WA 3: Solutions
Problem 1. Prove that:
(i) If f = u + iv is analytic and satises u2 = v in a domain D, then f is a
constant.
(ii) If f is a realvalued analytic function in a domain D, then f is a con
Page 41: 2,348
Page 44: 1,4
Page 47: 5,6,8
1 Page 41, Problem 2
Expand 27153 'in powers of z 1. What is the radius of convergence?
Let f (z) = 224433. By taking a few derivatives, we can quickly
Page 96: 1
Page 99: 1
Page 108: 2,3,4,5
1 Page 96, Problem 1
Map the common part of the disks z < 1 and [z 1 < 1 onto the inside of the unit circle. Choose the
mapping so that the two symmetries ar
Page 83: 4,5,7
Page 84: 1
Page 88: 1
Page 96: 2
1 Page 83, Problem 4
Find the linear transformation which carries the circle z = 2 into z + 1 = 1, the point 2 into the origin,
and the origin into
(1) (Page 120, item 1) Compute
/ edz.
. cfw_21:1 2
Solution. Let 7 represent the unit circle. Note that f (z) := 62 is
entire, and so it is analytic on an any open disk containing 7. Since
0 9.5 'y, C
(1) (Page 72, item 1) Give a precise denition of a singlevalued branch
of \/1 + 2 + V1 2 in a suitable region, and prove that it is analytic.
Solution. Let f1(2) = \/1+2 and f2(2) = \/1 2. To solve t
WA 8: Solutions
Problem 1. Find the Taylor series expansion of a function f (z ) about the point
z0 and determine the disk of convergence:
2
(a) f (z ) = (2z + 1)ez , z0 = 0;
(b) f (z ) =
z 2 +2z +3
(
WA 7: Solutions
4. (a) Let f be a nonconstant analytic function on the closed region R, and
f (z ) = 0 for all z R. Show that minzR f (z ) can be attained only at a boundary
point of R.
(b) Let f b
WA 6: Solutions
z  dz for
Problem 1. Evaluate the integral
(i) being the line segment [i, i];
(ii) being the lefthand half of the unit circle joining i to i;
(iii) being the righthand half of the
WA 5: Solutions
Problem 1. Find the general form of a linear fractional transformation of the
upper half plane Im z > 0 onto itself.
Solution. Since a linear transformation which maps the upper half p
WA 4: Solutions
Problem 1. Show that for any z such that Im z  , with some > 0,
tan z 
1
1+
sinh2
1
2
cot z 
,
1
1+
sinh2
1
2
.
Solution. We have
 sin z 2
sin2 x + sinh2 y
1 + sinh2 y
1

WA 9: Solutions
Problem 1. Suppose is an analytic function in C except for a set of nonpositive
integers, where it has simple poles. Suppose, moreover, that (1) = 1 and z (z ) =
(z + 1) for all regula
Final Exam  MATH 630: Solutions
ix
Problem 1. Find all x R satisfying exe = eix .
Solution. Comparing the moduli of both parts, we obtain ex cos x =
1, and therefore, x cos x = 0, which is possible o
WA 2: Solutions
z2
;
z 1+i z 1
Problem 1. Find the limits: (a) lim
z2
z 1+i z 1
Solution. (a) lim
z 3 2z +2
3
z (z +1)
(b) lim
z +1
2
z 2i z +4
(c) lim
= lim
=
1
z3
(1+i)2
1+i1
2
z +2
1
3
z 0 ( z +1)
(1) (Page 11, item 1) Prove that
awb
<1
lEb
ifa <1 and b< 1.
Proof. Write a = rem where 0 S r < 1, and let s = ewb. Note that
rrs _ IrsHewl
1 73 _ 1 r(e'9b)
_ rem _ sew
_