Complex Analysis General Exam Fall 2021

August 13, 2021

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. You may assume earlier parts of a problem on later parts. E.g. if you solve part (b) of a problem assuming part (a), but cannot solve part (a), you will get full points for part (b). Throughout $\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}$. Don’t use any of the Picard theorems.

Problem 1

Compute, for $\xi>0$,

$$\int_{-\infty}^{\infty} \frac{e^{-i x \xi}}{x^{2}-2 x+2} d x$$

Show all estimates.

Problem 2

Suppose that $f: \mathbb{C} \rightarrow \mathbb{C}$ is entire and that

$$\lim _{|z| \rightarrow+\infty} \frac{|f(z)|}{|z|}=0$$

show that $f$ is constant.

Problem 3

Suppose that $f: \mathbb{C} \rightarrow \mathbb{C}$ is entire and that there exists constants $R, C>0$ so that $|f(z)| \geq C$ if $|z| \geq R$. Show that $f$ is a polynomial.

Hint: it may be helpful to consider $g: \mathbb{C} \backslash\{0\} \rightarrow \mathbb{C}$ given by $g(z)=f(1 / z)$.

Problem 4

Let $U \subseteq \mathbb{C}$ be a nonempty connected and open set. Suppose $\left(f_{n}\right)_{n=1}^{\infty}$ is a sequence of holomorphic functions $f_{n}: U \rightarrow \mathbb{D}$ and that $\left(f_{n}\right)_{n}$ converges uniformly on compact sets to $f: U \rightarrow \mathbb{C}$. Show that if there is a $p \in U$ with $|f(p)|=1$, then $f$ is constant.

Problem 5

Let $f_{n}: \mathbb{D} \rightarrow \mathbb{D}$ be a sequence of holomorphic functions such that $f_{n} \rightarrow 0$ pointwise on $\{z \in \mathbb{C}$ : $|z| \leq 1 / 2\}$.

1. Suppose that $f: \mathbb{D} \rightarrow \mathbb{C}$ is a limit (uniformly on compact sets) of a subsequence of $\left(f_{n}\right)_{n}$. Show that $f=0$.

2. Show that $f_{n} \rightarrow 0$ uniformly on compact subsets of $\mathbb{D}$. (You are allowed to use that there is a metric so that convergence with respect to that metric is the same as uniform convergence on compact sets).