Math 502 - Complex Analysis
Homework 5
Zhaoning Yang
February 21, 2014
Problem 1.
(a) Calculate
C
f (z)dz where f (z) = z exp(z 2 ) and C is part of the curve y = x2 from 0 to 1 + i.
(b) Calculate C f
Math 502 - Complex Analysis
Homework 1
Zhaoning Yang
January 23, 2014
Problem 1.
Rewrite the following complex numbers in the form a + bi
z1 =
2+i
5
5
1
z2 =
+
z3 = 2013
1i
2 + i 1 + 2i
i
Solution:
2
Math 502 - Complex Analysis
Homework 4
Zhaoning Yang
February 14, 2014
Problem 1. Find a conformal one to one onto map from U = cfw_z |
D = cfw_z | |z| < 1.
(z) > 0, |z| < 5 to the unit disk
Solution.
Math 502 - Complex Analysis
Homework 2
Zhaoning Yang
February 1, 2014
Notation:
For f : C C, we will use
f and
f denote the real and imaginary part of f respectively.
Problem 1. Suppose h : D R is har
Math 502 - Complex Analysis
Homework 3
Zhaoning Yang
February 6, 2014
Notation:
We use H to denote the upper half plane in C. (i.e. H = cfw_z C | (z) > 0).
Problem 1. We want to dene Log(z) as the inv
Math 502 - Complex Analysis
Homework 9
Zhaoning Yang
April 4, 2014
Problem 1.
Let f (z) = 1/[(z 1)(z 2)]. Find the Laurent series for in the following cases
(a) = the unit disk cfw_z C | |z| < 1
(b) =
Math 502 - Complex Analysis
Homework 8
Zhaoning Yang
March 28, 2014
Problem 1.
the function
k
k=0 ak z
Suppose the function f (z) =
g(z) =
k=0
is holomorphic for every |z| r. Prove that
ak k
z
k!
is h
Math 502 - Complex Analysis
Homework 10
Zhaoning Yang
April 11, 2014
Problem 1. Find singular points (including 1) for each of the following functions in complex
plane. Determine which of them are iso
Math 502 - Complex Analysis
Homework 12
Zhaoning Yang
April 29, 2014
Problem 1.
k
k= ak z
Let f (z) =
for all |z| > R. We dene Res (f ) = a1 .
(a) Prove that the residue of f at is
1
2i
|z|=r
1
f
z2
1
Math 502 - Complex Analysis
Homework 11
Zhaoning Yang
April 18, 2014
Notation:
We will use Res(f, z0 ) to denote the residue of f at z = z0 .
Problem 1.
Calculate residues at the isolated singular poi
Math 502 - Complex Analysis
Homework 6
Zhaoning Yang
February 28, 2014
Problem 1. Let f : C C be a function. Show if f 0 then f a for some a C. Moreover, prove
if f (n) 0 then f is a polynomial of deg
Math 502 - Complex Analysis
Homework 7
Zhaoning Yang
March 13, 2014
Problem 1. Let BR C be the open ball centered at origin with radius R > 0. Prove if cfw_fn
n=1
is a sequence of functions holomorph