ERRATA AND ADDITIONS: SECOND SECOND EDITION EDITION September 2013 COMPLEX VARIABLES, INTRODUCTION AND APPLICATIONS
M.J. Ablowitz and A.S Fokas Cambridge University Press, 2003 Corrections and small additions p. 4 line 8 from top: after after “ ..by arg z.” add “We “We take the standard conve conventio ntion, n, countercounter-
clockwise is the positive direction.” p.7 Problem 2c, replace: cos(z cos( z ) = (eiz + e−iz )/(2) by cos z = = (eiz + e−iz )/2 p.8 Problem 4. Spelling: change: “Estabilish” to “Establish” p.19 First First two lines should read: “Use these results results to deduce deduce where the pow p ower er series for
sin2 z and and sechz sechz can can be expected to converge.” p.26 Section 1.3.1 should be titled 1.4 to be consistent with Table of Contents. p.33 16 lines from top: eliminate “exist and”; now reads: “...must be differentiable..” p.34: line 4 from top: eliminate eliminate “after “after noting noting Eq. (2.1.2) (2.1.2) and manipulating. manipulating.”” and replace replace
“∆z “∆z approaching approaching zero” by “∆x, “∆ x, ∆y approaching zero”. p.39 In Example Example 2.1.6, 2.1.6,
1st line line after after equations equations replac replacee “ψ “ψ(x)” by “ψ “ψ(y)”
p.45 Problem Problem 7, 4th line, before Establish Establish add: “Assuming “Assuming the necessary necessary differenti differentiabilit ability” y”
establish the following: p.47 line 8 from bottom: just before “The semiaxis..” add “In fact , for 0 < θ p < 2 < 2π, π, r = 0,
the function is analytic (from the Cauchy-Riemann conditions).”
p.48 4 lines from top, replace “On the other hand, if we took..” by “On the other hand we
reiterate reiterate,, if we took...” p.66 line 11 from top, replace: “Riemann surface with a cut along the positive x-axis.” by
“Riemann surface on which each sheet connects smoothly.” 1
p.71 lines 8,9 from bottom, replace “..there is a set of points...of the interval.” to: “..there
is a point (x(t), y(t)) that yields the image point z (t). p.76
line 4 from bottom, after dz = rieiθ dθ add: “ and since the curve is simple we can take 0 < θ < 2π (on same page) line 2 from bottom, replace “any closed curve” by “any simple closed curve” p.84 2 lines from bottom, replace “over z” by “ of f (z ) over the simple contour C p.88 line 10 from top, replace “a small, but finite circle of radius r” by “a small circle of
radius r > 0 p.98 line 8 from top: after harmonic functions” add: u = Ref or v = Imf, f = u + iv, p.99 In Theorem 2.6.7: change: “. . . bounded by a simple closed contour C , then at any
interior point z ” to “. . . bounded by a simple closed contour C , and if f is continuous on C , then at any interior point z ” p.111
line 6 in proof of Theorem 3.1.1, after “n > N,” add: “N independent of z ” (on same page) line 8 in proof of Theorem 3.1.1, before “Continuity” add: “Now take an n > N .” (on same page) line 10 in proof of Theorem 3.1.1, omit “n > N,”; line should now read: “Thus for z
| − z | < δ ” 0
p.113
Line 3 change: “Theboundedness. . . ” to “The boundedness. . . ” (on same page) In the two equations following line 3, change: “ b1 (z ) < B hence bn (z ) < BM n−1 ” to “ b1 (z ) B hence bn (z ) BM n−1 ”
|
|≤
|
|
|≤
|
|
|
(on same page) line 5 above Problems, after Theorem 3.1.2 add: “(proven in section 3.4)” p.114 Problem 5b, replace “R < Re z
|
| ≤ 1 by “R < Re z ≤ 1
p.131 line 10 from top: replace equation: f (z ) =
0 C n n=−∞ (z −z0 )n
p.135 Near top of page; in the formula for C n , change
1 2n+1
to
by f (z ) =
−1 2n+1
C n n=0 (z −z0 )n
∞
p.137 Problem 7; 2nd line: after the formula for E (z ) or to the right of E (z ), add: Re z > 0;
in the formula for R n (z ) change ( )n+1 to ( 1)n+1 ; in Problem 8 line 1: after “z = x real”,
−
−
2
add “ x > 0.” Line 5 omit: “Explain why this approximation holds true for Rez > 0.” p.139
line 1 of Theorem 3.4.3 replace “... by “...
≤ M in some region R with M a sequence of constants.” j
j
≤ M , with each M , j = 1, 2... constant, in some region R.” j
j
(on same page) last line of proof replace “Theorem 3.4.2 follows.” by “Theorem 3.4.3 follows.” p.141
line 6 from top replace “ ζ
| − z | > ν ” by “|ζ − z | ≥ ν ”
(on same page) line 11 from top after “uniformly” add tof (z ) and after “and” add “then employing” (on same page) line 14 from top eliminate “all” p.142 line 5 from top before “has no” add: “x = mπ, m integer,”
p.145 Example 3.5.2, first line. Replace: “Describe the singularities of the function ” by
“Describe the singularity of the function at z = 0 ” p.146 Example 3.5.3 The line after the formula, change: “Here the function f (z ) has simple
poles with strength 1..” to “Here we will show the function f (z ) has simple poles with strength -1..” p.148
3 lines above Eq. (3.5.5) after “. . . . . . f ( z ) 1.”
