Prelim in Complex Analysis, August 2003
Let C be the complex plane and D = {z ∈ C : |z| < 1} be the open unit disk.
1. Evaluate the following integrals.
∫
|z|=2
csc z dz,
∫
|z|=1
1− cos z
z2
dz,
∫ π
0
dθ
2 + cos θ
.
2. If f(z) is entire and |f(z)| ≤ |z|3/2 for all z, show that f is identically zero.
3. Show that a function f : D → C is constant if and only if both f and f are analytic
in D.
4. Show that the class X of analytic functions f in D with
∫
D
|f(z)| dx dy ≤ 1
is a normal family.
5. Find the real and imaginary parts of the complex number
z = Log (1 + i) + cos(1 + i),
where Log is the branch of the logarithm on C − {x : x ≤ 0} with Log (1) = 2πi.
6. If f : D → C is a bounded analytic function, show that
sup
z∈D
(1− |z|2)|f ′(z)| ≤ sup
z∈D
|f(z)|.
7. Show that if f : D → C is analytic and one-to-one, then f ′(z)
6= 0 for every z ∈ D.
8. If f : D → D is analytic and f(0) = f ′(0) = 0, show that |f(z)| ≤ |z|2 for all z ∈ D.
9. Find the Laurent series of f(z) = 1/[z(1 − z)] at z = 0, at z = 1, at z = 2, and at
z = ∞.
10. Let
B(z) =
n∏
k=1
ak − z
1− akz
,
where a1, · · · , an are distinct points in D − {0}. Show that
B(z) =
n∏
k=1
1
ak
+
n∑
k=1
1
akB′(ak)(1− akz)
.