 49

Jîp). 129 Jn,k(z )> 326 it), 327 /(*; i/,

p.

65]

(z\ 78 [see also p. 65]

v

N-n m

Dn

485

479

/<*)*, 526

£„(€*), 247

£n;M> , (0,293

the symbols are defined.]

0- n (0,276 <©» (0, 569

P, 547 P„ 227 P„(#),156 m-*(*), 295

P

P„-^(#),-51

P„(cos<£p), 129


790

Pn (r;

a lt

a.,..

,

o»),

420

2U(jr),343

P„(r;a 1 ,a.2 ,...,t 'n|p),421

7/n

P(x,v),205

?;,(*,*),

(a,

b,

c, "J

P{«, fry, s\, 158

k^,7.

(*,a),320

582

rw («,«;; #), 602 rn («, «|i2),606

J

0,(#), 71,340 0L,(jr),343

Q„"(*),156

tfn (w, z),

537

Q„(*),46 Q(x,v), 205

tf„ («/, z),

537

Fn (w, *), Vv (w, z),

537 538

606

294 £_,,,(*), 299 i2 TO> ,.(*),

r,

6

^n «*«* Sn (0,285 Sn (*), 591 A.W, 658 46 Sp,.(s), 347 *^n>

£„(*,#;#), 598 £n (a;|i2), 610 £i(*), 152 &'„ (a), 320

*«,

67

**,*(*).

34 5

^«, 33 Tn (s),340

@», 27; 291

0,112 Ajn,

577 38

26

fi,

;

/*i, A*2, /*3, •••, i>,

38

w 502 ,

HI Tn (s),64 5, *r,

% 514

339

42 552

88ft,

*•(*), 56

«/,

505 60 i/rn (s), 55 ^, 280 fl n (<), 290 v, 51 83 i/r,

^ (*),

Yn (z), 64 F„(s),64

F» (*),

67

F<">(*),

70

Y

(*),

;

59

100 (j», w), 198 01 43

Y»(*),58

(«)«,

Y-»(*), 59 Y„(s), 63 y*. y/, y*",

485

a)*"*

S^(z),S^(z), 263

Sn (s),353

B, 112

49

W„ (z),

>

^(4

k<*),56

*, 19; 112

Q„ 227

i2,

22

*«,

* (s), 204

,20

£(*),204 7,60 7 (i/, a?), 204

//

V„39

n, 1,103

e,6

J

51 '

488

AUTHOES QUOTED

LIST OF [The numbers refer

to the pages.

References are not given to entries in the bibliography,

pp. 753- -788.] Abel, N. H., 68, 616, 621

Borel, E., 8, 281, 536

Adamoff, A., 196

Bourget,

Aichi,

Airey, J.

R,

65, 142, 214, 247, 319, 502, 505,

Airy, Sir George B., 188, 189, 229, 249, 320, 321, 322. 654, 659

W.

S., 65,

Brajtzew, J. R., 169

Brassinne, E., 91

516, 659, 660

Aldis,

324, 325, 326, 484, 485, 517,

J., 6,

659, 660

K„ 60

655, 656, 657, 658, 662, 663,

Brenke, W. C, 23 Bridgeman, P. W., 597

Bromwich, T.

J. 1'a., 8, 11, 44, 68, 156, 187,

189, 191, 202, 203, 214, 230, 231, 234, 279,

664 P.. 579 Anding, E., 313, 657 Anger, C. T., 21, 22, 308, 309, 310, 311, 312 Anisimov* V. A., 92 Appell, P. E., 146, 371 Autonne, L., 94

Alexander,

302, 349, 360, 385, 391, 393, 399, 574, 575,

601 Bruns, H., 327 Bryan, G. H., 127, 480 Burkhardt, H. F. K. I/., 236, 280 Burnside, W. S., 305

C,

44, 149, 386, 395, 415, 437, 455,

Bach, D., 96, 102, 110 Bachmann, P., 197

Cailler,

Baehr, G. F. W., 479 Ball, L. de, 157 Barnes, E. W., 83, 100, 102, 104, 105, 156,

Callandreau, O., 196, 208, 387

536, 537

190, 192, 196, 196, 220, 221, 340, 351, 357,

Basset, A. B., 76, 77, 78, 80, 172, 173, 180,

385, 388, 395, 425, 426, 454 Bateman, H., 130, 131, 367, 370, 372, 373, 376, 379, 380, 389, 406, 417, 437, 456, 530, 533, 535

Bauer, G., 50, 128, 368, 370 Beltrami, E., 51, 358, 361, 374, 386, 389, 390, 391, 500, 579, 621 Bernoulli, Daniel (1700-1782), 2, 3, 4, 9, 85, 86, 87, 88, 111, 123, 478,

576

James (1654-1705),

1, 2, 3,

88, 90,

92 Bernoulli, John (1667-1748), 1, 2, 3 Bernoulli, Nicholas, (1687-1759), 2

177, 366, 395, 499, 500, 507,

S.,

509, 583

Catalan, E.

C,

21, 22, 27, 96, 173, 188

Cauchy, (Baron)

A

L., 7,

15, 16, 21, 150,

183, 230, 231, 232, 233, 247, 249, 259, 309,

319, 324, 449, 545, 554, 557, 579 Cayley, A., 88, 90, 96, 102, 103, 109, 188,

502 H. W., 91 S., 621 Chessin, A. S., 135, 175, 325, 346, 382 Chree, C, 6, 597 Christoffel, E. B., 154 Chrystal, G., 102, 288, 295 Challis,

Chapman,

Cinelli, M.,

633

Clebsch, R. F.

Bessel, F. W.,

Clifford,

13, 14, 15, 18, 19,

637

268, 572

Bernoulli, Nicholas, (1695-1726), 2 1, 3, 4, 5, 9,

P.,

Carlini, F., 6, 7, 194, 225, 226, 227, 249, 255,

Carslaw, H.

367, 383, 387, 402, 409

Bernoulli,

Cantor, G. F. L.

W.

A,

359, 363

K., 90, 91

21, 24, 25, 38, 84, 140, 148, 153, 160, 295,

Coates, C. V., 173, 180, 313, 622

308, 478, 551, 554, 654

Cotter, J. R., 41

27

Binet, J. P. M., 183

Crawford,

Bocher, M., 46, 57, 64, 376, 494, 495, 517 Bohmer, P. E., 142

Crelier, L., 286, 287, 288, 295, 300, 301,

Curtis, A. H., 96, 110

Boole, G., 27, 47, 110,627

Curzon, H. E.

L.,

J.,

395

302


792

D'Alembert, J. le Rond, 3 Dandelin, G. P., 503 Darboux, J. G, 233 Darwin, C. G., 437 Debye, P., 225, 235, 237, 240, 241, 247, 249, 250, 251, 255, 262, 263, 268, 513, 516 De la Vallee Poussin, Ch. J., 53, 160, 189 De Morgan, A., 188, 190 Dendy, A., 107 Dini, U., 10, 577, 578, 597, 600, 616, 651 Dinnik, A., 579, 659, 660 Dirichlet, P. G. Lejeune, 157, 230, 406, 581,

B23 Dixon, A. C, 35, 480, 481, 482 Donkin, W. F., 109 Dougall,

J., 65,

Eamshaw, Ellis,

S.,

P. D. G., 183, 455,

470

38, 49, 59, 68, 227

108

J. B.,

120

503

Graf, J. H., 32, 64, 75, 145, 153, 160, 165, 175, 197, 215, 227, 286, 287, 290, 295, 296, 299, 301, 302, 303, 341, 344, 345, 359, 360, 362, 398, 498, 502, 583

Greenhill, Sir A. George, 91, 96

Gregory, Duncan Farquharson, 391 Gregory, Walter, 224 Grunert, J. A., 27

145, 290

Ermakoff, W., 455 Escherich, G. von, 165 Euler, L.,

662

Gifford, E.,

Gilbert, L. P., 545, 548, 549 Giuliani, G., 155, 156, 324, 326, 327 Glaisher, J. W. L., 89, 96, 102, 103, 108, 109, 140, 171, 173, 183, 664

Gray, A., 64, 65, 78, 194, 206, 454, 480, 655 656, 657, 658, 660 Green, G, 124

248 Encke, J. F., 342 Enestrom, G, 92 Enneper, A., 173 S.,

383,

396,

Genocchi, A, 119 Gibson, G. A., 197

Graeffe, C. H., 502,

F.,

Epstein, S.

