Math 336 𝔐𝔦𝔡𝔱𝔢𝔯𝔪, April 30, 2012

Name:

One notebook sized page of notes is allowed on the test.

1. Let f : ℂ → ℂ be defined by f(z) = x^{3}y^{2} + ix^{2}y^{3}. Find the points where f is complex differentiable and the points where f is complex analytic. midterm 2

2. Suppose u is harmonic on ℂ and u(0) = 0. Prove that if u(z) → 0 as |z| → ∞ then u(z) = 0 for all z. midterm 3

3. Compute

∫_{|z−1|=2} \frac{sin(z)dz}{(z − 4)(z − 1)}.

4. Suppose f is analytic on D = {z : |z| < 1} and continuous on the closure \overline{D} of D. Suppose f(z) ≠ 0 in D. Prove that the minimum of |f(z)| on \overline{D} occurs on the boundary of D.