Homework 5: Complex Analysis

Due March 6th

Problem 1

Suppose $latex f\in H(\mathbb C)$, $latex f(z+1) = - f(z)$ for all $latex z\in\mathbb C$, $latex f(0) = 0$ and $latex |f(z)| \le e^{\pi|\Im z|}$ for all $latex z\in\mathbb C$. Then $latex f(z) = c\sin\pi z$ for some constant $latex c$. (Hint: Use Problem 4 of Homework 2.)

Problem 2

Suppose $latex f:\mathbb C\to\mathbb C$ is continuous, $latex f(z + 2) = f(z)$ and $latex |f(z)| \le e^{\pi|z|}$ for all $latex z\in\mathbb C$. Then there exists a constant $latex c$ such that $latex |f(z)| \le c e^{\pi|\Im z|}$ for all $latex z\in\mathbb C$.

Problem 3

Find $latex \displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$ using the series for $latex \cot\pi z$.

Problem 4

Prove that Theorem 7.5 in the text follows from Theorem 7.6.

Problem 5

Suppose $latex f\in H(V)$, $latex W\subset \mathbb C$ is open, and $latex h:W\to V$ satisfies $latex f(h(z)) = z$ for all $latex z\in W$.

1. Give an example where it is not true that $latex h\in H(W)$.
2. If $latex h$ is continuous, then $latex h\in H(W)$.