365,

430, 438, 439, 480, 508, 517, 522, 524, 525, 579

Goursat, Edouard

R. L., 95, 109, 110, 173

Emde,

274, 283, 284, 290, 293, 351, 362, 363, 366, 367, 368, 369, 370, 373, 378, 379, 384, 385, 386, 389, 390, 391, 393, 395, 398, 406, 407, 413, 414, 415, 418, 426,

Gordan, P. A., 55

411

Du Bois Reymond, Duhamel, J. M. C,

Gegenbauer, L. von, 50, 51, 129, 138, 151,

3, 4, 5, 6, 24,

49, 53, 60, 62, 87, 88,

92, 93, 123, 133, 183, 410, 498, 500, 501,

503, 576, 659

Gubler, E., 32, 64, 145, 160, 165, 177, 197, 215, 227, 286, 287, 301, 329, 341, 351, 398, 408, 410, 426, 498, 502, 583

Gunther, S., 153 Gwyther, R. F., 621, 636

J. H. M., 94 610 Feldblum, M., 92

Falkenhagen, Fejer, L.,

Hadamard,

Fields, J.

Haentzschel, E., 71, 96, 159 Hafen, M., 389

G, 110

Filon, L. N.

Ford,

W.

G,

51, 578, 622, 623, 625,

B., 578,

629

605

Hall,

Forsyth, A. R., 42, 57, 107, 109, 117, 346, 358, 400, 499

Fourier, (Baron) J. B. Joseph, 4,

Hague,

B.,

A,

204, 205, 527

J.,

656

15

Hamilton, Sir William Rowan, 12, 195, 655 Hankel, H, 10, 38, 57, 58, 61, 62, 63, 65, 73,

9, 10, 13,

75, 76, 77, 160, 163, 164, 165, 167, 175,

22, 84, 135, 449, 450, 454, 455, 456, 478,

195, 196, 203, 206, 208, 211,' 384, 386, 390, 393, 395, 424, 427, 428, 429, 430, 434, 453, 454, 456, 457, 458, 459, 462, 464, 465, 471, 488, 513, 514, 577, 579, 581, 582, 633 Hansen, P. A., 14, 20, 30, 31, 37, 152, 154,

482, 483, 501, 576, 577, 578, 616

Freeman, A., 501 Frenet, F., 27 Fresnel, A. J., 544, 545

Frobenius, F. Frullani,

G,

G,

57

14, 19

155, 158, 195, 292, 406, 655,

Hanumanta Rao, Hardy, G. H.,

Gallop, E.

G,

405, 421, 422

Gaskin, T., 109 Gasser, A., 509, 517

Gauss,

C F.

(

Johann Friedrich Carl), .191, 506

656

C. V., 437

8, 111, 180, 183, 188, 189, 200, 309, 320, 321, 322, 324, 373, 382, 386, 395, 406, 421, 422, 437, 441, 442, 463, 464, 542, 546, 547, 573, 575, 579, 606, 615, 621

Hargreave, C.

J., 88,

170, 171

LIST OF

AUTHORS QUOTED Lacroix, S. F., 27

Hargreaves, R., 538 Harnack, A., 577 Hams, R. A., 15

Lagrange,

Lamb,

Lambert,

485 159

J. H.,

Lame\ G,

Heaviside, O., 64, 65, 203, 367, 385, 387, 388,

Landau,

96,

E. G. H., 197

Laplace, P. S. de,

393, 395, 410, 426 4, 56, 65, 66, 84, 154, 155, 156,

551

J. L. de, 6, 27, 28,

H., 56> 96, 385, 416, 475, 502

Havelock, T. H., 125, 171 Hayashi, T., 165

Heine, H. E.,

793

6, 7, 8, 9, 53,

280, 395, 421,

450

Hermite, C, 55, 477

Largeteau, C. L., 501 Laurent, Paul Mathieu Hermann, 157

Hertz, H., 80, 81

Laurent, Pierre Alphonse, 100

Herz, N., 554, 555

Lebedeff,

94 Hobson, E. W., 10,

Lebesgue, Henri, 457 Lebesgue, Victor AmeMee, 110, 123 Lefort, F., 6 Legendre, A. M., 52, 90, 183, 204, 485, 557 l'Hospital, G. F. A. (Marquis de St Mesme),

157, 181, 358, 363, 365

Hill, C. J. D.,

33, 54„58, 125, 128, 129,

149, 172, 174, 280, 353, 363, 369, 385, 386,

387, 480, 485, 578, 586, 591, 602

Hopf,

L., 178,

Horn,

J.,

406

Wera

Myller-, 99

134

225, 526

Hurwitz, A., 9, 297, 302, 303, 304, 305, 306, 307, 483, 484 Ignatowsky, W. von, 365 Isherwood, J. G., 657, 658, 664

Leibniz, G. W.,

Le

Paige,

1, 2,

3

C, 96

Lerch, M., 382, 393, 433, 434, 617 Lindner, P., 484 Lindstedt, A., 545, 661 Liouville, J., 27, 28, 87, 111, 112, 116, 117,

119, 120, 123

Jackson, Dunham, 343 Jackson, Frank Hilton, 43, 44 Jackson, William Hartas, 177

Lipschitz, R. O.

Jacobi, C. G.

Lobatschevsky, N., 503

J., 6, 8, 14,

21, 22, 25, 26, 27,

28, 29, 84, 195, 379, 555,

572

Jamet, E. V., 94 Johnson, W. W., 92 Jolliffe, A. E., 528, 529 Julius, V. A., 65, 200

S., 11, 12,

195, 200, 206, 331,

339, 384, 386, 390, 633

Lobatto, R., 49, 90

Lodge, A., 224, 229

Lommel,

E. C, J. von, 13, 21, 23, 25, 30, 34,

38, 43, 45, 46, 47, 49, 53, 59, 62, 65, 66, 71, 73, 76, 77, 96, 97, 98, 99, 106, 107, 132,

133, 135, 136, 140, 142, 143, 145, 148, 151,

Kalahne, A., 505, 507, 660 Kapteyn, W., 35, 183, 200, 268, 279, 281,

152, 154, 196, 200, 294, 295, 296, 297, 298,

282, 292, 351, 370, 373, 376, 380, 386, 404,

350, 364, 374, 406, 478, 479, 482, 529, 531, 537, 538, 539, 540, 542, 543, 544, 545, 546,

413, 498, 499, 531, 532, 533, 535, 536, 538, 551, 559, 560, 562, 565, 566, 568, 569,

570 Kelvin (Sir William Thomson), Lord, 81, 124, 203, 225, 229, 230, 233, 248, 654,

299, 300, 303, 308, 315, 328, 341, 345, 348,

548, 549, 550, 576, 654, 655, 656, 658, 660,

661, 664

Lorenz,

L., 57, 96, 224, 229,

500

Love, A. E. H., 56, 226, 417, 449, 654, 662

658 Kepinski, Kepler,

S.,

Macdonald, H. M., 78,

99

J., 551,

552

Kirchhoff, G., 107, 196, 203, 389, 578, 616

Kluyver,

J.

C,

367, 419, 420

Kneser, J. C. C. A., 499, 578, 583 Knockenhauer, K. W., 545

Konig, J., 354, 523 Koppe, M., 247

Kummer,

E. E., 49, 90, 101, 102, 104, 105,

148, 185, 190, 191 r 196, 203, 394, 409

79, 80, 158, 170, 171,

225, 229, 233, 365, 377, 385, 386, 389, 395, 396, 412, 413, 439, 440, 482, 509, 511

Maclean, M., 658

McMahon,

J., 64,

195, 200, 505, 507, 581

MacRobert, T. M., 197 Maggi, G. A., 13 Malmsten, C. J., 99, 173, 185, 187, 188, 196, 203 Manfredius, G., 92


794

March, H. W., 56, 225, 449 Marcolongo, R., 135 Marshall, W., 505, 506 Mathews, G. B., 64, 65, 78, 194, 206, 454, 480, 655, 656, 657, 658, 660

Petzval,

J.,

49

Maxwell,

Phragmen,

E.,

Clerk, 125

J.

Pearson, Karl, 98, 99, 419, 421 Peirce, B. 0., 501, 660,

664

P^res, J., 44

Perron, O., 154

358

Mayall, R. H. D., 550

Picard, C. E., 93, 94

Mehler, F. G., 65, 155, 157, 169, 170, 180,

Pincherle,S., 190, 196, 271, 274, 386, 526, 528 Plana, G. A. A., 10, 38, 42, 45, 49, 53, 95, 96,

183, 425, 431, 455, 475, 476

Meissel, D. F. E., 7, 145, 204, 226, 227, 229, 232, 233, 234, 247, 391, 521, 557, 558, 561, 564, 572, 655, 656, 658, 660, 662

Mellin, R. Hj., 190, 196

M.

Mittag-Leffler,

G., 83,

497

99, 195,

554

Plummer, H. C, 270, 552, 555 Pochhammer, L., 100, 101, 297, 346, 410 Pocklington, H. C, 537 Poincare\ J. Henri, 236

Molins, H., 106

Poisson, S. D.,

6, 9, 10, 11, 12, 13, 24, 25, 38,

Moore, C. N., 479, 578, 579, 597, 649 Morton, W. B., 65, 66

47, 49, 52, 67, 68, 69, 73, 95, 96, 160, 173,

Murphy,

501

183, 185, 186, 187, 194, 195, 308, 369, 477,

R., 91, 156, 157

Myller-Lebedeff,

W.

