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Preface’ The present volume is an outgrowth of a Conference on Mathematical Tables held at Cambridge, Mass., on September 15-16, 1954, under the auspices of the National Science Foundation and the Massachusetts Institute of Technology. The purpose of the meeting was to evaluate the need for mathematical tables in the light of the availability of large scale computing machines. It was the consensus of opinion that in spite of the increasing use of the new machines the basic need for tables would continue to exist. Numerical tables of mathematical functions are in continual demand by scientists and engineers. A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, of the increasing use of automatic computers. In the latter connection, the tables serve mainly forpreliminarysurveys of problems before programming for machine operation. For those without easy access to machines, such tables are, of course, indispensable. Consequently, the Conference recognized that there was a pressing need for a modernized version of the classical tables of functions of Jahnke-Emde. To implement the project, the National Science Foundation requested the National Bureau of Standards to prepare such a volume and established an Ad Hoc Advisory Committee, with Professor Philip M. Morse of the Massachusetts Institute of Technology as chairman, to advise the staff of the National Bureau of Standards during the ~course of its preparation. In addition to the Chairman, the Committee consisted of A. Erdelyi, M. C. Gray, N. Metropolis, J. B. Rosser, H. C. Thacher, Jr., John Todd, C. B. Tompkins, and J. W. Tukey. The primary aim has been to include a maximum of useful information within the limits of a moderately large volume, with particular attention to the needs of scientists in all fields. An attempt has been made to cover the entire field of special functions. To carry out the goal set forth by tbe Ad Hoc Committee, it has been necessary to supplement the tables by including the mathematical properties that are important in computation work, as well as by providing numerical methods which demonstrate the use and extension of the tables. The Handbook was prepared under the direction of the late Milton Abramowitz, Its success has depended greatly upon the cooperation of and Irene A. Stegun. Their efforts together with the cooperation of the Ad HOC many mathematicians. Committee are greatly appreciated. The particular contributions of these and other individuals are acknowledged at appropriate places in the text. The sponsorship of the National Science Foundation for the preparation of the material is gratefully recognized. It is hoped that this volume will not only meet the needs of all table users but will in many cases acquaint its users with new functions. ALLEN V. ASTIN, L?imctor. Washington,
D.C.
Preface
to the Ninth
Printing
The enthusiastic reception accorded the “Handbook of Mathematical Functions” is little short of unprecedented in the long history of mathematical tables that began when John Napier published his tables of logarithms in 1614. Only four and one-half years after the first copy came from the press in 1964, Myron Tribus, the Assistant Secretary of Commerce for Science and Technology, presented the 100,OOOth copy of the Handbook to Lee A. DuBridge, then Science Advisor to the President. Today, total distribution is approaching the 150,000 mark at a scarcely diminished rate. The successof the Handbook has not ended our interest in the subject. On the contrary, we continue our close watch over the growing and changing world of computation and to discuss with outside experts and among ourselves the various proposals for possible extension or supplementation of the formulas, methods and tables that make up the Handbook. In keeping with previous policy, a number of errors discovered since the last printing have been corrected. Aside from this, the mathematical tables and accompanying text are unaltered. However, some noteworthy changes have been made in Chapter 2: Physical Constants and Conversion Factors, pp. 6-8. The table on page 7 has been revised to give the values of physical constants obtained in a recent reevaluation; and pages 6 and 8 have been modified to reflect changes in definition and nomenclature of physical units and in the values adopted for the acceleration due to gravity in the revised Potsdam system. The record of continuing acceptance of the Handbook, the praise that has come from all quarters, and the fact that it is one of the most-quoted scientific publications in recent years are evidence that the hope expressed by Dr. Astin in his Preface is being amply fulfilled. LEWIS M. BRANSCOMB,
Director
National Bureau of Standards November 1970
Foreword This volume is the result of the cooperative effort of many persons and a number of organizations. The National Bureau of Standards has long been turning out mathematical tables and has had under consideration, for at least IO years, the production of a compendium like the present one. During a Conference on Tables, called by the NBS Applied Mathematics Division on May 15, 19.52, Dr. Abramowitz of t,hat Division mentioned preliminary plans for such an undertaking, but indicated the need for technical advice and financial support. The Mathematics Division of the National Research Council has also had an active interest in tables; since 1943 it has published the quarterly journal, “Mathematical Tables and Aids to Computation” (MTAC),, editorial supervision being exercised by a Committee of the Division. Subsequent to the NBS Conference on Tables in 1952 the attention of the National Science Foundation was drawn to the desirability of financing activity in table production. With its support a z-day Conference on Tables was called at the Massachusetts Institute of Technology on September 15-16, 1954, to discuss the needs for tables of various kinds. Twenty-eight persons attended, representing scientists and engineers using tables as well as table producers. This conference reached consensus on several cpnclusions and recomlmendations, which were set forth in tbe published Report of the Conference. There was general agreement, for example, “that the advent of high-speed cornputting equipment changed the task of table making but definitely did not remove the need for tables”. It was also agreed that “an outstanding need is for a Handbook of Tables for the Occasional Computer, with tables of usually encountered functions and a set of formulas and tables for interpolation and other techniques useful to the occasional computer”. The Report suggested that the NBS undertake the production of such a Handbook and that the NSF contribute financial assistance. The Conference elected, from its participants, the following Committee: P. M. Morse (Chairman), M. Abramowitz, J. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, to help implement these and other recommendations. The Bureau of Standards undertook to produce the recommended tables and the To provide technical guidance National Science Foundation made funds available. to the Mathematics Division of the Bureau, which carried out the work, and to provide the NSF with independent judgments on grants ffor the work, the Conference Committee was reconstituted as the Committee on Revision of Mathematical Tables of the Mathematics Division of the National Research Council. This, after some changes of membership, became the Committee which is signing this Foreword. The present volume is evidence that Conferences can sometimes reach conclusions and that their recommendations sometimes get acted on. V
,/”
VI
FOREWORD
Active work was started at the Bureau in 1956. The overall plan, the selection of authors for the various chapters, and the enthusiasm required to begin the task were contributions of Dr. Abramowitz. Since his untimely death, the effort has continued under the general direction of Irene A. Stegun. The workers at the Bureau and the members of the Committee have had many discussions about content, style and layout. Though many details have had t’o be argued out as they came up, the basic specifications of the volume have remained the same as were outlined by the Massachusetts Institute of Technology Conference of 1954. The Committee wishes here to register its commendation of the magnitude and quality of the task carried out by the staff of the NBS Computing Section and their expert collaborators in planning, collecting and editing these Tables, and its appreciation of the willingness with which its various suggestions were incorporated into the plans. We hope this resulting volume will be judged by its users to be a worthy memorial to the vision and industry of its chief architect, Milton Abramowitz. We regret he did not live to see its publication. P. M. MORSE, Chairman. A. ERD~LYI M. C. GRAY N. C. METROPOLIS J. B. ROSSER H. C. THACHER. Jr. JOHN TODD ‘C. B. TOMPKINS J. W. TUKEY.
