Preliminary Examination in Complex Analysis June 4, 1996 Notation: Δ = {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 z_{0} ∈ U, then f is constant in U.

2. (a) Find all constants c_{1} and c_{2} so that the functions

f_{1}(z) = c_{1}z and f_{2}(z) = \frac{c_{2}}{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} \frac{1 − cos z}{(e^{z} − 1) sin z} dz

where the path γ is the circle |z| = e traversed once counterclockwise.

4. For n ∈ N show that

∫_{Δ} |\frac{1 − z^{n}}{1 − z} |^{2}dxdy = π(1 + \frac{1}{2} + \frac{1}{3} + … + \frac{1}{n})

5. Let f be analytic in Δ, and let f(Δ) ⊆ Δ. Prove that if f(0) = 0 and f(a) = a for some a ≠ 0, then f(z) = z.

6. Let f be analytic in Δ. Show that

sup_{z∈Δ} (1 − |z|^{2}) |f′(z)| ≤ sup_{z∈Δ} |f(z)| .

7. Let f(z) be analytic in Δ. Suppose

lim_{r↑1} ∫_{0}^{2π} |f(re^{iθ})|dθ = 0 .

Show that f ≡ 0.

8. Prove that the zero set S of e^{z} + z:

S = {z ∈ C : e^{z} + z = 0}

is nonempty: S ≠ ∅. \underline{Bonus}. Prove that S is an infinite set.

9. Find w = f(z) that maps Δ conformally onto the strip |Im w| < \frac{π}{2} so that f(0) = 0 and f′(0) > 0.