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 real-valued analytic function in a domain D, then f is a constant.
Solution. (i) By CR,
ux = vy =
2
(u ) = 2uuy ,
y
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 determine a formula for the nth derivative:
1 2 6 (1"n
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 are preserved.
Name the disks D0 := cfw_lzl < 1 and D1 :=
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 i.
Call the linear transformation f. First, we give nam
(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, Cauchys integral formula implies
/lz|=1ZZdZ /|z|:1 :(ji)
(1) (Page 72, item 1) Give a precise denition of a single-valued 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 the
problem, it sufces to take 9 to be a subset of (C wh
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
(z +1)(z 2 4) ,
(c) f (z ) = sin z , z0 =
z0 = 1;
2.
2
S
WA 7: Solutions
4. (a) Let f be a non-constant 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 be a nonconstant analytic function on and inside a simpl
WA 6: Solutions
|z | dz for
Problem 1. Evaluate the integral
(i) being the line segment [i, i];
(ii) being the left-hand half of the unit circle joining i to i;
(iii) being the right-hand half of the unit circle joining i to i.
1
3/2
/2
/2
3/2
|ei |iei d
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 plane onto
itself, maps its boundary, the real line, int
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
| tan z |2 =
=
=1+
.
2
2
2 x + sinh2 y
| cos z |
cos
sinh
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 regular points. Prove that
Resz=n (z ) =
(1)n
,
n!
n = 0, 1,
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 only if x = 0 or x =
+ k , k Z. Both parts of the equati
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)
z 3 2z +2
3;
z (z +1)
(b) lim
z +1
.
2
z 2i z +4
(c) l
(1) (Page 11, item 1) Prove that
awb
<1
lEb
if|a| <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
_ 1 ~ (re9!)
_ a b
_ 15b'
Thus, it suices to show that
r