Handbook
of Mathematical
Functions
with
Formulas, Edited
Graphs, by Milton
1.
and Mathematical Abramowitz
and Irene A. Stegun
Introduction
The present Handbook has been designed to provide scientific investigators with a comprehensive and self-contained summary of the mathematical functions that arise in physical and engineering problems. The well-known Tables of Funct.ions by E. Jahnke and F. Emde has been invaluable to workers in these fields in its many editions’ during the past half-century. The present volume ext,ends the work of these authors by giving more extensive and more accurate numerical tables, and by giving larger collections of mathematical properties of the tabulated functions. The number of functions covered has also been increased. The classification of functions and organization of the chapters in this Handbook is similar to that of An Index of Mathematical Tables by A. Fletcher, J. C. P. Miller, and L. Rosenhead. In general, the chapters contain numerical tables, graphs, polynomial or rational approximations for automatic computers, and statements of the principal mathematical properties of the tabulated functions, particularly those of computa-
2.
Tables
Accuracy
The number of significant figures given in each table has depended to some extent on the number available in existing tabulations. There has been no attempt to make it uniform throughout the Handbook, which would have been a costly and laborious undertaking. In most tables at least five significant figures have been provided, and the tabular’ intervals have generally been chosen to ensure that linear interpolation will yield. fouror five-figure accuracy, which suffices in most physical applications. Users requiring higher 1 The most recent, the sixth, with F. Loesch added as cc-author, was published in 1960 by McGraw-Hill, U.S.A., and Teubner, Germany. 2 The second edition, with L. J. Comrie added as co-author, was published in two volumes in 1962 by Addison-Wesley, U.S.A., and Scientific Computing Service Ltd., Great Britain.
tional importance. Many numerical examples are given to illustrate the use of the tables and also the computation of function values which lie outside their range. At the end of the text in each chapter there is a short bibliography giving books and papers in which proofs of the mathematical properties stated in the chapter may be found. Also listed in the bibliographies are the more important numerical tables. Comprehensive lists of tables are given in the Index mentioned above, and current information on new tables is to be found in the National Research Council quarterly Mathematics of Computation (formerly Mathematical Tables and Other Aids to Computation). The ma.thematical notations used in this Handbook are those commonly adopted in standard texts, particularly Higher Transcendental Functions, Volumes 1-3, by A. ErdBlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi (McGrawHill, 1953-55). Some alternative notations have also been listed. The introduction of new symbols has been kept to a minimum, and an effort has been made to avoid the use of conflicting notation.
of the Tables precision in their interpolates may obtain them by use of higher-order interpolation procedures, described below. In certain tables many-figured function values are given at irregular intervals in the argument. An example is provided by Table 9.4. The purpose of these tables is to furnish “key values” for the checking of programs for automatic computers; no question of interpolation arises. The maximum end-figure error, or “tolerance” in the tables in this Handbook is 6/& of 1 unit everywhere in the case of the elementary functions, and 1 unit in the case of the higher functions except in a few cases where it has been permitted to rise to 2 units.
IX /-
.
INTRODUCTION
X
3.
Auxiliary
Functions
One of the objects of this Handbook is to provide tables or computing methods which enable the user to evaluate the tabulated functions over complete ranges of real values of their parameters. In order to achieve this object, frequent use has been made of auxiliary functions to remove the infinite part of the original functions at their singularities, and auxiliary arguments to co e with infinite ranges. An example will make t fi e procedure clear. The exponential integral of positive argument is given by
4.
775 ;:; E
: 89608 89717
4302 8737
d0 g. I
ze*El . 89823 .89927 90029
(z) 7113 7306 7888
8: 4 ix
:.90227 90129
4695 60”3
[ 1 ‘453
The numbers in the square brackets mean that the maximum error in a linear interpolate is 3X10m6, and that to interpolate to the full tabular accuracy five points must be used in Lagrange’s and Aitken’s methods. 8 A. C. Aitken On inte elation b iteration of roportional out the use of diherences, ‘Brot Edin i: urgh Math. 8 oc. 3.6676
recludes direct interThe logarithmic singularity polation near x=0. The Punctions Ei(x)-In x and x-liEi(ln x-r], however, are wellbehaved and readily interpolable in this region. Either will do as an auxiliary function; the latter was in fact selected as it yields slightly higher accuracy when Ei(x) is recovered. The function x-‘[Ei(x)-ln x-r] has been tabulated to nine decimals for the range 05x<+. For +
Interpolation
The tables in this Handbook are not provided with differences or other aids to interpolation, because it was felt that the space they require could be better employed by the tabulation of additional functions. Admittedly aids could have been given without consuming extra space by increasing the intervals of tabulation, but this would have conflicted with the requirement that linear interpolation is accurate to four or five figures. For applications in which linear interpolation is insufficiently accurate it is intended that Lagrange’s formula or Aitken’s method of iterative linear interpolation3 be used. To help the user, there is a statement at the foot of most tables of the maximum error in a linear interpolate, and the number of function values needed in Lagrange’s formula or Aitken’s method to interpolate to full tabular accuracy. As an example, consider the following extract from Table 5.1. zez El (2) . 89268 7854 : 89384 89497 6312 9666
and Arguments
parts, with. (1932).
Let us suppose that we wish to compute the value of xeZ&(x) for x=7.9527 from this table. We describe in turn the application of the methods of linear interpolation, Lagrange and Aitken, and of alternative methods based on differences and Taylor’s series. (1) Linear interpolation. The formula for this process is given by jp= (1 -P)joSPfi where jO, ji are consecutive tabular values of the function, corresponding to arguments x0, x1, respectively; p is the given fraction of the argument interval p= (x--x0>/(x1-~0> and jP the required instance, we have
interpolate.
jo=.89717
ji=.89823
4302
In the present 7113
p=.527
The most convenient way to evaluate the formula on a desk calculating machine is.to set o and ji in turn on the keyboard, and carry out t d e multiplications by l-p and p cumulatively; a partial check is then provided by the multiplier dial reading unity. We obtain j.6z,E.‘;9;72;&39717
4302)+.527(.89823
7113)
Since it is known that there is a possible error of 3 X 10 -6 in the linear formula, we round off this result to .89773. The maximum possible error in this answer is composed of the error committed
INTRODUCTION
by the last roundingJ that is, .4403X 10m5, plus 3 X lo-‘, and so certainly cannot exceed .8X lo-‘. (2) Lagrange’s formula. In this example, the relevant formula is the 5-point one, given by
The numbers in the third and fourth columns are the first and second differences of the values of xezEl(x) (see below) ; the smallness of the second difference provides a check on the three interpolations. The required value is now obtained by linear interpolation :
f=A-,(p)f_z+A-,(p)f-1+Ao(p>fo+A,(p)fi +A&)fa Tables of the coefficients An(p) are given in chapter 25 for the range p=O(.Ol)l. We evaluate the formula for p=.52, .53 and .54 in turn. Again, in each evaluation we accumulate the An(p) in the multiplier register since their sum is unity. We now have the following subtable. x m=&(x) 7.952
.89772
9757
7.953
.89774
0379
fn=.3(.89772
.;
&
-2
1 2 3 4 5
7.9 8.1 7.8 8.2 7.7
0999
Yn=ze”G@) : : . .
89823 89717 89927 89608 90029 89497
7113 4302 7888 8737 7306 9666
Yo.I 89773 :89774
Yo, 1.2. I
Yo. 1, (I
44034 48264 2 90220 4 98773 2 35221
0379)
In cases where the correct order of the Lagrange polynomial is not known, one of the prelimina interpolations may have to be performed witT polynomials of two or more different orders as a check on their adequacy. (3) Aitken’s method of iterative linear interpolation. The scheme for carrying out this process in the present example is as follows:
10620 .89775
9757)+.7(.89774
= 239773 7192.