(see

Lebedeff)

Porter,

M.

B., 299, 477, 480, 485, 515,

517

Preece, C. T., 27

Nagaeka, H., 340, 633, 634

Neumann, Carl

Puiseux, V., 559

Gottfried, 16, 19, 22, 23, 30,

31, 32, 33, 34, 36, 37, 46, 59, 60, 65, 66, 67,

Raffy, L., 94

68, 69, 70, 71; 73, 128, 143, 150, 151, 155,

271, 273, 274, 276, 277, 278, 280, 281, 284,

Ramanujan, S., 382, 449 Rawson, R., 91

286, 290, 291, 292, 346, 358, 359, 361, 363,

Rayleigh

365, 386, 418, 424, 440, 441, 453, 455, 456,

(J.

W.

Strutt), Lord, 50, 55, 56, 74,

95, 137, 155, 157, 189, 230, 231, 233, 331,

470, 471, 473, 474, 475, 476, 522, 523, 524,

333, 374, 389, 395, 419, 421, 477, 502, 510,

525

511, 616, 618, 660

Neumann, Friedrich E., 154 Newman, F. W., 663, 664 Newton, Sir Nicholson,

Riccati, (Count) J. F.,

Riemann, G.

Isaac, 120

J.

235, 427, 457, 486, 623, 637, 649

W., 107, 108, 145, 146, 149,

150, 189, 226, 229, 231, 247, 248, 249, 260, 252, 262, 329, 332, 413, 415, 425, 426, 431, 440, 441, 446, 448, 505, 656

Nicolas,

J., 77,

84

455, 465, 522, 523, 525, 526, 527, 528, 571, 572, 574, 597, 622, 629, 636

NiemOller, F., 57, 68, 195

1

73

McF, 145,

Rodrigues, O., 27 ROhrs, J. H., 10 Rudski, P., 477, 508

Rutgers,

375, 376, 380, 579 von, 56, 225, 449

J. G., 373, 374,

Rybczyriski,

W.

Sasaki, S., 507

Savidge, H.

G,

82, 204,

658

Schafheitlin, P., 64, 137, 142, 168, 169, 207, 215, 373, 391, 392, 398, 401, 402, 406, 408, 421, 447, 477, 479, 482, 485, 487, 489, 490,

Olbricht, R., 158, 481

Orr, W.

Riesz, M., 606, 614

Russell, A., 81, 82, 204

Nielsen, N., 24, 44, 49, 64, 73, 74, 77, 82, 83, 132, 142, 145, 148, 149, 154, 169, 224, 297, 298, 299, 315, 350, 355, 357, 359, 392, 405,

Oltramare, G.,

1, 2, 3, 85, 86, 87, 88, 94 F. B., 80, 158, 172, 203, 229,

146, 206, 224, 454,455,

579

Otti, H., 71, 274, 286, 341

491, 492, 493, 494, 508, 510, 543 Scheibner, W., 6 Schlafli, L., 10, 14, 27, 28, 30, 32, 33, 63, 64,

65, 67, 72, 79, 90, 91, 143, 145, 151, 160,

Panton, A. W., 305 Paoli, P., 53, 95,

Parseval, M. A., 359, 384

171, 174, 175, 176, 178, 179, 181, 185, 195,

186 9, 21, 24, 68,

196, 215, 216, 228, 253, 274, 276, 278, 284, 105, 229, 358,

285, 286, 288, 289, 290, 341, 342, 344, 345,

508, 577, 579, 581, 582, 583, 585

LIST OF

AUTHORS QUOTED

Schlomilch, O. X., 14, 18, 33, 34, 35, 36, 153,

173,183,617,618,619,621,622,628,655,666 Schonholzer, J. J., 145 Schott, G. A., 551, 556, 572, 573

W.

Siacci, F.,

92

Siegel, C. L.,

F., 199, 454,

Thomson, Sir Joseph John, 65, 173 Thomson, Sir William (see Kelvin) Tisserand, F., 371

Todhunter,

I.,

27, 157, 199

Turriere, E., 15

Schwarz, K. H. A, 358, 643 Schwarzschild, K., 361 Schwerd, F. M., 477, 654 Searle, J. H. C, 199 Segar, H. W., 483 Serret, J. A, 171, 173, 188 Sharpe, H. J., 105, 157

Sheppard,

795

Unferdinger, F., 310

Valewink, G. C. A., 196

Vandermonde, A., 102 van Vleck, E. B., 480 Verdet, E., 477

579, 595, 615

Vessiot, E., 94

Volterra, V., 579, 621

485

Siemon, P., 328, 398 Smith, Bernard A., 655 Smith, Clara E., 621 Smith, Otto Andreas, 50 Sommerfeld, A. J. W., 56, 57, 178, 361, 389, 395, 406, 417, 464, 499 Sonine, N. J., 82, 83, 132, 137, 139, 143, 169,

Voronoi, G., 200 Voss, A., 406

Wagner, C,

290, 353, 354, 362, 363, 367, 373, 374, 375,

13, 142 Walker, Gilbert Thomas, 360, 361 Walker, James, 328, 331, 333, 537, 544 Wallenburg, G., 94 Waring, E, 503 Watson, G. N., 11, 105, 125, 158, 226, 231,

376, 377, 378, 383, 386, 391, 394, 395, 398,

249, 268, 355, 444, 483, 485, 513, 519, 566,

170, 171, 175, 176, 177, 180, 279, 280, 281,

401, 411, 415, 417, 418, 431, 432, 433, 434,

575

Webb, H. A,

439, 454

351, 523, 533, 536

Weber, Heinrich, 63, 64,

Spitzer, S., 68, 71, 153

67, 75, 165, 167, 195,

Stearn, H., 482, 621

196, 210, 211, 212, 386, 391, 392, 393, 394,

Steiner, L., 655, 661

395, 396, 398, 402, 405, 406, 408, 421, 450,

Steinthal, A. E., 171, 387

Stephenson, A., 579 Stern,

M.

A.,

500

Stieltjes, T. J., 195, 196, 207, 208, 209, 213,

214, 464 Stirling,

James,

7, 8,

214

Stokes, Sir George Gabriel, 8, 12, 53, 55, 68, 69, 70, 80, 95, 97, 188, 189, 195, 201, 202,

225, 229, 238, 320, 324, 336, 391, 405, 503, 505, 507, 605, 659 Strutt, J.

W.

(see

Whittaker, E.

Rayleigh)

Struve, H., 328, 329, 333, 337, 392, 396, 397, 417, 661

Sturm,

J. C. F., 304, 477, 479, 517, 518,

Suchar, P. J., 90 Svanberg, A. F., 173

451, 452, 453, 454, 455, 468, 469, 470, 495 Weber, Heinrich Friedrich, 308, 309, 310,311, 312, 315, 320 Weierstrass, C. T. W., 358 Wendt, Cacilie, 363 Weyl, H., 189, 454 Weyr, E, 93, 94 Whewell, W., 479 Whipple, F. J. W., 177, 313, 387 Whitehead, C. S., 81, 82, 132, 148, 203

521

T., 44, 50, 124, 125, 173, 197,

339, 503, 632, 633

Wigert, C. S., 200 Williamson, B., 110 Willson, R. W., 501, 660, 664

Wilton,

J. R.,

154

Wirtinger, W., 189, 190 Takeuchi, T., 313 Tannery, J., 11, 156, 302 Theisinger, L., 184, 185, 338

Young, W. H.,

10, 351, 406, 578, 579, 580,

582, 586, 596, 616, 617

GENERAL INDEX [The numbers refer

to the pages.]

