Math 336 𝔐𝔦𝔡𝔱𝔢𝔯𝔪, April 29, 2013

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

1. Let f(z) = u(z)+iv(z), u = Re(f(z)), v = Im(f(z)) be analytic on an open connected set Ω. Suppose there are real numbers a, b, c with a^{2} + b^{2} ≠ 0 so that au(z) + bv(z) = c for all z ∈ Ω. Prove that f is constant. midterm 2

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

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

midterm 4

4. Let f(z) = e^{−z^{−4}} if z ≠ 0, f(0) = 0. Prove that f is analytic at z if z ≠ 0 and that the Cauchy-Riemann equations are satisfied at 0. Is f analytic at 0?