10622 7.954
XI
X,-X
Yo.1.a.s.n
.0473 0527 . 1473 -. 1527 . 2473 -. 2527
-. .89773
71499 2394 1216 2706
. 89773
71938 89773 ii
71930 30
Here 20-x 1 Yo x,-x x.--20 Yn x,-x 1 Yo.1 Yo.1 ,n=x,-x G--z1 l/O.”
S2fl
yo,n=-
Yo.
1.
. .
., m--l.m.n--
1 ~n-%n
safz wa
l/0.1. . . ., n-1.98 Yo.1. . . -, m-1.n
x,-x x,-x
1
If the quantities Z.-X and x~--5 are used as multipliers when forming the cross-product on a desk machine, their accumulation (~~-2) -(x,-x) in the multiplier register is the divisor to be used at that stage. An extra decimal place is usually carried in the intermediate interpolates to safeguard against accumulation of rounding errors. The order in which the tabular values are used is immaterial to some extent, but to achieve the maximum rate of convergence and at the same time minimize accumulation of rounding errors, we begin, as in this example, with the tabular argument nearest to the given argument, then take the nearest of the remaining tabular arguments, and so on. The number of tabular values required to achieve a given precision emerges naturally in the course of the iterations. Thus in the present example six values were used, even though it was known in advance that five would suffice. The extra row confirms the convergence and provides a valuable check. (4) Difference formulas. We use the central difference notation (chapter 25),
Here Sf1l2=f1-f0, 8f3/a=fz-f1, . . . ,, a2/1=sf3ia-afiia=fa-2fi+fo ~af3~~=~aja-~aj~=fa-3j2+3fi-k 8'fa=~aj~fsla-6~3~2=f4-~f~+~ja-4f~+fo
and so on. In the present example the relevant part of the difference table is as follows, the differences being lace of the written in units of the last decimal function, as is customary. The sma Bness of the high differences provides a check on the function values 7:9 8.0
Applying, formula
xe=El(x) .89717 4302 . 89823 7113
for example,
j~=(l-P)fo+E2(P)~*jo+E4(P)~4jo+
-2 -2
SY 2754 2036
Everett’s
. . . +Pfl+F2(P)~afl+F4(P)~4fl+
S4f -34 -39
interpolation . . * *
and takin the numerical values of the interpolation toe flf cients Es(p), E4 ), F,(p) and F,(p) from Table 25.1, we find t!l at
We may notice in passing that Everett’s formula shows that the error in a linear interpolate is approximately mPwfo+
F2(P)wl=
m(P)
can be used. We first compute as many of the derivatives ftn) (~0) as are significant, and then evaluate the series for the given value of 2. An advisable check on the computed values of the derivatives is to reproduce the adjacent tabular values by evaluating the series for z=zl and x1. In the present
With linear interpolation there is no difference in principle between direct and inverse interpolation. In cases where the linear formula rovides an insufficiently accurate answer, two met fl ods are available. We may interpolate directly, for example, by Lagrange’s formula to prepare a new table at a fine interval in the neighborhood of the approximate value, and then apply accurate inverse linear interpolation to the subtabulated values. Alternatively, we may use Aitken’s method or even possibly the Taylor’s series method, with the roles of function and argument interchanged. It is important to realize that the accuracy of an inverse interpolate may be very different from that of a direct interpolate. This is particularly true in regions where the function is slowly varying, for example, near a maximum or minimum. The maximum precision attainable in an inverse interpolate can be estimated with the aid of the formula
AxmAj/df dx in which Aj is the maximum possible error in the function values. Example. Given xe”Ei(z) = .9, find 2 from the table on page X. (i) Inverse linear interpolation. The formula for v is In the present ‘=.90029
With x0=7.9 and x-x0= .0527 our computations are as follows: an extra decimal has been retained in the values of the terms in the series to safeguard against accumulation of rounding errors.
~(x,=~(xo)+(x-x,,~~+(x-xo,~~~ +(~-x,)q$+
we have
f(x) =xeZEt(x)
Since the maximum value of IEz(p)+Fz(p)I in the range O
(5) Taylor’s series. In cases where the successive derivatives of the tabulated function can be computed fairly easily, Taylor’s expansion
example,
(x--so) y’k’(x0)/k!
.89717
4302
.89717
4302
- .00113 .01074 .00012
0669 7621 1987
-.ooooo .00056 .ooooo .a9773
6033 3159 7194 0017
53 9
Interpolation The desired z is therefore z=zQ+p(z,--2,,)=8.1+.708357(.1)=8.17083
57
To estimate the possible error in this answer, we recall that the maximum error of direct linear interpolation in this table is Aj=3X lOwe. An approximate value for dj/dx is the ratio of the first difference to the argument interval (chapter 25), in this case .OlO. Hence the maximum error in x is approximately 3XlO-e/(.OlO), that is, .0003. (ii) Subtabulation method. To improve the ap roximate value of x just obtained, we interpo Pate directly for p=.70, .7l and .72 with the aid of Lagrange’s 5-point formula, xe=El (x)
X
8. 170
.
89999 -.-_
8.171
.
90000
6
QQ
1 0151 3834
-2 1 0149
8. 172
Inverse gives
90001
linear
Hencex=8.17062 An estimate
3983
interpolation
in the new
table
23. of the maximum
error in this result
is df~1x10-8_1x10-7 Ajl z .OlO (iii) Aitken’s method. This is carried out in the same manner as in direct interpolation.
INTRODUCTION n 0
yn=xeeZE1(x) . 90029 7306
2, 8. 2
4 3
: 89927 90129 . 89823
6033 7888 7113
8. 31 8. 0
8. 17083 17023 8. 17113
5712 1505 8043
%
: 90227 89717
4302 4695
7. 49 8.
8. 16992 17144
0382 9437
Z0.n
QJ.98 8. 1706,l
8. 17062
21 8142 7335
6. Bivariate Bivariate interpolation is generally most simply performed as a sequence of univariate interpolations. We carry out the interpolation in one direction, by one of the methods already described, for several tabular values of the second argument in the neighborhood of its given value. The interpolates are differenced as a check, and
Generation
a.1 2.n
9521 2 5948
The estimate of the maximum error in this result is the same as in the subtabulation method. An indication of the error is also provided by the
7.
XIII
of Functions
Many of the special mathematical functions which depend on a parameter, called their index, order or degree, satisfy a linear difference equation (or recurrence relation) with respect to this parameter. Examples are furnished by the Legendre function P,(z), the Bessel function Jn(z) and the exponential integral E,(x), for which we have the respective recurrence relations
zo.l.2.3.74
2244 231 415
8. 17062
discrepancy in the highest case xo .I ,2.3 A, and ZLI .2.8 .s.