Addition theorems, 358-372 (Chapter xi) for Bessel coefficients of order zero, 128, 359; for Bessel coefficients of order n, 29; for Bessel functions of the first kind (Gegenbauer's type), 362, 367; for Bessel functions of the first kind (Graf's type), 130, 143, 359 ; for Bessel functions or cylinder functions of any kind (Gegenbauer's type), 363 for Bessel functions or cylinder functions of any kind (Graf's type), 143, 361 for hemi-cylindrical functions, 354; for Lommel's functions of two variables, 543; for Schlafli's function fn (z), 344; for Schlafli's polynomial, 289; integrals derived from, 367 ; physical significance of, 128, 130, 361, 363, 366; special and degenerate forms of, 366, 368 Airy's integral, 188 expressed in terms of Bessel functions of order one-third, 192 generalised by Hardy, 320; Hardy's expressions for the generalised integral in terms of the functions of Bessel, Anger and Weber, 321 references to tables of, 659 Analytic theory of numbers associated with asymptotic expansions of Bessel functions, 200 Anger's function 3 v {z), 308; connexion with Weber's function, 310; differential equation satisfied by, 312; integrals expressed in terms of, 312; recurrence formulae for, 311; representation of Airy's integral (generalised) by, 321 with large argument, asymptotic expansion of, 313 with large argument and order, asymptotic expansion of, 316 Approximations to Bessel coefficients of order zero with large argument, 10, 12 to Bessel functions of large order (Carlini), 6, 7 (extensions due to Meissel), 226, 227, 232, 247, 521 (in transitional regions), 248; to functions of large numbers (Darboux), 233; (Laplace), 421 to Legendre functions of large degree, 65, 155, 157, 158 to remainders in asymptotic expansions, 213; to the sum of a series of positive terms, 8. See also Asymptotic expansions, Method of stationary phase and Method of steepest descents Arbitrary functions, expansions of, see Neumann series and Kapteyn Beries (for complex variables); Dini series, Fourier-Bessel series, Neumann series and Schlbmilch series (for real variables) Argument of a Bessel function defined, 40 Asymptotic expansions, approximations to remainders in, 213; conversion into convergent series, 204; for Bessel coefficients of order zero with large argument, 10, 12, 194; for Bessel functions of arbitrary order with large argument, 194-224 (Chapter vn) (functions of the first and second kinds), 199 (functions of the third kind), 196; (functions of the third kind by Barnes' methods), 220; (functions of the third kind by Schlafli's methods), 215; (functions with imaginary argument), 202; for Bessel functions with order and argument both large, 225-270 (Chapter vni); (order greater than argument), 241 (order less than argument), 244 (order nearly equal to argument), 245 (order not nearly equal to argument, both being complex), 262; for combinations of squares and products of Bessel functions of large argument, 221, 448; for Fresnel's integrals, 545 for functions of Anger and Weber (of arbitrary order with large argument), 313 (with order and argument both large), 316; for Lommel's functions, 351; for Lommel's functions of two variables, 549; for Struve's function (of arbitrary order with large argument), 332; (with order and argument both large), 333 for Thomson's functions, ber (z) and bei (z), 203; for Whittaker's function, 340; magnitude of remainders in, 206, 211, 213, 236, 314, 332, 352, 449; sign of remainders in, 206, 207, 209, 215, 315, 333, 449. See also Approximations ;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

Basic numbers applied to Bessel functions, 43 Bateman's type of definite integral, 379, 382 Bei (z), Ber (z). See Thomson's functions

Bernoullian polynomials associated with Poisson's integral, 49 Bernoulli's (Daniel) solution of Biccati's equation, 85, 89, 123 Bessel coefficient of order zero, J (z), 3, 4; differential equation satisfied by, 4, 5 (general solution of), 5, 12, 59, 60; expressed as limit of a Legendre function, 65, 155, 157; oscillations of a uniform heavy chain and, 3, 4 Parseval's integral representing, 9 with large argument, asymptotic expansion of, 10, 12, 194; zeros of, 4, 5. See also Bessel coefficients, Bessel functions and Bessel's differential equation Bessel coefficients J„ (z), 5, 6, 13, 14-37 (Chapter n); addition theorem for, 29; Bessel's integral for, 19; expansion in power series of, 15; generating function of, 14, 22, 23; inequalities satisfied by, 16, 31, 268; notations for, 13, 14; order of, 14; (negative), 16; recurrence formulae for, 17; square of, 32; tables of (of orders and 1), 662, 666-697; (of order »i), 664, 730-732; (with equal order and argument), 664, 746; tables of (references to), 654, 655, 656, 658. See also Bessel coefficient of order zero, Bessel'B differential equation and Bessel functions Bessel functions, 38-84 (Chapter m) argument of, defined, 40 differential equations of order higher than the second satisfied by, 106 expressed as limits of Lame functions, 159 expressed as limits of P-functions, 158 history of, 1-13 (Chapter i) (compiled by Maggi and by Wagner), 13 indefinite integrals containing, 132-138 ; order of, defined, 38, 58, 63, 67, 70 rank of, defined, 129 relations between the various kinds of, 74 representation of cylinder functions in terms of, 82; solutions of difference equations in terms of, 83, 355; solutions of Laplace's ;

;

;

;

;

;

;

;

;

;

;

;

;

;;;

GENERAL INDEX

797

threeequation containing, 83, 124 ; solutions of the equation of wave motions containing, 125 ; and ten term relations connecting, 300 ; with negative argument, 75. See also the two preceding following entries, and Cylinder functions 367, 368 Bessel functions of the first kind, JV (*), 38 ; addition theorems for, 143, 359, 362, 363, cut in Barnes' type of integral representing, 190 ; Bessel's type of integral representing, 176 ; expansion of, plane to render uniform, 45 differential equation (Bessel's) satisfied by, 38 expressed ascending series, 40 ; expansion of, in descending series, see Asymptotic expansions limit of a hypergeometnc as a generalised hypergeometric function, 100, 101 ; expressed as the significance of), 155; function, 154 ; expressed as the limit of a Legendre function, 156; (physical generalisations expressed as the limit of a Lommel polynomial, 302 functional properties of, 45 infinite integrals containing, of, 43, 44, 308-357 ; inequalities satisfied by, 49, 255, 259, 270, 406 ; Chapter xm, passim ; of complex order, 46 of order n + *, 41 (expressed in finite terms), 52,

m

;

;

;

;

;

;

;

(notations for), 55, 80 ; Poisson's integral representing, 47, 48 ; (modifications of), 161, 163, of two, expressed as a 164, 169, 170 ; products of, see Products of Bessel functions ; quotient Lommel s continued fraction, 153, 154, 303 ; recurrence formulae for, 45, 294 ; relations with polynomial, 297; represented by integrals containing Legendre functions, 173, 174; symbolic formulae for, 170; tables of (of orders and 1), 662, 666-697 ; (of order n), 664, 730-732 ; (of order n+4), 664, 740-745; (of order $), 664, 714-729; (of order - *, method of computing), 664; to), (with equal order and argument), 664, 746 ; (zeros of), 664, 748-751 ; tables of (references with large argument, see 654, 655, 656, 658, 659, 660 ; Weierstrassian product representing, 497 Asymptotic expansions ; zeros of, see Zeros of Bessel functions

55

;

;

Bessel functions of the second (after Neumann), 67; for, 144, 361, 365, 368

Yv (z)

G

(z) (after Hankel), 57, 63; n (z) (after Heine), 65 ; F<»> (after Weber-Schlafli, the canonical form), 63; addition theorems

Hud, Y„ (z)

Bessel's type of integral representing, 177 component parts of, 71, 72, 340; continuity of (qua function of their order), 63; differential equation (Bessel's) satisfied by, descending series, 59, 63 expansion of, in ascending series, 59, 60, 61, 69, 72 expansion of, in the first kind, 5, see Asymptotic expansions ; expressed as an integral containing functions of 133, 382, 433 infinite integrals containing, 385, 387, 393, 394, 424, 425, 426, 428, 429, 430, 433; of), 169, 170; products of, Poisson's type of integral representing, 68, 73, 165; (modifications 149; (represented by infinite integrals), 221, 441,' 446 ; (asymptotic expansions of), 221, 448; recurrence formulae for, 66, 71 represented by integrals containing Legendre functions, 174 symbolic formulae for, 170 tables of (of orders and 1), 662, 666-697 (of order n), 664, 732735 ; (of order J), 664, 714-729; (of order -J, method of computing), 664; (with equal order and argument), 664, 747 (zeros of), 748-751 tables of, references to, 655, 656, 658 ; with large zeros of, set. Zeros of argument, see Asymptotic expansions ; with negative argument, 75 Bessel functions. See also Neumann's polynomial w (z), v ™ (z), 73; Barnes' integrals representing, 192 v Bessel functions of the third kind, Bessel's type of integral representing, 178 ; Poisson's type of integral representing, 166 (modifications of), 168, 169, 170; represented by integrals containing Legendre functions, 174; symbolic formulae for, 170 ; tables of (of orders O'and 1), 662, 666-697 ; (of order £), 664, 714729 ; tables of (references to), 657 with large argument, asymptotic expansions of, 199, 210, 215 with large argument and order, asymptotic expansions of, 244, 245, 262 ; with negative ;

;

;

;

;

;

;

;

;

;

;

H

H

;

;

;

argument, 75 Bessel functions whose order and argument are equal, approximations to, 229, 231, 232, 259, 260, 448, 515 ; asymptotic expansions of, 245 ; integrals representing, 258 ; tables of, 658, 664, 746,

747

;

tables of (references to),

658

fraction. Of orders ±J (and Airy's integral), 190; (and the tables of, 664, 714-729 ; tables of (references to), 659 ; zeros of, orders ±|, tables of (references to), 659. Of orders ± J, ±j . tables of (references to), small fractional orders, tables of zeros of (references to), 502, 660. See also Bessel

