Complex Analysis - May 2021

Note: all statements require proofs. You can make references to standard theorems from the course; however, you need state the relevant part of the theorem in your own words, unless it is a well-known named theorem. For example, “we had a theorem in the class that said that any continuous function on a compact subset of $\mathbb R^n$ is uniformly continuous” is a good reference, and “by the uniqueness theorem from the class, $f$ is unique” is not a good reference.

Problem 1

Let $z_1,\ldots,z_n\in\mathbb C$ be distinct points and suppose $f$ is injective and holomorphic on $\mathbb C\setminus\{z_1,\ldots,z_n\}$. Show that $f(z)=\frac{az+b}{cz+d}$ for some $a,b,c,d\in\mathbb C$.

Problem 2

Compute the following integral

$$\int_{-\infty}^{\infty}\frac{x^2}{(x^2+1)(x^2+4)}\,dx.$$

Problem 3

Determine the number of zeros the polynomial $p(z)=z^7+16z^2+1$ has in the open annulus $\{z\in\mathbb C:\frac12<|z|<2\}$.

Problem 4

Suppose $f$ is a conformal map (i.e. a bijective holomorphic function) from the punctured disc $D\setminus\{0\}$ to itself. Show that $f(z)=e^{i\theta}z$ for some $\theta\in\mathbb R$.

Problem 5

Let $f$ be an entire function such that for each $n\in\mathbb N$

$$\frac12 e^{2^{(1.5n)}}\le\max_{|z|=2^n}|f(z)|\le e^{2^{(1.5n)}}.$$

Show that $f$ has infinitely many zeros.