7306
-. . 00072 00129 -. 00176
2112 6033 2887
-. .00227 00282
5G98 4695
interpolates,
in this
I
Interpolation interpolation is then carried out in the second direction. An alternative procedure in the case of functions of a complex variable is to use the Taylor’s series expansion, provided that successive derivatives of the function can be computed without much difficulty.
from
Recurrence
Relations
(iii) the direction in which the recurrence applied. Examples are as follows. Stability-increasing Pm(x),
p:(2)
Qnb),
Q:(x)
y&9,
KG)
J-n-&), &Cd
P”(X), nE,+,+xE,,=e-=.
is being
n
(x
z-t44 (n
Stability-decreasing
Jn+*-~Jn+J.-l=O
Particularly for automatic work, recurrence relations provide an important and powerful computing tool. If the values of P&r) or Jn(z) are known for two consecutive values of n, or E',(z) is known for one value of n, then the function may be computed for other values of n by successive applications of the relation. Since generation is carried out perforce with rounded values, it is vital to know how errors may be propagated in the recurrence process. If the errors do not grow relative to the size of the wanted function, the process is said to be stable. If, however, the relative errors grow and will eventually overwhelm the wanted function, the process is unstable. It is important to realize that st,ability may depend on (i) the particular solution of the difference equation being computed; (ii) the values of x or other parameters in the difference equation;
2318 265
l/n-u .00029
P.,(z)
7t @
Qnh), Q:(x) J&4, Jn+Hcd Em(z) F,,(t,
Z.@) , Zn+&) (n >r) p) (Coulomb
wave
function)
Illustrations of the generation of functions from their recurrence relations are given in the pertinent chapters. It is also shown that even in cases where the recurrence process is unstable, it may still be used when the starting values are known to sufficient accuracy. Mention must also be made here of a refinement, due to J. C. P. Miller, which enables a recurrence process which is stable for decreasing n to be applied without any knowledge of starting values for large n. Miller’s algorithm, which is wellsuited to automatic work, is described in 19.28,
Example
1.
INTRODUCTION
XIV
8.
Acknowledgments
The production of this volume has been the result of the unrelenting efforts of many persons, all of whose contributions have been instrumental in accomplishing the task. The Editor expresses his thanks to each and every one. The Ad Hoc Advisory Committee individually and together were instrumental in establishing the basic tenets that served as a guide in the formation of the entire work. In particular, special thanks are due to Professor Philip M. Morse for his continuous encouragement and support. Professors J. Todd and A. Erdelyi, panel members of the Conferences on Tables and members of the Advisory Committee have maintained an undiminished interest, offered many suggestions and carefully read all the chapters. Irene A. Stegun has served eff ectively as associate editor, sharing in each stage of the planning of the volume. Without her untiring efforts, completion would never have been possible. Appreciation is expressed for the generous cooperation of publishers and authors in granting permission for the use of their source material. Acknowledgments for tabular material taken wholly or in part from published works are iven on the first page of each table. Myrtle R. Ke Yilington corresponded with authors and publishers to obtain formal permission for including their material, maintained uniformity throughout the
bibliographic references and assisted in preparing the introductory material. Valuable assistance in the preparation, checkin and editing of the tabular material was receive IFi from Ruth E. Capuano, Elizabeth F. Godefroy, David S. Liepman, Kermit Nelson, Bertha H. Walter and Ruth Zucker. Equally important has been the untiring cooperation, assistance, and patience of the members of the NBS staff in handling the myriad of detail necessarily attending the publication of a volume of this magnitude. Especially appreciated have been the helpful discussions and services from the members of the Office of Technical Information in the areas of editorial format, graphic art layout, printing detail, preprinting reproduction needs, as well as attention to promotional detail and financial support. In addition, the clerical and typing stafI of the Applied Mathematics Division merit commendation for their efficient and patient production of manuscript copy involving complicated technical notation. Finally, the continued support of Dr. E. W. Cannon, chief of the Applied Mathematics Division, and the advice of Dr. F. L. Alt, assistant chief, as well as of the many mathematicians in the Division, is gratefully acknowledged. M.
ABRAMOWITZ.
1.
Mathematical DAVID
Constants
S. LIEPMAN ’
Contents Page
Mathematical
Table 1.1.
+i,nprime
Constants
20s.
...............
2
..................
2
Some roots of 2, 3, 5, 10, 100, 1000, e, 20s ..........
Common Units and Conversion Factors . . . . . . . . . Names and Conversion Factors for Electric and Magnetic Units . . . . . . . . . . . . . . . . . . . . . . . Adjusted Values of Constants . . . . . . . . . . . . . Miscellaneous Conversion Factors. . . . . . . . . . . . Conversion Factors for Customary U.S. Units to Metric Units . . . . . . . . . . . . . . . . . . . . . . . Geodetic Constants . . . . . . . . . , . . . . . . . .
* National Bureau of Standards.
Page 6 6 7 8 8 8
2. Physical
Constants
and
Table
Quantity
and
Conversion
Common
Factors
for
Electric
and
=
SI
emu name
name
I
Magnetic
=
force
Magnetomotive force Magnetic flu* Magnetic flux density Electric displacement
ampere coulomb volt ohm henry farad amp. turns/ meter amp. turns weber tesla
I tbampere 1tbcoulomb
abvolt abohm centimeter
Conversion
statampere statcoulomb statvolt statohm
10-l LO-’ 108 100 100 10-g 4*x
centimeter gil bert maxwell gauss --._-_-______
Units
=
SI unit/ esu unit
-
oersted
-
Example: If the value assigned to a current *Divide this number by 4?r if unrationalized 6
and
SI unit/ emu unit
esu name
-
Current Charge Potential Resistance Inductance Capacitance Magnetizing
Units Factors
The SI unit of electric current is the ampere defined by the equation 2r,,,Z1ZJ4~= F giving the force in vacua per unit length between two infinitely long parallel conductors of infinitesimal cross-section. If F is in newtons, and rrn has the numerical value 477 X lo-‘, then I1 and Zr are in amperes. The customary equations define the other electric and magnetic units of SI such as the volt, ohm, farad, henry, etc. The force between electric charges in a vacuum in this system is given by Q, Qn/4nrerg= F, re having the numerical value 10r/4nc2 where c is the speed of light in meters per second (r,= 8.854 x 10-12). The CGS unrationalized system is obtained by deleting 4n in the denominators in these equations and expressing F in dynes, and r in centimeters. Setting r,,, equal to unity defines the CGS unrationalized electromagnetic system (emu), re then taking the numerical value of 1/c2. Setting re equal to unity defines the CGS unrationalized electrostatic system (esu), r,,, then taking the numerical value of l/cz.
Mass-the kilogram -fixed by the international kilogram at S&vres, France. Time-the second- fixed as l/31,556,925.9747 of the tropical year 1900 at 12” ephemeris time, or the duration of 9,19‘2,631,770 cycles of the hyperfine transition frequency of cesiurn 133. Temperature-the degree-fixed on a thermodynamic basis by taking the temperature for the triple point of natural water as 273.16 “K. (The Celsius scale is obtained by adding -273.15 to the Kelvin scale.) Other units are defined in terms of them by assigning the value unity to the proportionThe ality constant in each defining equation. entire system, including electricity units, is called the Systi.?me International d’unitds (SI). Taking the l/100 part of the meter as the unit of length and the l/1000 part of the kilogram as the unit of mass, similarly, gives =
2.1.
often used in physics
~
(1 meter - 1650763.73h).
2.2. Names
Factors
rise to the CGS system, and chemistry.
The tables in this chapter supply some of the more commonly needed physical constants and conversion factors. All scientific measurements in the fields of mechanics and heat are based upon four international arbitrarily adopted units, the magnitudes of which are fixed by four agreed on standards: Lengththe meter -fixed by the vacuum wavelength of radiation corresponding to the transition 2Plu-5Da of krypton 86
Table
Conversion
-3x 100 -3 x 109 -(1/3)X 10-Z -(1/9)X 10-u %(1/9)X 10-l’ -9x 10” -3 x loo*
IO-3*
4rX lo-I* 108 10’
__---___----__-______-_____
-3/10** -(1/3)X 10-z -(1/3)X 10-B -3x 105*
10-J*
I_-..____..____
-
-
is 100 amperes its value in abamperes is 100X10-‘=lO. system is involved; other numbers are unchanged.