→ 0 as r → 0.” add: “Also for θ = ±π/2, |f (z )| =
(on same page) 2 lines above Eq. (3.5.5) change “. . . namely, r = (1/R)cos θ (i.e the points...” to “...namely, r = (1/R)cos θ, R = 0, (i.e., the points. . . ”
(on same page) last two lines, change: “Thus f (z ) may take on any positive value other
|
|
than zero by the appropriate choice of R” to “Thus f (z ) may take on any positive value in the neighborhood of z = 0”. p. 160 Line 2 replace
∞ M < k k=1
p. 162 First line; change:
|a (z )| < M ...” k
∞ by
|
|
∞ M < k k=N
∞ “If we assume that | a (z )| ≤ M ... k
k
3
k
” to “If we assume that
p.167
In the denominator of the right side of the first equation, change: (z 2 3 (z π z + )
−
3!
···
−
(πz )3 3!
+
·· ·) to
p.181
In Theorem 3.7.3: change: “. . . simply connected domain D, then the linear. . . ” to “. . . simply connected domain D containing z 0 , then the linear. . . ” p.185
line after Eq. (3.7.41), before: “(z = 0 can be translated to z = z 0 if we wish) insert: “z = 0, ωm,n ”
(on same page) line after Eq. (3.7.43), before “The function . . . ” insert: “Alternatively, by taking the derivative of Eq.(3.7.42) w satisfies “w = 6w2 g2 ”.
−
2
p.186 line immediately after Eq. (3.7.45) insert (no new paragraph): “Also note that w1
satisfies the second order ODE w1 = 2k 2 w13
2
− (1 + k )w .” 1
p.192 Problem 5 part (a)line 3 (following equation w = ...), after “for z near z 0 ’ ’add: “with
r > 0.” 2
2
p.193 Equation in problem 5 part (d); change “ ddzw2 = zw3 + 2w” to “ ddzw2 = 2w3 + zw.” p.198 2nd line above Example 3.8.2 change “. . . time T with . . . ” to “.. . distance with
...” p.201 In Fig. 3.8.5 vertical axis label should have Log 10 of Error””. p.206 3rd line from bottom change “... lying in D.” by “...lying in D and enclosing z 0 .” p.207
Inside Fig. 4.1.1: the contours (on same page)
N should
−C , −C ,..., −C 1
2
be replaced by
−C˜ , −C˜ , ..., −C˜ 1
2
N
In Theorem 4.1.1 equation (4.1.4) replace loop integral with loop integral with subscript C such as the one in eq. (4.1.3) (on same page) Lines 3,5 from the bottom, the contours:
N should
C , C ,..., C 1
2
be replaced by ˜1 , ˜2 ,..., ˜N
C C
C
p.208
In the integral in the right hand side of eq. (4.1.5) the contour 4
C should be replaced by C˜ j
j
p.240 after z = z 0 + eiθ and before “to find” insert: “ and taking θ from 0 to φ, ” p.256 Problem 7a replace 0 < k < 2 by 0 < k < 1 p.257 Problem 14, 3rd line, change: “. . . where C R is the . . . ” to “. . . where C R is the outside
part of the . . . ” p.258 Problem 14, part (c): left hand side add dx in the integrand; change the sign of the
right hand side: from “= πbn+2 ” to “=
−πb
n+2 ”
p.260 1 [arg f (z )]C to 2πi 1 change 2πi [arg w]C ˜ to
In equation (4.4.3) the furthest righthand side member, change In equation (4.4.4) the furthest righthand side member,
1 [arg f (z )]C 2π 1 [arg w]C ˜ 2π
1 ˜ ...” In the line below equation (4.4.4) change “ 2πi [arg w]C ˜ is called the winding number of C ˜ ...” to “ 1 [arg w] ˜ is called the winding number of C C
2π
p.266 problem 6. Change the last two lines from: “Consider the two functions
f (z )
− f , and use . . . 0
0
and
to deduce that f (z ) = f 0 .” to: “Consider the two functions
0
and f (z ). Then Rouch´e’s Theorem implies that the functions number of zeroes.