Bessel functions stability of

whose order

a

is

a vertical pole), 96

;

751. Of 659. Of funetions

whose order is ± (n + £) Bessel functions whose order is large, 225-270 (Chapter vm) asymptotic expansions of, 241, 244, 245, 262 Carlini's approximation to, 6, 7 ; (extended by Meissel), 226, 227 ; Horn's (elementary) approximation to, 225 ; Laplace's approximation to, 7, 8, 9 ; method of stationary phase applied to, 232 ; method of steepest descents applied to, 237 ; miscellaneous properties of, 252-261 tables of (reference to), 658 ; transitional formulae for, 248 ; zeros of, 513, 516, 517, 518. See ;

;

also Bessel functions whose order and argument are equal Bessel functions whose order is =(n t J), 10, 52, 80 ; expressible in finite terms, 52 ; notations for, 55, 80 ; tables of, 664, 740-745 ; tables of (references to), 658, 659 Bessel functions with Imaginary argument, I„{z), K (z), K v (z), 77, 78; differential equation integrals representing (of Bessel's type), 181 ; (of Poisson's* type), 79, 171, 172; satisfied by, 77 (proof of equivalence of various types), 185-188 monotonic property of, 446 ; of order ± (», + 1), 80 ; recurrence formulae, 79 tables of (of orders and 1), 663, 698-713 ; (of order J), 664, 714729 ; (of various integral orders), 664, 736, 737-739 ; tables of (references to), 657, 658; with large argument, asymptotic expansions of, 202 ; zeros of, 511 ; (computation of), 512 ; (references to),

K

;

;

;

660 Bessel's differential equation, 1, 19 ; (generalised), 38 ; for functions of order zero, 5, 12, 59, 60; for functions with imaginary argument, 77 ; fundamental system of solutions of, 42, 75 ; has no


798

algebraic integral, 117 ; soluble in finite terms when and only when the functions satisfying it are of order n + %, 52, 119; solution of, in ascending series, 39, 40, 57, 59-61 ; solution of, in descending series, see Asymptotic expansions ; symbolic solution of, 41 ; transformations of, 94, 97. See also Bessel coefficients and Bessel functions Bessel's integral representing Bessel coefficients, 19, 21; generalisations

and extensions

of, see

Anger's function, Bourget's function, Brans' function and Weber's function ; modifications of, to represent Bessel functions of arbitrary order, 175, 176, 177, 178, 181 ; Theisinger's transformation of, 184 ; used in theory of diffraction, 177 ; used to obtain asymptotic expansions, 215. See also Parseval's integral

Bounds, upper,

see Inequalities'

Bourget's function

326

326 Brans' function J(z;v,

differential equation satisfied by,

;

approximation for Bessel functions of large order, 6, 7 324 ; recurrence formulae for, 325 OT Cayley's solution of Biccati's equation, 88 Chain, oscillations of a uniform heavy, 3, 4, 576 Cognate Riccati equations, 91 ,

,

;

recurrence formulae for,

327

k),

Carlini's

Cauchy's numbers i^_ n i

327

;

extended by Meissel, 226, 227

,

Complex variables, expansions of arbitrary functions of, see Kapteyn series and Neumann series Complex zeros of Bessel functions, 483 of Bessel functions with imaginary argument, 511 of Lommel's polynomials, 306 ;

;

Composition of Bessel functions of the second kind of integral order, 340 Computation of zeros of Bessel functions by various methods (Oraeffe's), 500, 502; (Stokes'), 503; (Sturm's, for the smallest zero), 516. See also Zeros of Bessel functions Constant phase, Schlafli's method of, 216 Constants, discontinuity of arbitrary (Stokes' phenomenon), 201, 203, 238, 336 Continuants, connected with Schlafli's polynomial, 288 Continued fractions representing quotients of Bessel functions, 153 convergence of, 154, 303 Convergent series, Hadamard's conversion of asymptotic expansions into, 204 Crelier's integral for Schlafli's polynomial, 288. See also Neumann's integral for Neumann's polynomial ;

Cross-ratio of solutions of Biccati's equation, 94

Cube of a Bessel function, expansion of, 149 Cat necessary for definition of Bessel functions, 45, 77 Cylinder (circular), motion of heat in, 9, 10, 576, 577 Cylinder functions, 9£v {z), 4, 82, 480; addition theorems, 143, 361, 365; connexion with Bessel functions, 83 origin of the name, 83 rank of, 129; solutions of differential equations of order ;

;

higher than the second by, 106 three-term relations connecting, 300. See also Bessel functions and Hemi-cylindrical functions ;

Darboux' method of approximating to functions of large numbers, 233 Definite integrals, containing Bessel functions under the integral sign, 373-382 (Chapter xu); evaluated by geometrical methods, 374, 376, 378 ; the Ramanujan-Hardy method of evaluation, 382. See also Infinite integrals Definite integrals representing special functions, see Bessel functions and Integrals Determinants, representing Lommel's polynomials, 294 ; Wronskian, 42, 76, 77 Difference equations (linear with linear coefficients) solved by means of Bessel functions, 83. See also Functional equations and Recurrence formulae Differentiability of Fourier-Bessel expansions, 605 ; of special Schlomilch series, 635 Differential coefficients, fractional, 107, 125 Differential equations (ordinary), linear of the second order, equivalent to the generalised Riccati equation, 92 of order higher than the second solved by Bessel functions, 106 oscillation of solutions of, 518 satisfied by the product of two Bessel functions, 145, 146 solved by elementary transcendants, 112; symbolic solutions of, 41, 108. See also under the names of special equations, such as Bessel's differential equation, and under the names of various functions and polynomials satisfying differential equations, such as Anger's function Differential equations (partial), solution of by an integral containing Bessel functions, 99 see also Laplace's equation and Wave-motions, equation of Diffraction, theory of, connected with Airy's integral, 188 with Bessel's type of integral, 177 with Schlomilch series, 633 ; with Struve's functions, 417 Diffusion of salts in a liquid, and infinite integrals containing Bessel functions, 437 ;

;

;

;

;

;

Dini expansion, 580.

See also Dini series

;

;;

GENERAL INDEX

799

Dlni series, 577, 580, 596-605, 615-617 (Chapter xvra), 651-653 ; expansion of an arbitrary function of a real variable into, 580, 600 ; methods of theory of functions of complex variables applied of, to, 596, 602 ; Riemann-Lebesgue lemma, analogue of, 599 ; Riemann's theorem, analogue 649; summability of, 601, 615 ; uniformity of convergence of, 601, 604 ; uniqueness of, 616, 651 value at end of range, 602 Dirichlet's discontinuous factor, 406

Discontinuity of arbitrary constants (Stokes' phenomenon), 201, 203, 238, 336 Discontinuous factor (Dirichlet's), 406; (Weber's), 405 Discontinuous integrals, 398, 402, 406, 408, 411, 415, 421

Domain

K (Kapteyn's),

559

;

diagram

270

of,

Du Bois Reymond's integrals with oscillatory integrands expressed in terms of

Bessel functions, 183

Electric waves, 56, 226, 449 Electromagnetic radiation, 551, 556 Elementary transcendants, definition of, 111 order of, 111 ; solution of differential equations by, 112 Equal order and argument, Bessel functions with, 231, 232, 258, 260 ; tables of, 746, 747 ; tables of (references to), 658, 664 Buler's solution of Riccati's equation, 87 Exponential function, tables of, 698-713 tables referred to, 663, 664 ;

;

Factors, discontinuous (Dirichlet's), 406 ; (Weber's), 405 ; Neumann's en of Bessel functions as products of Weierstrassian, 497 Fejer's theorem, analogue of, for Fourier-Bessel expansions, 610