3. Elementary
Analytical
MILTON
Methods
ABRAMOWITZ l
Contents Elementary 3.1.
3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.3. 3.9. 3.10.
Analytical
Methods
Page 10
.................
Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means ............... Inequalities ...................... Rules for Differentiation and Integration ......... Limits, Maxima and Minima .............. Absolute and Relative Errors .............. Infinite Series ..................... Complex Numbers and Functions ............ Algebraic Equations .................. Successive Approximation Methods ........... Theorems on Continued Fractions ............
Numerical 3.11.
Methods
Table
3.1.
19 19 23
............................
Powers and Roots
13 14 14 16 17 18 19 19
.......................
Use and Extension of the Tables ............ 3.12. Computing Techniques. ................
equality holds if and only if g(z) =cf(x) stant>O).
Sums
(c=con-
fj>l 3.3. Rules
for
Differentiation
and
Integration
Derivatives
equality holds if and only if jbkl=cIuEIP-’ stant>O). If p=q=2 we get Cauchy’s
(c=con-
$; (cu) =c $9 c constant
3.3.1
3.3.2
Inequality
-g (u+v)++g
3.2.9 [&
akb,]2<&
a; &
b: (equality
for &=Cbk,
& (uv) =u E+v
3.3.3
2
c constant). Hiilder’s 4
Inequality
for
Integrals
3.3.4
& (u/v)=
vdu/dx-udv/dx v2 -
,
1r;++=1,p>1,
q>l
.g u(v) =g
3.3.5
gi
3.2.10
& (u”) =uD e %+1.n U 2)
3.3.6 equality holds if and only (c=constant>O). If p=p=2 we get Schwa&s
3.2.11
Inequality
if
jgCs)I=clflr)Ip-’
Leibniz’s
Theorem
for
Differentiation
of an Integral
3.3.7 b(c)
d &
s
a(c)
f
(2,
wx b(c)
3
a(cj
s
b ,J(x,c)dx+f@,
4
$--f
(a,
4
2
12
ELEMENTARY
Leibniz’s
Theorem
for
Differentiation
ANALYTICAL
The following
of a Product
3.3.8 s
METHODS
formulas
S
P(x)dx
n>l
is an integer.
where P(x)
(ux”+ fJx+cy
(t&g
u+(y)
g
g
+(g
ds
g
are useful for evaluating is a polynomial
and
3.3.16 +...+~)d~rg+...+cg
dx 2 (ax2+br+c)=(4c&c-bz)~
S dX -&=1g
3.3.9
(b2--4uc
a
2az+b-
3.3.17
I 2azfbf
#x -d2y dy -3 dy2=dr2 0 zc
3.3.10
$=
-B
g-3
Integration
3.3.12
(g--j
($)-"
-2
S
by Parts
S
3.3.20
jidu=w+dti
(a+ bx$c+dxj=k
jkudx=(jidx) Integrals
v-s(judx)
of Rational
(Integration
Algebraic
2 dx
dx 1 =- arctan E! u2+b2;C2 ub U
Functions
3.3.22
const,ants are omitted)
S(ux+f,)"dx=(ux+6)"+1 4n+1> S S S (n#-1)
3.3.24
3.3.15
$&In
)ax+b)
S
t(u+bx)
of Irrational
dx =h2 (c+dx)11’2
3.3.27
=+
3.3.28
=h2
3.3.29 3.3.30
(x2;1-“a2)2=&
arctan
~+20~(xf+U2)
3.3.25
Integrals
3.3.26
c+dx bc In I-a+bx I
dx (a+bx)“P(c+dx)=[d(bc~ud)]1~2
S
=[d(ad&]1/2
Algebraic
arctan arcsin
(ad # bc)
S___ lnb2+ b2x21 S S
3.3.21
3.3.23
3.3.14
(P-4ac=O)
3.3.19
S 3.8.13
(b2-4m>O)
=2az+b
3.3.18 3.3.11
(b2--4uc)t (P-4acy
Functions
-d(a+
1 W
bx) 1’2
C b(c+dx) 2bdx+ad+ bc-ad
bc >
ln J[bd(n+ bx)]1/2+ b(c+dx)1/21 arctan ~~~~-J” d(u+bx)1’2-[d(ad-bc)]1’2 In d(a+bx)1’2+[d(ud-bc)]“2
(b>O, dO)
(d(u&-bc)
I
@b-J--4>O)
ELEMENTARY
If zn=un+ivn, 3.7.23
then ~~+l:=u,+,+iv~+~
u,+~=xu~-~v,;
ANALYTICAL
where
3.8. Algebraic Solution
v,+,=xv,+yu,
9?z” and 92” are called harmonic
17
METHODS
polynomials.
3.7.24
Equations
of Quadratic
3.8.1
Given az2+ bz+c=O,
21,2=-
0-
2”, j-k
gf, p= b2-4ac, .zi+,za= -b/u,
3.7.25
If a>O, two real roots, p=O, two equal roots, a
Roots
3.7.26
z*=&=rte+rs=r+
cos @+iri
sin $0
Solution
If --?r<~< ?r this is the principal root. The other root has the opposite sign. The principa: root is given by 3.7.27
d=[+(r+x)]+&-i[$(r--x)]*=ufiv
where sign is taken tc
2uv=y and where the ambiguous be the same as the sign of y. 3 . 7 . 28
I
l&l-I221
I
Let sl=[r-+(q3+r2)q+,
Catwhy-Riemann
Equations
au -=-,
av
au -=--
av
a~
by
by
ax
;,
; f$=-$
a
If z=Tefff, 3.7.31
sz=[T’-((p3+?3*]*
then
f(z)=f(x+iy)=u(x,yy)+iv(x,y)whereu(x,y),v(z,y aA real, is unaly& at those points z=z+$ which 3.7.30
Equations
one real root and a pair of complex c.onjugate roots, $+9=0, all roots real and at least two are equal, . p3+r2<0, all roots real (irreducible case).
_<1z1~~2111211+I~21
Functions,
roots.
$+P>O,
root if - ?r<0 5 7r) (k=l,2,3, a . ., n-1)
Inequalities
Complex
If
of Cubic
Z~Z~=C/U
Given Z3+a2z2+ulz+a0==0, let
3.8.2
Zl/n,Tl/nefe/n , (principal
Other roots are Pet(B+2rn’ln
3.7.29
Equations
If zl, z2, z3 are the roots of the cubic equation g=;
Z~+Z~+Z~=-CIC~
Laplace’s
~~~2+~$,+&~,:=~~
Equation
The functions U(X, y) and v(x, y) are callec harmonic functions and satisfy Laplace’s equation Cartesian
Polar
3.7.33
r ;
(r g)+$=r
of Quartic
Equations
Given 24+~323+a3~~+~1l~+ag=O, real root u1 of the cubic equation
3.8.3
Coordinates
r
i!?+$E~2+g2=o
3.7.32
Solution
U3
-
a2U2
i-
(~23
-
4ao)U
-
(a; j- Uoa$ - kW3)
find the =
and determine the four roots of the quartic solutions of the two quadratic equations
Coordinates
;
(r g+g=o
0
as
18
ELEMENTARY
ANALYTICAL
If all roots of the cubic equation arc real, USC the value of U, which gives real coefficients in the *quadratic equation and select signs so that if
METHODS Method
of Iteration
then
-a3,
4142==0.
Convergence
of
an
fcxk)
xk
2, . . .).
