𝔐𝔞𝔱𝔥336 Midterm, May 21, 2012

Name: One notebook sized page of notes is allowed.

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

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. Suppose f is analytic on the disk D = {z : |z| < 1} and suppose f is

continuous on the closure of D. Suppose

|f(e^{it})| ≤ R_{j},

when (j − 1)\frac{π}{2} < t < j \frac{π}{2} for j = 1, 2, 3, 4. Prove |f(0)|^{4} ≤ R_{1}R_{2}R_{3}R_{4}.

Midterm 4

4. Using a contour integral compute

∫_{−∞}^{∞} \frac{dx}{x^{4} + 1}