COMPLEX ANALYSIS GENERAL EXAM FALL 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

For $\xi \in \mathbb{R}$, compute

$$\int_{-\infty}^{\infty} \frac{\cos (\xi x)}{x^{2}+4 x+5} d x$$

You must justify all estimates.

Problem 2

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function whose derivative satisfies

$$\left|f^{\prime}(z)\right| \leq e^{\sqrt{\log (|z|+1)}}$$

for all $z \in \mathbb{C}$. Show that there exist constants $a, b \in \mathbb{C}$ with $|a| \leq 1$ such that $f(z)=a z+b$.

Problem 3

Let $U:=\{z \in \mathbb{C}: \operatorname{Re}(z)>0\}$. Suppose that $f: \bar{U} \rightarrow \mathbb{C}$ is continuous, $\sup _{z \in \partial U}|f(z)| \leq 1,\left.f\right|_{U}$ is holomorphic, and that there exist $M, \alpha>0$ such that $|f(z)| \leq M e^{\alpha \sqrt{|z|}}$ for all $z \in U$. Show that $|f(z)| \leq 1$ for all $z \in U$. (Hint: for fixed $\varepsilon>0$, show that the desired estimate holds for $f_{\varepsilon}(z):=f(z) e^{-\varepsilon z^{3 / 4}}$.)

Problem 4

Given, $n, m \in \mathbb{N}$ with $m<2 n$ and $a, b \in \mathbb{R}$ with $b>0$ and $a \in \left(-b^{1-\frac{m}{2 n}}, b^{1-\frac{m}{2 n}}\right)$, show that the polynomial $P(z):=z^{2 n}+a z^{m}+b$ has exactly $n$ zeroes (counted with multiplicity) with positive imaginary parts. (Hint: consider the case $a=0$ first.)

Problem 5

Let $\left(f_{n}\right)$ be a sequence of holomorphic functions from $\mathbb{D}$ to $\mathbb{C}$ that is locally bounded, where $\mathbb{D}:=\{z \in \mathbb{C}:|z|<1\}$ denotes the unit disc centred at the origin. Suppose that there exists a holomorphic function $f: \mathbb{D} \rightarrow \mathbb{C}$ such that the set $A=\left\{z \in \mathbb{D}: \lim _{n \rightarrow \infty} f_{n}(z)=f(z)\right\}$ has a limit point in $\mathbb{D}$. Show that $\left(f_{n}\right)$ converges uniformly on compact sets to $f$.