Math 336 Midterm, May 18, 2009

Name: One notebook sized page of notes is allowed.

1. Let P (z) be a polynomial of degree n. Prove that

∫_{|z|=2} \frac{P (z)}{(z − 1)^{n+2}} dz = 0.

2. Let D = {z : |z| < 1}. Suppose f is analytic on an open set that includes the closure of D and suppose |f(z)| < 1 if |z| = 1. Prove that there is a unique ζ ∈ D such that f(ζ) = ζ.

Midterm 3

3. Suppose f is an entire function so that

|f(z)| ≤ |z|^{4}/ log |z|, for |z| > 1.

Prove that f is a polynomial of degree at most 3.

Midterm 4

4. Use the residue theorem to compute

∫_{|z|=1} \frac{|dz|}{|z − a|^{4}} , where a > 1.