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## Due February 28th

### Problem 1

For any positive integer $n$, the polynomial $z^n(z-2)-1$ has $n$ roots in the disk $\mathbb D = D(0,1)$.

### Problem 2

Suppose $f:\overline{\mathbb D}\to\overline{\mathbb D}$ is continuous and holomorphic in $\mathbb D$. Then $f$ has a fixed point in $\overline{\mathbb D}$.

### Problem 3

Prove Hurwitz's Theorem: If $V$ is an open set, $f_n\in H(V)$ is a sequence such that $f_n\to f$ uniformly on compact subsets of $V$, $\overline{D(z,r)}\subset V$, and $f$ has no zeroes on $\partial D(z,r)$, then there exists $N$ such that, for all $n\ge N$, $f_n$ and $f$ have the same number of zeroes in $D(z,r)$.

### Problem 4

1. The series $\displaystyle \sum_{n=0}^\infty \frac{1}{z-n}$ diverges for all $z\in\mathbb C\setminus\mathbb Z$.
2. The series $\displaystyle \sum_{n=0}^\infty \Big( \frac{1}{z-n} + \frac{1}{z+n} \Big)$ converges absolutely for all $z\in\mathbb C\setminus\mathbb Z$

### Problem 5

Prove Wallis' formula: $\displaystyle \frac{\pi}{2} = \prod_{n=1}^\infty \frac{4n^2}{(2n-1)(2n+1)} = \frac{2}{1}\frac{2}{3}\frac{4}{3}\frac{4}{5}\cdots$.
Warning: It is not enough, to conclude the second inequality, to observe that $\displaystyle \frac{4n^2}{(2n-1)(2n+1)} = \frac{2n}{2n-1}\frac{2n}{2n+1}$. For example, it is also true that $\displaystyle \frac{4n^2}{(2n-1)(2n+1)} = \frac{4n}{2n-1}\frac{n}{2n+1}$, but it is not true that
$\displaystyle \frac{\pi}{2} = \frac{4}{1}\frac{1}{3}\frac{8}{3}\frac{2}{5}\cdots$.