Math 336 Midterm, May 23, 2008

Name:

One notebook sized page of notes is allowed.

1. Let u and v be harmonic on an open connected set W. Suppose that u(z)v(z) = 0 on an open subset of W. Prove that either u or v is identically 0 on W.

Midterm 2

2. Using Rouché’s theorem determine the number of roots of the equation z^{7} − 4z^{3} + z + 1 = 0 inside the disk {z : |z| < 1}.

Midterm 3

3. Let f be analytic on D = {|z| < 1}. Suppose f(0) = f′(0) = 0 and |f(z)| < M for z ∈ D. Prove that |f(z)| ≤ M|z|^{2} for z ∈ D.

4. Let Ω be a bounded connected open set. Suppose 0 ∈ Ω and that f is an analytic function on Ω such that if z ∈ Ω, f(z) ∈ Ω. Suppose also that f(0) = 0, f′(0) = 1. Prove that f(z) = z. Hint: use Cauchy’s inequalities on the functions obtained by composing f with itself k times.