Math 336 Midterm, April 27, 2009

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

1. Let u and v be harmonic on an open set. (a) Is it necessarily true that u^{2} is harmonic? What about uv?

(b) Answer the same question in the case that v is the harmonic conjugate of u.

2. Let f(z) = u(x, y)+iv(x, y) where u and v are real, twice-differentiable on an open set. Suppose that the real and imaginary parts of f(z) and zf(z) are harmonic. Prove that f is analytic. midterm 3

3. Compute ∫_{|z|=1} \frac{dz}{2z^{2} − 5z + 2}.

midterm 4

4. Let f be analytic on D = {z : |z| < 1}. Let d = sup_{z,w∈D}{|f(z) − f(w)|}.

Prove that

2|f′(0)| ≤ d.

Hint: For 0 < r < 1, prove

2f′(0) = \frac{1}{2πi} ∫_{|z|=r} \frac{f(z) − f(−z)}{z^{2}}dz,

And let r → 1.