𝔐𝔞𝔱𝔥336 Midterm, May 20, 2013

Name: One notebook sized page of notes is allowed.

1. Using Rouché’s theorem, show that z^{7} + 5z^{3} + z − 2 has four roots in the set {z : 1 < |z| < 2}.

Midterm 2

2. Let f be an analytic function on an open connected set W. Suppose 0 ∈ W and suppose |f(\frac{1}{n})| < e^{−n} for all n > 0. Prove that f(z) = 0 for all z ∈ W.

Midterm 3

3. Let f(z) = e^{z−1/z} = ∑_{n=−∞}^{n=+∞} a_{n}z^{n}. Prove that

a_{n} = \frac{1}{2π} ∫_{0}^{2π} cos(2 sin(θ) − nθ)dθ.

Midterm 4

4. Using a contour integral prove

∫_{0}^{2π} \frac{dt}{1 + sin^{2} t} = \sqrt{2}π.