Approximution
Process
(n=l,
bk)
will converge quadratically to x=5: (if instead of the condition (2) above), (1) Monotonic convergence, f(zO)r’(zo) >0 and f’(s), j”(z) do not change sign in the interval (Q, t), or (2) Osdato y conwgence, f(xJf” (x0)<0 and f’(s), f”(z) do not change sign in the interval (x0, x1), xo
3.9.6 x=N’l”
3.9.2 Let zl, z2, x3, . . . be an infinite sequence of approximations to a number f. Then, if 1%n+~-~I<&n-tlk,
Rule
f’
Then, if f’(z)>0 and the constants cn are negative and bounded, the sequence x,, converges monotonically to the root [. If c,,=c=constantO, then the process converges but not necessarily monotonically. of
Approximations
Methods
3.9.1 Let z=zl be an approximation to x=[ where f(t) =0 and both x1 and [ are in the interval a$r
Degree
of Successive
xk+l=
(n=l,
b.
If z=zk is an approximation to the solution z= I of f(z) =0 then the sequence
Comments
GI+1=G+C&n)
for aLzSb,
Newton’s
Approximation
General
Method
will
3.9.5
-&,
z1z2z3z4=ao.
Czj2,=u2,
3.9. Successive
Newton’s
z2 j2,2t=
IF’(s)J
(2) a
If zl, z2, z3, z4 are the roots, z2 j=
Substitution)
3.9.4 The iteration scheme Q+~=F(z~) converge to a zero of z=F(z) if (1)
pl+p2=a2,plp2+pl+q2=a2,p~q2+p2q~=al,
(Successive
Method
Applied
to Real
nth
Roots
Given x”=N, if zk is an approximation then the sequence xk+l=-
;
[$i+(n-l)xk]
2, . . .) will converge quadratically
to a.
where A and k are independent of n, the sequence is said to have convergence of at most the kth degree (or order or index) to [. If k=l and A<1 the convergence is linear; if k=2 the convergence is quadratic. Regula
Falsi
(False
Position) Aitken’e
3.9.3 Given y=f(z) to find 5 such that f(.$)=o, choose ~0 and x1 such that f(rO) and f(zl) have opposite signs and compute x*=x,
-Hi
f,=
f 1~o-JoX, jlVfO
for
Acceleration
of
Sequences
3.9.7 If 2k, &+I, zri+2 are three successive iterates in p, sequence converging with an error which is approximately in geometric progression, then
* &=xk-
Then continue with x2 and either of x0 or x1 for which f(;ro) or j(zl) is of opposite sign to f(zl). Regula falsi is equivalent to inverse linear interpolation,
G-Process
(5k--k+1)*=;tk~k+2-2:+1.
A*&
is an improved
estimate
OGtk) then Z=s+O(P),
A*Xk
’
of x. Ix\<~.
In fact, if zk”x+*
ELEMENTARY
ANALYTICAL
METHODS
* 3.10.
Theorems
on Continued
Fractions
A,B,_l-A,-lB,=(-l)n-’
(4)
(5)
Definitions
kiI al;
For every n>O, j,=b,
1
claI
ClC&
c2c3a3
&I-lW%
c,bl+ czbz+ caba+ ’ * * c,b,’ (6)
l+b,+b,b,+
. . . +bzb3. . . b, bz =-- 1 l- b,+l-
=b,,+&e&.
..
. . . +;=-&
d+$+ If the number of terms is finite, j is called a ternlinating continued fraction. If the number of ternls is infinite, j is called an infinite cont’inued fraction and the terminating fraction
1 --a0
x+A ...l
_- b3 b,+l-
--&
I
. . . $yu
1
2
t(-1,n----5
aoGa2
aof
1
=-
aox
n1
n
_ . . . a, _- a12
___ al-x+
uo+
b, * ’ ‘--b,+l
I12-xf
%-1X
* . . +un-2
is called the nth convergent of j. A (2) If lim -A exists, the infinite continued fracIt-+- 88 If uf= 1 and the tion j is said to be convergent. bt are integers there is always convergence. Theorems (1)
Example 7. MultiplyM=20243 97459 71664 32102 by m=69732 82428 43662 95023 on a 10X10X20 desk calculating machine. Let MO=20243 97459, Ml=71664 32102, mO= 69732 82428, ml=43662 95023. Then Mm= M0m0102’+ (Mom,+Mlmo) 101o+M~ml. (1) Multiply ,W1m1=31290 75681 96300 28346 and record the digits 96300 28346 appearing in positions 1 to 10 of the product dial. (2) Transfer the digits 31290 75681 from positions 11 to 20 of the product dial to positions 1 to 10 of the product dial. (3) Multiply cumulatively M,mo+Mom,+31290 75681=58812 67160 12663 25894 and record the digits 12663 25894 in positions 1 to 10. (4) Transfer the digits 58812 67160 from positions 11 to 20 to positions 1 t,0 10. (5) Multiply cumulatively Mom,+58812 67160 =14116 69523 40138 17612. The results as obtained are shown below, 9630028346 1266325894 14116695234013817612 141166952340138176121266325894963~?28346
If the product Mm is wanted to 20 digits, only the result obtained in step 5 need be recorded. Further, if the allowable error in the 20th place is a unit’, the operation MImI may be omitted. When either of the factors M or m contains less than 20 digits it is convenient to position the numbers as if they both had 20 digits. This multiplication process may be extended to any higher accuracy desired. 8. Divide N=14116 69523 40138 17612 by d=20243 97459 71664 32102. Method (1)--linear interpolation.
Method @)--If N and d are numbers each not more than 19 digits let N=N1+NolOQ, d=dI+ dolO where No and do contain 10 digits and N, and dl not more than 9 digits. Then N NolOQ+N, zs- 1 [.N-y] d=,lOQ+d, dolO Here N= 14116 69523 40138 1761, d=20243# 97459 71664 3210 No= 14116 69523, do=20243 97459, d,=71664 3210 (1) NodI= 10116 63378 42188 8830 (productdial). (2) (Nod,)/do=49973 55504 (quotient dial). (3) N- (N&/d,= 14116 69!;22 90164 62106 (product dial). (4) [N- (NodI)/do]/dolOQ= .69732 82428=first 10 digits of quotient in quotient dial. Remainder =r=O8839 11654, in positions 1 to 10 of product dial. (5) r/(d010Q)=.43662 9502.10-“O=next 9 digits of quotient. N/d=.69732 82428 43662 9502. This method may be modified to give the quotient of 20 digit numbers. Method (1) may be extended to quotients of numbers containing more than 20 digits by employing higher order interpolation. 9. Sum the series S= l-&+*-i to 5D using the Euler transform. + The sum of the first 8 terms is .634524 to 6D. If u,=ljn we get Example
n 9
%7 . 111111
10
. 100000
Difference X.71664 32102=24685 64402&10-*O (note this is an 11 X 10 multiplication). Quotient= (69732 82430 90519 39054-246856 44028).10-20 =.69732 82428 43662 95028 There is an error of 3 units in the 20th place due to neglect of the contribution from second differences.
Evaluating the integrals in numerical integration we get
(-l)f
g
dt.
where f(k)
the last
sum
by
$, k-2=1.54976 s
= (k+10)-2.
...
Thus,
7731+.1
- .005 + .00016 6667 - .OOOOO0333 = 1.64493 4065,
Ic
as compared
1.85194
Example
.43379
12.
A
A2
A3
A4
4067.
