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 holomorphic on Br such that they converge uniformly on |z| = r w
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. Let U = cfw_z | (z) > 0, |z| < 5, note the conformal m
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 (z)dz where f (z) = z |z| and C is the closed curve co
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 harmonic on the open disk D = cfw_z C | |z| < 1. Let f : D
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 inverse function of z ez . Its easy to check e2z = (ez )2
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) = the annulus cfw_z C | 1 < |z| < 2
(c) = cfw_z C | |z|
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 holomorphic for every z C and there exists M R, M 0 such
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 isolated. And for isolated singularities nd if they are re
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
z
dz =
1
2i
f (z)dz
|z|=R
where R > R and r = 1/R . T
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 points.
(a) f (z) =
1
z(1 z 2 )
(d) f (z) = z 9 e1/z
1
sin
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 degree at most n 1.
Proof. Let f (z) = f (x + iy) = u(x, y