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Homework 4: Complex Analysis

Due February 28th

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

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

Problem 5

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