Compute
arctanx=$3+g7+
.25661 . 18260
with $=1.64493
to 5D for x= .2. Here n>l, &,=O, b,=2n-1,
x2 4x2 9x2 al=x, A-l=l,
. . .
an=(n-l)2x2 for Bdl=O, A,,=O,
. 14180
-2587 .11593
799 - 1788
-321
478
.09805
-1310
- 168
.08495
A0 -= Bo ’
153
310 - 1000
A -r,*g
.07495 The sum to k=3 is 1.49216. Applying Euler transform to the remainder we obtain
Bl
the
A=.197368
f (.14180)-h
(-.02587)+&
B2
(.00799)
A3
-;
(-.00321)+$
= .07090 + .00647 + .00100+ .00020 + .00005 = .07862 We obtain the value of the integral compared with 1.57080. Example
11.
as 1.57018 as
Sum the series $I kep==f P the Euler-Maclaurin summation formula. From 3.6.28 we have for n= a,
B=.197396
(.00153)
using
3
[II A4
3.032
Bq = 15.36
Note that in carrying out the recurrence method for computing continued fractions the numerators A, and the denominators B, must be used as originally computed. The numerators and denominators obtained by reducing An/B, to lower terms must not be used.
ELEMENTARY
ANALYTICAL
23
METHODS
References Texts
[3.1] R. A. Buckingham, Numerical methods (Pitman Publishing Corp., New York, N.Y., 1957). [3.2] T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948). [3.3] L. Fox, The use and construction of mathematical tables, Mathematical Tables, vol. 1, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1956). [3.4] G. H. Hardy, A course of pure mathematics, 9th ed. (Cambridge Univ. Press, Cambridge, England, and The Macmillan Co., New York, N.Y., 1947). [3.5] D. R. Hartree, Numerical analysis (Clarendon Press, Oxford, England, 1952). [3.6] F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1956). [3.7] A. S. Householder, Principles of numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [3.8] L. V. Kantorowitsch and V. I. Krylow, Naherungsmethoden der Hoheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1956; translated from Russian, Moscow, U.S.S.R., 1952). [3.9] K. Knopp, Theory and application of infinite series (Blackie and Son, Ltd., London, England, 1951). [3.10] Z. Kopal, Numerical analysis (John Wiley & Sons, Inc., New York, N.Y., 1955). [3.11] G. Kowalewski, Interpolation und genaherte Quadratur (B. G. Teubner, Leipzig, Germany, 1932). [3.12] K. S. Kuns, Numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1957). [3.13] C. Lanczos, Applied analysis (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1956). [3.14] I. M. Longman, Note on a method for computing infinite integrals of oscillatory functions, Proc. Cambridge Philos. Sot. 52, 764 (1956). [3.15] S. E. Mikeladze, Numerical methods of mathematical analysis (Russian) (Gos. Izdat. TehnTeor. Lit., Moscow, U.S.S.R., 1953). [3.16] W. E. Milne, Numerical calculus (Princeton Univ. Press, Princeton, N.J., 1949). [3.17] L. M. Milne-Thomson, The calculus of finite differences (Macmillan and Co., Ltd., London, England, 1951).
[3.18] H. [3.19]
(3.201 [3.21]
[3.22] [3.23]
[3.24] [3.25]
Mineur, Techniques de calcul numerique (Librairie Polytechnique Ch. B&anger, Paris, France, 1952). National Physical Laboratory, Modern computing methods, Notes on Applied1 Science No. 16 (Her Majesty’s Stationery Office, London, England, 1957). J. B. Rosser, Transformations to speed the convergence of series, J. Research NBS 46, 56-64 (1951). J. B. Scarborough, Numerical mathematical analysis, 3d ed. (The Johns Hopkins Press, Baltimore, Md.; Oxford Univ. Press, London, England, 1955). J. F. Steffensen, Interpolation (Chelsea Publishing Co., New York, N.Y., 1950). H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Inc., New York, N.Y., 1948). E. T. Whittaker and G. Robinson, The calculus of observations, 4th ed. (Blackie and Son, Ltd., London, England, 1944). R. Zurmtihl, Praktische M.athematik (SpringerVerlag, Berlin, Germany, 1953).
Mathematical
Tables
and
Collections
of Formulas
[3.26] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution, Wa,shington, D.C., 1957). [3.27] L. J. Comrie, Barlow’s tables of squares, cubes, square roots, cube roots a,nd reciprocals of all integers up to 12,500 (Chelmical Publishing Co., Inc., New York, N.Y., 1954). [3.28] H. B. Dwight, Tables of integrals and other mathematical data, 3d ed. (The Macmillan Co., New York, N.Y., 1957). [3.29] Gt. Britain H.M. Nautical Almanac Office, Interpolation and allied tables (Her Majesty’s Stationery Office, London, England, 1956). [3.30] B. 0. Peirce, A short table of integrals, 4th ed. (Ginn and Co., Boston, Mass., 1956). [3.31] G. Schulz, Formelsammlung zur praktischen Mathematik (de Gruyter and Co., Berlin, Germany, 1945).
24
ELEMENTARY
ANALYTICAL
POWERS
Table 3.1
AND ROOTS nk
k : 3 4 : 7 fz
10 24
nl=
for use
See Examples 1-5
T&L n3, n‘L 5 12= .L
16
n7==
128
of the table. Floating decimal notation: 910=3486784401 = (9)3.4867 84401
The numbers in square brackets at the bottom of the page mean that the maximum error in a linear interpolate is a X 10-P (p in parentheses), and that to interpolate to the full tabular accuracy 11~ poJnts must be used in Lagrange's and Aitkens methods for the respective functions W'. *See page xx.
The author acknowledges the assistance of Lois K. Cherwinski io the preparation and checking of the tables.
and Elizabeth
F. Godefroy
.
4. Elementary Logarithmic,
Transcendental
Exponential,
Circular
Mathematical 4.1. Logarithmic Integral
4.1.1
and Hyperbolic
IFunctions
Properties
Function
Representat
In z=
Functions
Logarithmic
4.1.6
ion
Identities
Ln (z,z2)=Ln
zl+Ln
z2.
(i.e., every value of Ln (z1z2) is one of the values of Ln zr+Ln z2.)
z dt 17
S
4.1.7
In (z1z2)=ln q+ln
z2
iY
(--?r
n
0
1
x
-I
4.1.8
Ln z=Ln
4.1.9
In z=ln
zl--Ln z,-ln
z=r+iy
FIGURE
4.1.
(a not an integer or zero.)
where the path of integration does not pass through the origin or cross the negative real axis. In z is a single-valued function, regular in the z-plane cut along the negative real axis, real when z is positive. z=x+iy=reie. 4.1.2 4.1.3
In z=ln r+iO r= (x2+y2)*,
(-*
Ln zn=n Ln 2
4.1.11
In P=n In 2 (n integer,
4.1.13
In 0=-
4.1.14
ln (-l)=*i
4.1.15
In (&i)=f& In e=l,
03
6 dt
1 t=’
“=2.71828 18284.. . (see 4.2.21) Logarithms
4.1.18
S
arg 25~)
S
The general logarithmic function is the manyvalued function Ln z defined by Lnz=
-s
e is the real number such that
e=arctan f.