(=1 or 2), 22

;

expression

Finite terms, Bessel functions of order Ht (n+i) expressed in, 52 ; Bessel functions of other orders not so expressible, 119 ; solutions of Riccati's equation in, 85, 86, 89 ; the solution of Riccati's equation in, not possible except in Daniel Bernoulli's cases and their limit, 123 Flights, problem of

random, 419

Fourier-Bessel expansion, 580. Fourier-Bessel functions,

4,

See also Fourier-Bessel series

84

Fourier-Bessel integrals, see Multiple infinite integrals Fourier-Bessel series, 576-617 (Chapter xvm), 649-651 ; expansion of an arbitrary function of a real variable into, 576, 580 ; Fejer's theorem, analogue of, 610 ; Kneser-Sommerfeld expansion of a combination of Bessel functions into, 499 ; methods of theory of functions of complex variables applied to, 582, 607 ; order of magnitude of terms in (Sheppard's theorem), 595 ; RiemannLebesgue lemma, analogue of, 589 ; Riemann's theorem, analogue of, 649 ; summability of, 578, 606, 613 ; term-by-term differentiation of, 578, 605 ; uniformity of convergence of, 593, 594 (near origin), 615 ; uniformity of summability of, 612 ; uniqueness of, 616, 649 ; value at end of range, 594, 603 Fractional differential coefficients, 107, 125 Fresnel's integrals, 544 ; asymptotic expansion of, 545 ; tables of, 744, 745 ; tables of maxima and minima of, 745 ; tables of (references to), 660, 661, 664 Functional equations denning cylinder functions, 82 ; generalised by Nielsen, 355 Functions of large numbers, approximations due to Darboux, 233 ; approximations due to Laplace, 8, 421 . See also Approximations, Asymptotic expansions, Method of stationary phase and Method of steepest descents Fundamental system of solutions of Bessel's differential equation, 42, 75, 78 Gallop's discontinuous infinite Integrals, 421 Gamma functions, representation of Bessel functions by integrals containing, 190, 192, 221 ; applications to determination of asymptotic expansions, 220, 223 ; applications to evaluation of infinite integrals, 383, 434, 436 Gamma functions, representation of Lommel's functions by integrals containing, 351 ; applications to determination of asymptotic expansions, 352 Gegenhauer's addition theorem for Bessel functions, 362, 363, 367 Gegenbauer's discontinuous infinite integrals, 415, 418 Gegenbauer's function Cn " (z), 50, 129, 363, 365, 367, 368, 369, 378, 407 Gegenbauer's polynomial A nyV (t), 283 ; contour integrals containing, 284, 524 ; differential equation satisfied by, 283 equivalence with special" forms of Lommel's function, 351 ; recurrence formulae for, 283 Gegenhauer's polynomial B n>lkiV [t), 293, 525 Gegenbauer's- representation of Jv \z) by a double integral resembling Poisson's integral, 51 ;

Gegenhauer's type of definite integral, 378 Generalised'hypergeometric functions, see Hypergeometric functions (generalised)

; ;;


800

Generalised Integrals (with implied exponential factor), 188, 441, 463, 464 Generating function of Bessel coefficients, 14, 22, 23 ; of Neumann's polynomials, 281, 282 Gilbert's integrals, 548, 549 Giuliani's function, see Bourget's funotion Graeffe's method of calculating zeros, 500, 502 Grafs addition theorem for Bessel functions, 359, 361 Group velocity, 229 Growth of zeros of Bessel functions, 485

Hankel's infinite integrals, 384, 386, 389, 390, 393, 395, 424, 427, 428, 434 Hansen's upper bound for Jn (x), 31 generalised, 406 Hardy's functions Ci n (a), Si n (a), Ein (a), (generalisations of Airy's integral), 320; expressed in terms of functions of Bessel, Anger and Weber, 321, 322 Hardy's integrals representing Lommel's functions of two variables, 546 Hardy's method of evaluating definite integrals, 382 Heat, conduction of, 9, 10, 450, 576, 577, 616 ;

Hemi-cylindrical functions S w (z), defined, 353 353 addition theorem for, 354

expressed in terms of the function of order zero,

;

;

Hypergeometric functions, limiting forms expressed as Bessel functions, 154 Hypergeometric functions (generalised), 90, 100 Bessel functions expressed in terms of, 100, 101 notations for, 100 relations between (Rummer's formulae), 101, 102 Sharpe's differential equation solved by, 105 ;

;

;

Imaginary argument, Bessel functions with,

gee Bessel functions with imaginary argument; Struve's functions with, 329, 332 Indefinite integrals containing Bessel functions under the integral sign, 132-138, 350, 581 ; tables of, 744, 745, 752 ; tables of (references to), 660, 661, 664 Inequalities satisfied by Bessel functions, 16, 31, 49, 255, 259, 268, 406; by Neumann's polynomial, 273, 282 ; by Struve's function, 328, 337, 417 ; by zeros of Bessel functions, 485, 489, 490, 492, 494, 515, 516, 521

under the integral sign, 383-449 (Chapter xm) ; discontinuous, 398, 402, 406, 408, 411, 415, 421 ; generalised, 441 ; methods of evaluating, described, 383 ; Bamanujan's type (integrals of Bessel functions with respect to their order), 449. See also under the names of various integrals, e.g. Lipschitz-Hankel infinite integral Infinity of the number of zeros of Bessel functions and cylinder functions, 4, 478, 481, 494, 495 Integrals, expressed in terms of Lommel's functions of two variables, 540 ; expressed in terms of the functions of Anger and Weber, 312 ; Fresnel's, 544, 545, 660, 661, 664, 744, 745 ; Gilbert's, 548, 549 values of, deduced from addition theorems, 367 ; with oscillatory integrands, 183 with the polynomials of Neumann and Gegenbauer under the integral sign, 277, 285. See also Definite integrals and Infinite integrals Interference, 229 Infinite Integrals containing Bessel functions

;

Interlacing of zeros of Bessel functions and of cylinder functions, 479, 480, 481 Irrationality of *-, 90, 485 Jacobl's transformation connecting sinnl with the (n - l)th differential coefficient of sin 2B_1 respect to cos d, 26 ; erroneously attributed to Bodrigues, 27 ; various proofs of, 27, 28

with

Kapteyn's domain K, 559 diagram of, 270 Kapteyn series, 6, 13, 551-575 (Chapter xvn) ; connexion with Kepler's problem, 551 ; expansions into, derived from Kepler's problem, 554, 555 expansion of an arbitrary analytic function into, 570; fundamental expansions into, 557, 559, 561, 564, 566, 568, 571 ; Kapteyn's domain K, of convergence of, 559 (diagram of), 270 nature of convergence outside and on the boundary of K, 574 ; second kind of, 572 Kapteyn's polynomial % (t), 568 ; expressed in terms of Neumann's polynomial, 569 Kapteyn's type of definite integral, 380 Kepler's problem, 6, 551, 554 ; Bessel's solution of, 13 ; Lagrange's solution of, 6 Kinds of Bessel functions, (first) 40 ; (second) 58, 63, 64, 65, 67 ; (third) 73 Kneser-Bonunerfeld expansion of a combination of Bessel functions as a Fourier-Bessel series, 499 Hummer's formulae connecting generalised hypergeometric functions, 101, 102 ;

;

;

;

®

Lame functions, limiting forms expressed as Bessel functions, 159 Laplace's equation, general solution due to Parse val, 9 ; general solution due to Whittaker, 124 solutions involving Bessel functions, 83, 124 ; used to obtain addition theorems for Bessel functions,

127

;

GENERAL INDEX

801

Laplace's methods of approximating to functions of large numbers, 8, 421 Laplace's transformation, 280, 395 Large numbers, methods of approximation to functions of (Darboux), 233 ; (Laplace), 8, 421. also

Sec

Approximations and Asymptotic expansions

Large order, see Anger's function, Bessel functions whose order is large, Struve's function and Weber's function Lebesgue's lemma, see Riemann-Lebesgue lemma Legendre functions, Barnes' notation for, 156; integrals containing, 50, 173, 174, 339, 475; limits (physical significance of), 155 of large degree, expressed as Bessel functions, 65, 155, 157 approximations to, 158 relation between two kinds of, 174 Whipple's transformation of, 387. See also Oegenbauer*s function Cn v (z) Lipschltz-Hankel infinite integral, 384 ; generalised, 389 Lommel's functions 8 Ml (z), £p,i> (z), 345, 347 ; cases of expression in finite terms, 350 integrals special cases expressible by the polynomials representing, 346, 350 recurrence formulae, 348 of Gegenbauer, Neumann and Schlfifli, 350; special cases with /jl±v an odd negative integer, 348 with large argument, asymptotic expansion of, 351 Lommel's functions of two variables, Uv (w, z), Vv (w, z), 537, 538 addition formulae for, 543 integrals representing, 540, 546 reciprocation formulae, 542 recurrence formulae, 539 special case of, 581, 752 ; tables of, 752 ; tables referred to, 660 with large argument, asymptotic expansions of, 549 Lommel's polynomial J?wt , „ (z), 294, 295 differential equation satisfied by, 297 ; Hurwitz' notation 9m, v (z), 303 limit of, expressed as a Bessel function, 302 of negative order, R^ m v (z), 299 recurrence formulae, 298 recurrence formulae in Hurwitz' notation, 303 ; relations with Bessel functions, 295, 297, 302 ; three-term relations connecting, 300, 301 zeros of, 304, 305, 306 of,

;

;

;

;

;

i;

;

;

;

;

;

;

;

;

;

;

;

,

;

;

Magnitudes of remainders in asymptotic expansions, 206, 211, 213, 236, 314, 332, 352, 449 MaTJitia of Bessel functions, 488; of FresnePs integrals, table of, 745 of integrals of Bessel func;

