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## Due March 6th

### Problem 1

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

### Problem 2

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

### Problem 3

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

### Problem 4

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

### Problem 5

Suppose $f\in H(V)$, $W\subset \mathbb C$ is open, and $h:W\to V$ satisfies $f(h(z)) = z$ for all $z\in W$.
1. Give an example where it is not true that $h\in H(W)$.
2. If $h$ is continuous, then $h\in H(W)$.

### Comentarios

1. algún Hint para el 2?

1. Las hipótesis implican que $|f(z)| \le e^{|z+2n|}$ para todo $n\in\mathbb Z$.

2. Quise decir $e^{\pi|z+2n|}$, claro.