4.1.4
(n integer)
In l=O
x=r cos 0, y=r sin 0:
z dt 1t
z2
(see chapter 1)
Values
4.1.12
4.1.16
zl-arg
;z2
4.1.10
Special
z217r)
z2
(-r
Brunch cut for, In z and 9.
zl+arg
to General
log, z=ln
Base
z/in 1s
4.1.19
where the path does not pass through the origin. 4.1.20 4.1.5 Ln (re@)=ln (refe)+2kti=ln k being an arbitrary integer. principal branch of Ln z.
r+i(e+akr), In z is said to be the
4.1.21 4.1.22
log, z=ln 2 log,, z=ln z/in lO=log,, e In z = (.43429 44819 . . .)ln 2 67
68
ELEMENTARY
TRANSCENDENTAL
4.1.23 ln z=ln 10 log,, z= (2.30258 50929 . . .) loglo z (log, x=ln x, called natural, Napieriun, or hyperbolic logarithms; log,, x, called common or Briggs logarithms.) Series
4.1.24
Expansions
FUNCTIONS
4.1.35 4.1.36 4.1.37
In (1+2)=2-~2z++23-. (/z/11
Iln (l--x)\<
. . and z#-1)
4.1.38
, . .
4.1.39
111
In x
($J+
/In
4.1.26 In z=(z-1)-~(z-1)*+Q(z-l)3-.
(1+2)(:2-ln
(Iz-1111,
ZZO)
(sy+.
. .]
(sy+i
(922 In (%)=2
(k+&+&,+
a+2
(z in the cut plane of Figure Polynomial
zffl)
Approximations
log10 x=alt+u3t”+t(x),
(&>
(a>O, Limiting
4.7.)
*
t=(x-1)/(x+1)
al= .86304
(&J5+
* . .]
a,=.36415
4.1.42
921--a#z) log10 x=a~t+a~t~+a~t~+a,t’+u~t”+~(x)
Values
lim x-0 In x=0 z+m (a constant,
4.1.31
..
le(x)l56XlO-’ [(&)+f +f
4.1.30
2 42 42 92
422 922 . . =---2;: 22 --. l-- 3- 5- 7-
4.1.29 In (z+a)=ln
Fractions
4.1.41
. . .) jl421,
W<1)
(l-120
4.1.40
z#O)
0,
n
(z in the plane cut from - 1 to - co)
. .
4.1.27 In 2=2 [(s)+i
(XX)
I’*-- 1) for any positive
In (1+z)=*&GG5+6+.
(92 13)
4.1.28
2_<2-1
Continued
(?J+a
(O
(XX)
4.1.25 In z=(G)+:
g
lim xa In x=0 z--10 (CYconstant,
t=(x-.1)/(x$1) je(x)/_
9&~>0) aI=.
1718
u3= .28933 5524
2&~>0)
a’= .09437 6476 f&=.19133
7714
as= .17752 2071
4.1.32 4.1.43 lim $I i-ln m-+m(
m =y (Euler’s >
=.57721 56649 . . . (see chapters 1, 6 and 23) Inequalities
4.1.33
ex
U+X)-1,
4.1.34
z<-In
(l--2)<
01:x<
1
constant)
x#O)
& (x
‘XZO)
In (1+x)=u1x+~x2+u3x3+u4x4+usx5+~(x) Is(x)/ 5 1 x 10-s UI= .99949 556
a4= - .13606 275
a2= - .49190 896
as= .03215 845
a3= .28947 478
* The approximations ings, Jr., Approsimations ,Univ. Press, Princeton,
4.1.41 to 4.1.44 are from C. Hastfor digital computers. Princeton N.J., 1955 (with permission).
(See chapters 5, 7 and 29 for other integrals involving exponential functions.) 4.3. Circular
4 The a proximations 4.2.43 to 4.S1.45 are from B. Carlson, M. E oldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LIA-1943, Los Alamos, N. Mex., 1955 (with permission). 5 The approximations 4.2.46 to 4.2.47 are from C. Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N.J., 1955 (with permission).
6 The approximations 4.2.48 are from C. W. Clenshaw, Polynomial approximations to cxlementary functions, Math. Tables Aids Camp. 8, 143-147 (1954) (with permission).
72
ELEMENTARY
TRANSCENDENTAL
4.3.17
tan 2=%
4.3.3
FUNCTIONS
43.18 csc
4.3.5
SW z=-
4.3.6
1 cot z=- tan 2 Periodic
4.3.7
cos
(2
2
4.3.19
1 cos 2
z2
tan
2
2
(zl+z2,=“~~t~;“;i”,,“zz’
cot
2
Half-Angle
1
Formulas
4.3.20
sin $= &(qc)’
4.3.21
cos ;= *(
4.3.22
t,an i= &(~~>1=~=~2
Properties
2 +2/h)
(k any integer) = cos
2
tan (z+&r)=tan
4.3.9
tan z,+tan 1
(zl+z2)=cAtan
z1 sin z2
sin 2
sin (2+2h)=sin
4.3.8
tan
q cos z2-sin
(z~+~~)=c~s
1
4.3.4
2=-r--
cos
2
+=)”
The ambiguity in sign may be resolved with the aid of a diagram.
V
,
I Transforkation
of Trigonometric
Integrals
If t,an %=z then 4.3.23
22
sinx co3 x tonx
FIGURE 4.3. Relations
Circularfunctions.
Between
Circular
Functions
4.3.10
sin2 z+cos2 z=l
4.3.11
sec2 z-tan2
2=1
4.3.12
csc2 z-cot?
z=l
Angle
sin (--)=-sin
4.3.14
cos (- z)=cos tan (--)=-tan Addition
4.3.16
sin 22=2 sin ,a cos 2=,“,“t”,“,~2
4.3.25
cos
4.3.26
tan 22=--= 2 tan 2 l-tan22
sin (zl + z2)=sin
co9 z-1=1-2
sin2 z l-tan2 2 =cos2 z-sin2 z= 1+tarP 2
2~2
2 cot
2
cot22-1
2
=-cot z-tan
4.3.27
sin 3z=3 sin z-4 sin3 2
4.3.28
cos 3z=-3
4.3.29
sin 42=8 cos” 2 sin 2-4 cos 2 sin 2
4.3.30
cos 42=8 co& z-8 Products
cos z+4 cosa 2
of Sines
cos2 23-1 and
Cosines
2 sin z1 sin zz=cos (ZI-2&-co8
(21+22)
Formulas
4.3.13
4.3.15
Formulas
4.3.24
4.3.31 Negative
du
cos u=l+zz
Multiple-Angle
1 70
-------
l-z2
sin u=--, 1+z2
2 2 2
Formulas
z1 cos z2+cos z1 sin 22
4.3.32
2 cos
4.3.33
2 sin
Addition
and
z1 z1
cos Z:~=COS (a-z2)+cos
cos z:!=sin (zl-z2)+sin
Subtraction
of
Two
Circular
4.3.34
sin zl+sin
z2=2 sin(‘+)
cOS(y)
(21+22) (21+22)
Functions
2
ELEMENTARY
TRANSCENDENTAL
4.3.41 cos2 a-cos2
4.3.35
sin zl-sin
z2=2 cos(F)
sin(v)
73
FUNCTIONS
z2=-sin
(zl+z2) sin (~~-2~)
4.3.36
4.3.42 cos2 zl--sin2 z2=cos (z1+z2) cos (zl-z2) cos 21--cos z2=-2
sin(Tk)
sin(y)
4.3.43 Signs of the Circular Functions in the Four Quadrants
4.3.38
z2= sin (215 z2) cos 21 cos 22
tan z,i-tan
Quadrant
4.3.39
sin csc ___~-
(~2fzJ cot 21&cot z2= sin . sin z1 sin z2 Relations
