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 u2 is harmonic in Ω, then u is constant. 3. By using complex integration verify the formula ∫ 2π 0 1 a + sin θ dθ = 2π √ a2 − 1 where a > 1. 4. Let f(z) = ez + z and for 0 ≤ θ < 2π let R(θ) = {z : z = reiθ, r ≥ 0}. Show that lim z→∞ z∈R(θ) |f(z)| = ∞ for all θ. Does this imply that lim z→∞ 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)| ≤ Im f(z) Im z for z ∈ H, and determine when equality holds in this inequality. 6. Let ∆ denote the open unit disk in C and let A = {z : 3 4 < |z| < 1}. Show that the function f(z) = 1 z − 12 cannot be approximated uniformly on compact subsets of A by functions analytic in ∆. 7. For |z| < 1 let f(z) = 1 1− z exp [ − 1 1− z ] , and for 0 ≤ θ < 2π let `θ = {z : z = reiθ, 0 ≤ r < 1}. Show that f is bounded on each set `θ. Is f bounded in ∆? Explain. 8. Find all conformal automorphisms of the annulus {z : 1 < |z| < 2}.