Preliminary Examination in Complex Analysis June 6, 1994

1. Show that every function that is meromorphic on the extended complex plane is rational.

2. Show that if u is a real-valued harmonic function in a domain Ω ⊂ C such that u^{2} is harmonic in Ω, then u is constant.

3. By using complex integration verify the formula ∫_{0}^{2π} \frac{1}{a + sin θ} dθ = \frac{2π}{\sqrt{a^{2} − 1}} where a > 1.

4. Let f(z) = e^{z} + z and for 0 ≤ θ < 2π let R(θ) = {z : z = re^{iθ}, r ≥ 0}. Show that lim_{{z→∞ \atop z∈R(θ)}} |f(z)| = ∞ for all θ. Does this imply that lim_{z→∞} \frac{1}{f(z)} = 0? Explain.

5. Suppose that f is an analytic function in H = {z : Im z > 0} and Im f(z) > 0 for z ∈ H. Show that |f′(z)| ≤ \frac{Im f(z)}{Im z} for z ∈ H, and determine when equality holds in this inequality.

6. Let Δ denote the open unit disk in C and let A = \{z : \frac{3}{4} < |z| < 1\}. Show that the function f(z) = \frac{1}{z − \frac{1}{2}} cannot be approximated uniformly on compact subsets of A by functions analytic in Δ.

7. For |z| < 1 let f(z) = \frac{1}{1 − z} exp [ − \frac{1}{1 − z} ], and for 0 ≤ θ < 2π let ℓ_{θ} = \{z : z = re^{iθ}, 0 ≤ r < 1}. Show that f is bounded on each set ℓ_{θ}. Is f bounded in Δ? Explain.

8. Find all conformal automorphisms of the annulus {z : 1 < |z| < 2}.