−f
−f
−f , f (z ) − f have the same 0
0
p.268
line 10-11 from top, omit: “(sometimes referred to as bounded mean oscillations (BMO))”; (on same page) line 12 from top, omit “(i.e. in BMO)” p.270 In the right hand side of of eq. (4.5.10) replace δ (x
− x ) by ∆(x − x ; ) 0
0
p.272 In eq. (4.5.17) 2nd line replace e ikx g(x ) by e −ikx g(x )
(on same page) p.272 2 lines after eq. (4.5.18) replace f (x) = δ (x
− x) by f (x) = δ (x)
p.280
In problem 7a replace e ωx by e −ωx p.282 In Problem 13a,b omit a > 0 p.292
Line below eq. (4.4.6.28) change sin kx = (eikx
5
ikx
− e−
)/2 to sin kx = (eikx
− e−
ikx
)/(2i)
2 π
(on same page) in the first line of eq. (4.6.30) change
2 ∞ −η2 ∞ dη to √ 2π √ e−η dη √ e
x
x
t
2
2
t
p.329 line 8 from top before “(See also...).” add “Note, in the above Eq. when x 2 + y 2
for y > 0, y < 0, tan−1 [ ]
· → 0, π respectively.” √ p.339 Problem 9 line 4 replace ζ ≡ (z + z − c ) by ζ = 1 2
2
2
1 (z + (z 2 2
2
−c )
1 2
→ 1,
)
p.341
line 10 from bottom replace “one value for g(z ) = w inside” to “one value z for which g(z ) = w
−w
0
= a corresponding to every z
− w = a inside” 0
(on same page) replace ‘Rouche” by “Rouch´e” everywhere p.342
line 14 from bottom replace “< 1 and therefore F (w) is continuous.” by “< 1 . Now let δ 1 be small enough so that w
| − w | < δ is in P and |F (w) − F (w )| < . Since is arbitrary 1
1
1
1
1
and there is a corresponding δ 1 > 0, therefore F (w) is continuous.”
(on same page) line 6 from bottom after “only one solution” add “ counting multiplicity” p.343 line 4 in proof replace “to an arbitrary point z 0
point z 0
∈ D and is not a point ...” to “to a
∈ D which is not a point ...”
p.349 line 6 from top change “...a point z = z l ..” to “...a point z = a l ..” z
p.361: Problem 3 should read: “Show that the function w =
0
of the unit circle in z to a regular hexagon in w.”
dt (1
z
p.365: Problem 12 should read: “Show that the function w =
6
−t ) (1
1 3
maps the interior
4 1/2
− ζ )
dζ maps the ζ 2 interior of the unit circle in the z -plane to the exterior of a square in the w-plane.” 1
p.371 Bottom p. 371– 2nd line from bottom, after “w plane” add: “obtained under the
bilinear transformation”. p.372 In property (vii) replace “Then Either...” by “Then under a bilinear transformation
either...” p.401 line 6 from bottom in the equation for g(z ) + 1 replace (z z 2 )3 by (z
·
t k
p.426 In Example 6.2.4 replace e− by e −
3
− z ) 2
t
p.459: Before line 10, “Next we outline the procedure...”, add the paragraph: “From the
6
above result, we can can determine the asymptotic behavior of the Hankel function of the (2)
(1)
second kind: H ν (z ) = H ν (z ) and similarly of the Bessel functions of 1st, 2nd kind J ν (z ), (1)
Y ν (z ) from H ν (z ) = J ν (z ) + iY ν (z ).” p.463 In Example 6.4.5 line above first set off equation: replace y
→ ±iπ by y → ±π
p.471 In two places on the page (middle of the page and after formulae for I 1 (k)) spelling
correction: change “principle value” to “principal value” p.481: line 7, replace “Equation (6.6.19)” with “Equation (6.5.19).” p.515 in Eq. (7.1.8) replace z
− t by t − z in the denominator of the integral p.520 line 4 from bottom replace t → ±∞ by τ → ±∞ p.522 line below Eq. (7.2.16) replace “part 1 ” by “part 1” –i.e. eliminate bold on 1. p.529 line 2 from bottom replace “g(t) is κ” by “g(t) is κ = 0”
p.528 line 1 of Example 7.3.1 replace “inside a closed contour” by “inside a simple closed
contour” p.533 Line 2,3 of Section 7.3.2 change “but now the closed contour C is replaced by...” to
“but now the smooth closed contour C is replaced by...”. p.534 5 lines below eq. (7.3.12) change: “Equation (7.3.13) implies...” to “Equation (7.3.12)
implies...” p.544: Replace lines 3-5 “where p + (z ) and q + (z ) are ... enclosed by C ” to “where p + (z ) and
q + (z ) are polynomials with zero’s inside the region enclosed by C , while p − (z ) and q − (z ) are polynomials with zero’s outside the region enclosed by C .” p.595: Problem 3 part (b) change “...equation (2), and use (1) and the resulting eq. to
establish...” to “...equation (2) and use (1) and (3) to establish...” p.563 First equation in 2nd paragraph for Φ(k). Inside integral (add a left parens.): change f (l) X + (l)l−k)
to
f (l) X + (l)(l−k)
p.606 line 5 from bottom replace g
necessary.
ˆ → ˆf and → hˆ by g → ˆg and h → h –note: no boldface
p. 619 in the equation 3 lines below equation (7.7.26b) replace
7
∼ (
Ψ1
1 θ ) k
by
∼ (
Ψ1
1 θ ) k
1
0
p.630 In the solution 3b, move parentheses: from left of summation to just inside summation;
∞ i.e. from (Σ∞ n=0 ... to Σn=0 (... p.639 In ref. correct misspelling: Trefethan should be Trefethen p.640 index: omit reference to ‘BMO (bounded mean oscillations), 268”
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