Math 336 Midterm, May 17, 2010

Name: One notebook sized page of notes is allowed.

1. Let a be a complex number with |a| > e. Prove that aze^{z} = 1 has exactly one solution in {z : |z| < 1}.

2. Let D = {z : |z| < 1}. Let f be analytic and non-constant on the open connected set W, and suppose \overline{D} ⊂ W. Suppose |f| is constant on ∂D. Prove that f has at least one zero in D.

Midterm 3

3. Suppose f is an entire function so that

|f(z)| ≤ a + b|z|, where a > 0, b > 0.

Prove that f is a polynomial of degree at most 1.

Midterm 4

4. Compute ^{∫_{−∞}^{∞} \frac{dx}{x^{4} + 16}}