Preliminary Exam
Complex Analysis
August 1999
1. Find an analytic function f(z) whose real part is (z = x + iy).
Re f(z) = xy − 10 .
Does such a function exist? Justify your answer.
2. Find the general form of an entire function f(z) satisfying
|f(z)| ≤ A + B|z|3/2, where A and B are constants .
3. Find the general form of a function f(z) which is analytic inside the ellipse D (z =
x + iy)
x2
16
+
y2
9
= 1 ,
continuous in D, and
Im f(z) = −5 (z ∈ ∂D)
4. Find a conformal mapping from C\{[0,+∞)} to the unit disk.
5. 1. Prove that for any polynomial p and any a ∈ ∆
p(a) =
1
2π
∫ 2π
0
p(eiθ)
1− e−iθa
dθ
2. Deduce from 5.1 that
|p(a)| ≤
[
1
(1− |a|2)
1
2π
∫ 2π
0
|p(eiθ)|2dθ
]1/2
6. Let f be analytic in the unit disk and map the unit disk into itself given f(1/2) = 0.
Prove that |f ′(1/2)| ≤ 4
3
.
7. Let
f(z) =
1
z
· 1− 2z
z − 2
· . . . · 1− 10z
z − 10
Find
∫
|z|=100
f(z)dz.
8. Let f(z)
6≡ 0 be a meromorphic function in C such that
|f(z)| = 1 (|z| = 1)
and
f
(
1
2
)
= 0 .
Can f be an entire function?