COMPLEX ANALYSIS HOMEWORK ASSIGNMENT 1
Due Friday, January 25, 2013, at the beginning of class.
Please write clearly, and staple your work !
1. Problem
Assume that the power series aj z j has convergence radius 0 < R < .
j =0
(a) Prove that the series con
COMPLEX ANALYSIS MIDTERM 1 SOLUTIONS
1. Problem
Assume that f H(C) is an entire function whose imaginary part is bounded. Prove that
f must be a constant.
Hint: Consider eif .
Solution: Let f = u + iv . Then, eif = eiu ev . But |eiu | = 1, therefore |eif
COMPLEX ANALYSIS PRACTICE PROBLEMS
1. Problem
1
(a) Prove that if f H(), then g (z ) := f ( z ) is holomorphic in = cfw_z C | z . In
particular,
g (z ) = f (z ) .
P
(b) What is the general form of a rational function R = Q (where P, Q are polynomials)
whi
COMPLEX ANALYSIS PRACTICE SOLUTIONS
1. Problem
1
(a) Prove that if f H(), then g (z ) := f ( z ) is holomorphic in = cfw_z C | z . In
particular,
g (z ) = f (z ) .
P
(b) What is the general form of a rational function R = Q which has absolute value 1 on
t
COMPLEX ANALYSIS HOMEWORK ASSIGNMENT 6
Due Friday, March 8, 2013, at the beginning of class.
Please write clearly, and staple your work !
1. Problem
What can you say about an entire function whose real part is always less than its imaginary
part? Justify
COMPLEX ANALYSIS HOMEWORK ASSIGNMENT 5
Due Friday, February 22, 2013, at the beginning of class.
Please write clearly, and staple your work !
1. Problem
Assume that f H( \ cfw_z0 ), and that limz z0 (z z0 )f (z ) = 0. Dene g (z ) := (z z0 )f (z )
for z \
COMPLEX ANALYSIS HW 5 SOLUTIONS
1. Problem
Assume that f H( \ cfw_z0 ), and that limz z0 (z z0 )f (z ) = 0. Dene g (z ) := (z z0 )f (z )
for z \ cfw_z0 , and g (z0 ) := 0. Prove that g is holomorphic in .
Solution: We prove that the contour integral of g
COMPLEX ANALYSIS HOMEWORK ASSIGNMENT 4
Due Friday, February 15, 2013, at the beginning of class.
Please write clearly, and staple your work !
1. Problem
Find all automorphisms of D, H, and C.
Hint: For an arbitrary automorphism f : D D, show that you can
COMPLEX ANALYSIS HW 4 SOLUTIONS
1. Problem
Find all automorphisms of D, H, and C.
Hint: For an arbitrary automorphism f : D D, show that you can assume f (0) = 0
via composition with a fractional linear transformation. Then, apply the Schwarz lemma.
For t
COMPLEX ANALYSIS HOMEWORK ASSIGNMENT 3
Due Friday, February 8, 2013, at the beginning of class.
Please write clearly, and staple your work !
1. Problem
Expand
2z +3
z +1
in powers of z 1. What is the radius of convergence?
2. Problem
Assume that denotes t
COMPLEX ANALYSIS HW 3 SOLUTIONS
1. Problem
Expand
2z +3
z +1
in powers of z 1. What is the radius of convergence?
Solution:
2z + 3
1
1
1
1
1
(1)n
=2+
=2+
=2+
=2+
(z 1)n
z+1
z+1
2 + (z 1)
2 1 + z 1
2
2n
2
n=0
The convergence radius is 2, from | z 1 | < 1.
COMPLEX ANALYSIS HOMEWORK ASSIGNMENT 2
Due Friday, February 1, 2013, at the beginning of class.
Please write clearly, and staple your work !
1. Problem
(a) For which A =
ab
cd
GL(2, C) are the Mbius transformations
o
az + b
cz + d
automorphisms of C, res
COMPLEX ANALYSIS HW 2 SOLUTIONS
1. Problem
(a) For which A =
ab
cd
GL(2, C) are the Mbius transformations
o
TA : z
az + b
cz + d
automorphisms of C, respectively automorphisms of C ?
Solution: The Mbius transforms TA are automorphisms of C matrices of t
M361 FALL 2015 UNIQUE NUMBER 53490 HOMEWORK 7
This homework is due on Friday, October 23, 2015, before lecture begins. Late
homework is not accepted. In order to receive credit for this assignment you must
write your name and UT eid at the top of first pa