PhD COMPREHENSIVE EXAM—COMPLEX ANALYSIS—August 1996

Notation: B = {z ∈ C: |z| < 1}, the open unit disc

1. a) Let f be an entire function, and fix p, q ∈ C with p ≠ q. If r > 0 is chosen so that |p| < r and |q| < r, evaluate

∫_{γ_{r}} \frac{f(z)}{(z − p)(z − q)} dz,

where γ_{r}(t) = re^{it} on [0, 2π].

b) Use part a) to give a proof of Liouville’s Theorem.

2. Let f and g be analytic functions on B so that f(z)g(z) = 0 for all z ∈ B. Show that either f(z) = 0 for all z ∈ B or g(z) = 0 for all z ∈ B.

3. Show that the polynomial 2z^{5} − 6z^{2} + z + 1 has exactly three zeros (counting multiplicities) in {z: 1 < |z| < 2}.

4. Suppose u is harmonic on the punctured unit disc B \ {0}. Does there exist an analytic function f on B \ {0} so that u is the real part of f there? Give a proof or a counterexample.

5. a) State carefully the Riemann Mapping Theorem.

b) Let D = {x + iy: x, y ∈ R and x^{2} < y^{2} + 1} and f an entire function so that f(C) ⊂ D. Show that f is constant.

6. Let E be a set with no limit point in C. Show that there exists a meromorphic function f on C so that f has a pole of order two at each point of E and f ≠ 0 on C.