Complex Analysis General Exam - January 2020

Problem 1

Compute

$$\lim _{R \rightarrow \infty} \int_{-R}^{R} \frac{\sin (x)}{x} d x$$

using Residue theory.

Problem 2

Let $f$ be a nonconstant entire function. Show that $f(\mathbb{C})$ is dense in $\mathbb{C}$. You are not allowed to use any of the Picard’s theorem.

Problem 3

Let $f$ be an entire function. Assume there is a sequence $r_{n} \in(0, \infty)$ with $r_{n} \rightarrow \infty$, and constants $C, \alpha \in(0, \infty)$ with

$$\sup _{z \in \mathbb{C}:|z|=r_{n}}|f(z)| \leq C r_{n}^{\alpha}$$

Show that $f$ is a polynomial.

Problem 4

Let $\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}$. Let $\left(f_{n}\right)_{n}$ be a sequence of holomorphic functions and suppose that $f_{n}$ converge uniformly on compact subsets of $\mathbb{D}$ to a function $f$. Suppose $f$ has no zeroes on $\left\{z \in \mathbb{C}:|z|=\frac{1}{2}\right\}$. Show that there exists an $N \in \mathbb{N}$ so that if $n, m \geq N$, then $f_{n}, f_{m}$ have the same number of zeroes in $\{z \in \mathbb{C}:|z|<1 / 2\}$ (counting with multiplicity). (This is Hurwitz’s theorem, you are not allowed to just quote this theorem).

Problem 5

Let $U$ be an open subset of $\mathbb{C}$. Let $f_{n}: U \rightarrow \mathbb{C}$ be a sequence of analytic functions. Suppose that $f_{n}$ converge pointwise to a function $f: U \rightarrow \mathbb{C}$. If $\left|f_{n}\right| \leq 1$ for every $n$, show that $f$ is analytic.