Preliminary Examination in Complex Analysis
June 4, 1996
Notation: ∆ = {z ∈ C : |z| < 1} is the open unit disk
1. Let f be analytic in a nonempty connected open set U . Let F be a nonconstant entire
function. Show that if F (f(z)) = 0 for all z in a neighborhood of some z0 ∈ U , then
f is constant in U .
2. (a) Find all constants c1 and c2 so that the functions
f1(z) = c1z and f2(z) =
c2
z
define conformal self-maps of the annulus A = {z ∈ C : a < |z| < b} (0 < a < b are
given constants).
(b) Prove that there are no other conformal self-maps of A.
3. Evaluate
∫
γ
1− cos z
(ez − 1) sin z
dz
where the path γ is the circle |z| = e traversed once counterclockwise.
4. For n ∈ N show that∫
∆
∣∣∣∣∣1− zn
1− z
∣∣∣∣∣
2
dxdy = π
(
1 +
1
2
+
1
3
+ . . . +
1
n
)
5. Let f be analytic in ∆, and let f(∆) ⊆ ∆. Prove that if f(0) = 0 and f(a) = a for
some a
6= 0, then f(z) = z.
6. Let f be analytic in ∆. Show that
sup
z∈∆
(1− |z|2) |f ′(z)| ≤ sup
z∈∆
|f(z)| .
7. Let f(z) be analytic in ∆. Suppose
lim
r↑1
∫ 2π
0
|f(reiθ)|dθ = 0 .
Show that f ≡ 0.
8. Prove that the zero set S of ez + z:
S = {z ∈ C : ez + z = 0}
is nonempty: S 6= ∅.
Bonus. Prove that S is an infinite set.
9. Find w = f(z) that maps ∆ conformally onto the strip |Im w| < π
2
so that f(0) = 0
and f ′(0) > 0.