Complex Analysis - August 2023

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’s 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

Suppose that $f(1/n)=e^{-n}$ for all $n\in\mathbb N$. Show that $f$ cannot be holomorphic in a neighborhood of 0.

Problem 2

For $a>0$, evaluate

$$\int_0^{+\infty}\frac{\cos(ax)}{(1+x^2)^2}\,dx.$$

Problem 3

Let $f$ be an entire function satisfying $|f(z)|\ge1$ for all $z\in\mathbb C$ with $|z|\ge10$. Show that $f$ is a polynomial.

Problem 4

Let

$$U=\{z\in\mathbb C:|z|<1\}\setminus[0,1].$$

In other words, $U$ is the open unit disk with the line segment between 0 and 1 removed. Find a conformal mapping between $U$ and the upper half plane.

Problem 5

Show that there does not exist a function $f$ holomorphic in $\{z\in\mathbb C:|z|>3\}$ with

$$f'(z)=\frac{z^2+1}{z(z-1)(z-2)}$$

for all $z$ from the above set.