COMPLEX ANALYSIS GENERAL EXAM SPRING 2023

Solve as many problems as you can. Full solutions on a smaller number of problems will be worth more than partial solutions on several problems. Don't use any of the Picard theorems.

Problem 1

Compute

$$\int_{-\infty}^{\infty} \frac{\sin ^{2} x}{x^{2}+1} d x$$

You must justify all estimates.

Problem 2

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function for which there exist $M, R>0$ such that $|f(z)| \geq M|z|^{2}$ for all $|z|>R$. Show that $f$ is a polynomial of degree at least 2 .

Problem 3

Let $f, g: \mathbb{C} \rightarrow \mathbb{C}$ be entire functions for which there exists $\lambda \in \mathbb{R}$ such that $\operatorname{Im}(f(z)) \leq \lambda \operatorname{Im}(g(z))$ for all $z \in \mathbb{C}$. Show that there exist constants $a, b \in \mathbb{C}$ such that $f(z)=a g(z)+b$ for all $z \in \mathbb{C}$.

Problem 4

Let $P(z):=z^{9}+z^{5}-8 z^{3}+2 z+1$. Determine the number of zeroes (counted with multiplicity) of $P$ that lie in the annulus $A_{1,2}(0):=\{z \in \mathbb{C}: 1<|z|<2\}$.

Problem 5

Let $\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}$ denote the unit disc, and let $\mathcal{F}$ denote the family of holomorphic functions $f: \mathbb{D} \rightarrow \mathbb{C}$ satisfying $|f(z)|>1$ for all $z \in \mathbb{D}$ and $f(0)=2 i$. Show that $\mathcal{F}$ is normal, so that every sequence ( $f_{n}$ ) in $\mathcal{F}$ has a subsequence that converges uniformly on compact sets to a holomorphic function $f: \mathbb{D} \rightarrow \mathbb{C}$.