752 Mean anomaly, expansions of elements of an orbit in trigonometrical series of, 6, 13, 552, 554, 556 Mehler-Dirichlet integral representing Legendre functions, limiting form expressed as Poisson's integral, 157 Mehler-Sonine integrals representing Bessel functions, 169, 170 Meissel's approximations to Bessel functions of large order, 226, 227, 232, 247, 521 ; types of Kapteyn series, 557, 561, 564, 566 Membrane, vibrations of a circular, 5, 576, 618 ; vibrations of a sectorial, 510 Method of constant phase (Schlafli's), 216 Method of stationary phase, 225, 229 applied to Bessel functions, 231, 233 tions, table of,

;

Method of steepest descents, 235 applied to Bessel functions, 237, 241, 244, 245, 262 applied to functions of Anger and Weber, 316; applied to Struve's function, 333; connexion with Laplace's method of approximation, 421 Minima of Bessel functions, 488 of Fresnel's integrals, table of, 745 of integrals of Bessel func;

;

;

;

tions, 752

Monotonic properties of

Jv (vx)jjv (*),

257; of

Jv (v) and

,TV

'

(v),

260

;

of

K

(.r),

446

Multiple infinite Integrals, 450-476 (Chapter xrv) ; investigated by Neumann, 453, 470 (generalised by Hankel), 453, 456, 465; (generalised by Orr), 455; (modified by Weber), 468; RiemannLebesgue lemmas, analogues of, 457, 471 Weber's type of, 450 ;

;

Neumann

series, 522-537 (Chapter xvi); expansion of an arbitrary analytic function into, 523; generalised, 525 ; (special series), 30, 31, 36, 69, 71, 151 Laurent's expansion, analogue of, 524; Pincherle's theorem on the singularities of, 526; special series, 18, 23, 25, 33, 34, 35, 128, 130, 138, 139, 140, 527, 581 ; Webb-Kapteyn (real variable) theory of, 533. See also Addition theorems ;

and Lommel's functions of two variables Neumann's factor eB (=1 or 2), 22 Neumann's Integral for <7m2 (z), 32 for Neumann's polynomial, 278, 280 Neumann's polynomial On (t), 271, 272, 273 connected with Kapteyn's polynomial, 569 connected with Neumann's polynomial O n [t), 292 connected with Schlafli's polynomial, 285, 286 contour integrals containing, 277 differential equation satisfied by, 276 expressed in terms of Lommel's ;

;

;

;

;

;

;

formerly called a Bessel function of the second kind, 67, 273 generalised by ; Gegenbauer, see Gegenbauer*s polynomial A n
;

;

;

;

;

;

,

;

;;


802

Nielsen-Hankel functions,

see Bessel

functions of the third kind

Null-functions, Lerch's theorem on integrals representing, 882 634, 636, 642, 647

;

represented by Schloniilch series,

Numbers, analytic theory of, associated with asymptotic expansions Numbers, Cauchy's, 324 recurrence formulae for, 325

of Bessel functions,

200

;

Order of a Bessel function denned, 38, 58, 63, 67, 70 integrals with regard to, 449 Ordinary differential equations, see Differential equations Oscillation of solutions of linear differential equations, 518 Oscillations of membranes, 5, 510, 576, 618 of uniform heavy chains, 3, 4, 576 Oscillatory integrands, Du Bois Beymond's integrals with, expressed in terms of Bessel functions, 183 ;

;

P-functions, limiting forms expressed as Bessel functions, 158 Parseval'8 integral representing J (z), 9, 21 ; modifications of, 21

Partial differential equations, see Differential equations Phase, method of stationary, general principles of 225, 229; applied to Bessel functions, 231, 233 Phase, Schlafli's method of constant, 216 ,

Pincherle's

theorem on

singularities of functions defined by

Neumann

series,

526

Poisson's integral for Bessel coefficients, 12, 24, 25 ; for Bessel functions, 47, 48, 49 ; (generalised by Gegenbauer), 50 ; (symbolic form of), 50 ; for Bessel functions of imaginary argument, 80 ; for Bessel functions of the second kind, 68, 73 ; limit of the Mehler-Dirichlet integral for Legendre functions as, 157 ; transformation into contour integrals to represent Bessel functions of any order (of the first kind), 161, 163, 164 ; (of the second kind), 165 ; (of the third kind), 166, 167; (with imaginary argument), 171, 172; transformations of the contour integrals, 168, 169, 170. See also Parseval's integral and Struve's function

Polar coordinates, change of axes of, used to obtain transformations of integrals, 51, 374, 376, 378 used to express Bessel functions as limits of Legendre functions, 155

Probleme de moments of Stieltjes, 464 Products of Bessel functions, 30, 31, 32, 82, 146, 147, 148, 149; Bateman's expansion of, 130, 370 expansions of arbitrary functions into series of, 525, 572 integrals representing, 31 150, 221 438, 439, 440, 441, 445, 446, 448 ; series of, 30, 151, 152 with large argument, asymptotic expansions of, 221, 448 Products of Weierstrassian factors, Bessel functions expressed as, 497 ;

,

,

;

Quotient of Bessel functions expressed as a continued fraction, 153, 154, 303

Radius vector of an orbit, expansion as trigonometrical series of the mean anomaly, 6, 13,552,553,554 Ramanujan's integrals of Bessel functions with respect to their order, 449 Bamanujan's method of evaluating definite integrals, 382 Random flights, problem of, 419 Rank of Bessel functions and cylinder functions, 129 Real variables, expansions of arbitrary functions of, see Dini series, Fourier-Bessel series, Neumann series (Webb-Kapteyn theory), and 8chl6mllch series Reality of zeros of Bessel functions, 482, 483, 511 Reciprocation formulae for Lommel's functions of two variables, 542 Recurrence formulae for Anger's functions, 311 ; for Bessel coefficients, 17 for Bessel functions of the first kind, 45 ; for Bessel functions of the second kind, 66, 71 for Bessel functions of the third kind, 74 for Bessel functions with imaginary argument, 79 for Bourget's functions, 326 for Cauchy's numbers, 325 ; for cylinder functions, 82 for Gegenbauer's polynomials, 283 for ;

;

;

;

;

tions, 329 ; for Weber's functions, 311 ; for Whittaker's functions, 339. equations, Heml-cylindrlcal functions and Three-term relations Reduced functions, Cailler's, 536

Remainders in asymptotic expansions, magnitudes 207, 209, 215, 315, 333

;

Stieltjes'

See also Functional

206, 211, 236, 314, 332, 352

of,

approximations

;

to,

;

signs of, 206,

213

Repetition of zeros of Bessel functions and cylinder functions, impossibility

of,

479

Riccati's differential equation, 1, 2, 85-94; connexion with Bessel's equation, 1, 90} equation cognate to, 91; limiting form of, 86; soluble cases .of (D. Bernoulli's), 85; soluble cases of, exhausted by D. Bernoulli's formula and its limit, 123 ; solutions by various mathematicians (D. Bernoulli), 2, 85, 89 ; (Cayley), 88 ; (Euler), 87 ; (Schlafli), 90 ; solved by means of infinite series by James Bernoulli, 1 ; transformations of, 86

Riccati's differential equation generalised, 3, 92, 94

;

cross -ratio of solutions. 94

;

equivalence

GENERAL INDEX with the linear equation of the second order, 92 ; singularities bers (two, one or none) of quadratures, 3, 93 Riemann-Lebesgue lemma, analogues of the, 457, 471, 589, 599

803 of,

94

Riemann's theorem on trigonometrical series, analogues for Schlomilch for series of Fouricr-Bcsscl and Dini, 649

;

soluble by various

series, 642,

647

;

num-

analogues

Rodrigues' transformation, see Jacobl's transformation Schafheitlln's discontinuous Infinite integral, 398, 402, 405, 406, 408, 411

Bcbafheitlln's integrals representing Bessel functions

Tn (z)

and cylinder functions, 168, 169, 490, 491, 493

Vn (z),

and

71, 340, 343; addition theorems for, 344, 345; differential equations satisfied by, 342, 343; of negative order, 343 ; recurrence formulae for, 71, 342, 343

Bchlafli's functions

hypergeometric function, 90 polynomial .S'„ (t), 284, 286 addition theorem for, 289 connexion with Neumann's polynomial On (t), 285, 286; Crelier's integral representation of, 288; differential equation satisfied by, 285 expression by means of Bessel functions, 287; expression in terms of Lommel's function, 350 integrals evaluated in terms of, 350 recurrence formulae for, 285 Schlafli's solution of Riccati's equation, 90 Schlomilch series, 618-649 (Chapter xix) definition of, 621 ; definition of generalised, 623 expansion of an arbitrary function of a real variable into, 619, 623, 629 ; nature of convergence of, 637, 645 null-functions expressed by, 634 Riemann's theorem on trigonometrical series (analogue of), 642, 647; special cases of, 632; symbolic operators in the theory of, 626, 627; theory of functions of complex variables connected with, 623 uniqueness of, 643, 647 8chlafli's

