Suggested solution of HW4
Chapter 5 Q14: Assume F has finitely many zeros. By Hadamards factorization theorem,
F (z) = eP (z) z m
N
Y
Ek (z/an )
n=1
for some a1 , a2 , .aN , where P is a polynomial of degree k < . But then F is of order
k instead of . Con

Elementary proof of Picard little theorem
We present the proof of Picard little theorem by John L. Lewis in [1]. Here we only extract
the essential part for the proof in the paper.
For any harmonic function u on C, by Poisson formula, we have
u(a + rei )

Suggested solution of HW6(Sketch)
Ch1-Q7: (a) If |w| < 1, consider the holomorphic map z 7 (w z)/(1 wz)
on D. By
maximum principle,
wz
sup w z .
1 wz
D 1 wz
On D, z = 1/z,
wz w
z
w z zw
1
=
= 1.
1 wz
1 w
z
1 wz
zw
If max point is attained at int

Suggested solution of HW2
Chapter 3 Q12: By Residue formula, for any N |u|
I
X
1
cot z
dz
=
2i |z|=N +1/2 (u + z)2
Res
|n|N +1/2
cot z
cot z
, n + Res
, u .
(u + z)2
(u + z)2
At z = n,
d
sin z = (1)n .
dz
n
Thus,
Res
cot z
1
,n =
.
(u + z)2
(u + n)2
A

Suggested solution of HW1
Chapter 1 Q13: Write f = u + iv where u and v are a pair of differentiable functions satisfying the
cauchy Riemann equation.
(a) If u = Re(f ) is constant, we have
v
u
=
=0
y
x
v
u
=
=0
x
y
Thus, u and v are constant functions, f

Suggested solution of HW3
Chapter 2 Q11: (a) By scaling, we assume R = 1. Consider g = f where (z) =
z+a
.
1+a
z
By Cauchy formula, we get
1
f (a) = g(0) =
2i
I
B(1)
g(z)
1
dz =
z
2i
I
B(1)
f (z)
dz.
z
We now perform change of coordinate w = (z). We have

THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH4060 (First term, 2015-16)
Complex Analysis
This is a second course in complex analysis. Topics to be covered include the Poisson summation
formula, Weierstrass infinite products, and funct

Alternative proof of the example
For a domain C, z0 , and constants a 6= 0 and w0 with Re(
aw0 ) > 0,
F = cfw_f H() : f (z0 ) = w0 , Re(
af (z) > 0, z
is normal.
Proof.
Method 1: It suffices to show that it is locally bounded. Let K be a compact set of c

Math 4060
Supplementary notes 1
REVIEW OF HOLOMORPHIC FUNCTIONS
PO-LAM YUNG
Theorem 1. Let be an open set in C, and f : C be a complex-valued function on . Then
the following are equivalent:
(a) f is holomorphic on ;
f (z + h) f (z)
exists for every z ;
h

Partial solution to midterm (revised)
1. (a) We claim that f is differentiable at a for arbitrary a C. For a C, there exists
compact set K containing a. Since fn converge to f uniformly on K, for any
triangle 4 K,
Z
Z
fn dz
f dz.
4
4
But
Z
fn dz = 0
4
as

Sample questions for MATH4060 Midterm Exam
1. State and prove Rouches theorem, about the comparison of the number of zeroes between
two holomorphic functions.
2. Suppose f is holomorphic on C \ cfw_0, and has an essential singularity at 0. If > 0, let
D0

Sample questions for MATH4060 Final Exam
Note: I have included only questions about the part of the course that was not covered by the
midterm exam. You should also consult the homework assignments, for some excellent questions
about the topics we covered