Schlafli's

;

;

;

;

;

;

;

;

;

;

Series containing Bessel functions, see Dini series, Fourier-Bessel series, series and Schlomilch series

Kapteyn series, Neumann

Series of Bessel functions, definition of, 580

sum of (greatest term method), 8 Sharpe's differential equation, 105 solution by generalised hypergeometric functions, 105 Sign of remainders in asymptotic expansions, 206, 207, 209, 215, 315, 333, 449 ; of Struve's function, 337, 417 Sine-Integral expressed as a series of squares of Bessel coefficients, 152 Singularities of functions defined by Neumann series (Pincherle's theorem), 526 of the generalised Riceati equation, 94 Smallest zeros of Bessel functions, 5, 500, 516 Sommerfeld's expansion, see Kneser-Sommerfeld expansion Sonine-Mehler integrals representing Bessel functions, 169, 170 Series of positive terms, approximation to the ;

;

Bonlne's definite integral, 373

;

generalised, 382

Bonlne's discontinuous Infinite Integrals, 415

Bonlne's infinite integrals, 432 Spherical geometry used to obtain transformations of integrals, 51, 374, 376, 378 ; used to express Bessel functions as limits of Legendre functions, 155 Sound, Sharpe's differential equation in the theory of, 105 Squares of Bessel functions, see Products of Bessel functions Stability of a vertical pole associated with Bessel functions of order one-third, 96

Stationary phase, method of, 225, 229 ; applied to Bessel functions, 231, 233 Steepest descents, method of, 235 ; applied to Bessel functions, 237, 241, 244, 245, 262 ; applied to functions of Anger and Weber, 316 ; applied to Struve's function, 333 connexion with Laplace's method of approximation, 421 Stokes' method of computing zeros of Bessel functions and cylinder functions, 503, 505, 507 ;

Stokes'

phenomenon

of the discontinuity of arbitrary constants, 201, 203, 238, 336

Struve's function H„ (z), 328 ; connexion with Weber's function, 336 ; differential equation satisfied by, 329 ; inequalities connected with, 328 ; infinite integrals containing, 392, 397, 417, 425, 436; integral representations of, 328, 330 ; occurrence in generalised Schlomilch series, 622, 623, 631, 645, 646, 647 ; of order ± (n -(• $), 333 ; recurrence formulae for, 329 ; sign of, 337, 417 ; tables of, 663, 666-697; Theisinger's integral for, 338; with imaginary argument, 329, 332 ; with large

argument, asymptotic expansions sions of, 333 ; zeros of, 479

of,

332

;

with large argument and order, asymptotic expan-

Struve's infinite integrals, 396, 397, 421 Sturm's methods applied to determine the reality of zeros of Bessel functions, 483 ; of Lommel's polynomials, 304, 305, 306 ; applied to estimate the value of the smallest zero of Bessel functions and cylinder functions, 517, 518 Symbolic operators in expressions representing Bessel functions, 50, 170 ; in expressions representing solutions of various differential equations, 41, 51, 108; in the theory of Schlomilch series,

627


804

Tables of Bessel coefficients (of orders and 1), 662, 666-697; (of order n), 664, 730-732 (with - n - £), equal1 order and argument), 664, 746 of Bessel functions of the first kind (of orders n + J, and 664, 740-741 (of order i), 604, 714-729 of Bessel functions of the second kind (of orders (of order ^), 664, 714-729 ; (with equal order and (of order n), 664, 732-735 1), 662, 666-697 and 1), 662, 666-697 (of argument), 664, 747 ; of Bessel functions of the third kind (of orders "' "" ** " " order.Jj), 698-7i3; ;

.-„,...

;

i

;

;

;

;

"

;

'

integrals, and 1), 663, 666-697 tions (of orders order n and of order $, 664, 748-751

;

and functions

of zeros of Bessel coefficients

of integral

Tables (references to) of Airy's integral, 659 of Bessel coefficients and iunctions derivable from them, 654, 655, 656, 658 of Bessel functions (of orders n + £, - n - h), 658, 659 (of orders ± I, ±3), 659; (of orders ±J, ±£), 659; of Bessel functions of the second kind, 655, 656, 658; of Bessel functions of the third kind, 657 ; of Bessel functions with imaginary argument, 657, 658; of ;

;

;

Fresnel's integrals, 661 of integrals of Bessel functions and Struve's functions, 661 ; of LommePs functions of two variables, 660 ; of Thomson's functions ber r and bei x, etc., 658 ; of zeros of Bessel coefficients, functions and associated functions, 659, 660 Theisinger's integral representation of Bessel functions, 184 ; of Struve's and Weber's functions, ;

338

Thomson's

William) functions, ber 2, beiz, 81 ; connexion with Bessel functions, 81 ; generalireferences to tables of, 658 ; squares and products of, 82, 148 ; with large argument,

(Sir

sations, 81

;

asymptotic expansions of, 203 Three-term relations connecting Bessel functions, cylinder functions and Lommel's polynomials, 300, 301 Transcendants, elementary, definition of, 111 order of, 111 ; solutions of differential equations by, 112 Transitional regions associated with Bessel functions of large order, 248 ;

Uniformity of convergence of Dini series, 601 of Fourier-Bessel series, 593, 594 of Kapteyn series, 575 ; of Schlomilch series, 632 Uniqueness of Fourier-Bessel and Dini series, 616, 649, 651 of Schlomilch series, 643, 647 Upper bounds, see Inequalities ;

;

;

Viscous

fluid,

motion

of, associated

with Airy's integral, 189

Wave-motions, equation of, general solutions, 125 generalised to p dimensions, 128 ; used to obtain addition theorems for Bessel functions, 129 Waves, electric, 56, 226, 446 ; on water, and the method of stationary phase, 229 Weber's (H.) discontinuous factor, 405 Weber's (H.) infinite integrals, 391, 393, 395, 396; (discontinuous types of), 398, 402, 405, 406, 408, 411 Weber's (H. F.) function E„ (z), 308 connexion with Anger's function, 310 connexion with Struve's function, 336 ; differential equation satisfied by, 312 integrals expressed in terms of, 312 recurrence formulae for, 311 representation of Airy's integral (generalised) by, 321 tables of, see Struve's function Theisinger's integral for, 338 with large argument, asymptotic expansion of, 313 ; with large argument and order, asymptotic expansion of, 316 Weierstrassian products, expression for Bessel functions as, 497 Whipple's transformation of Legendre functions, 387 recurrence formulae for, differential equation satisfied by, 339 Whittaker's function v (z), 339 339 ; with large argument, asymptotic expansion of, 340 ;

;

;

;

;

;

;

;

;

W

Wronskian determinant,

;

;

42, 76,

77

Zeros of Bessel functions, 477-521 (Chapter xv) computation of (various methods of), 142, 500, 502, 503, 516 inequalities connected with, limits of, rates of growth of, 485, 489, 490, 491, 494, 507, 513, 516, 518 infinity of, 4, 478 interlacing of, 479, 480, 481 ; non-coincidenee of (Bourget's hypothesis), 484 non-repetition of, 479 ; number of, in a strip of arbitrary width, 495 reality of, 482, 483; tables of, 664, 748-751 tables of (references to), 659 ; values of, 4, 5, 512, 516 with imaginary argument, 511 with unrestrictedly large order, 513, 516 Zeros of Lommel's polynomials (reality of), 304, 305, 306 ;

;

;

;

;

;

;

;

Zeros of Struve's function, 479

;

Generalized

Hypergeometric Functions

LUCY JOAN SLATER of generalized hypergeometric functions is fundamental in the field of mathematical physics, since all the commonly used functions of analysis (Bessel Functions, Legendre Functions, etc.) are special cases of the general functions. The unified theory provides a means for the analysis of the simpler functions and can be used to solve the more complicated equations in physics. The generalized Gauss

The theory

mathematical statistics and the basic analogues of the Gauss functions have applications in the

function field of

is

also used in

number

theory.

treatment leads on from a discussion of the Gauss functions to the basic hypergeometric functions, the

Dr

Slater's

hypergeometric integrals, bilateral series and Appel series. This book was planned jointly with the late Professor

an extended revision of his Cambridge Mathematical Tract (1985) on the subject and Dr Slater has

W. N. Bailey continued

it

as

single-handed since Professor Bailey's death,

incorporating in

it

the results of

many

of her

own

researches.

r

CAMBRIDGE UNIVERSITY PRESS Bentley House, 200 Euston Road, London, N.W.I American Branch: 82 East 57th Street, New York, N.Y. 10022 West African Office: P.M.B. 5181, Ibadan, Nigeria

Printed in Great Britain

Watson ATreatiseOnTheTheoryOfBesselFunctions Text

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