Complex Analysis - August 2021

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 is 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

Compute the following integral

$$\int_{-\infty}^{\infty}\frac{x^2}{(x^2+1)(x^2+9)}\,dx.$$

Problem 2

Consider the polynomial

$$p(z):=z^6+az^2+1$$

Find an $a>0$ so that $p(z)$ has exactly 4 zeros in the annulus $\{z\in\mathbb C:\frac12<|z|<2\}$.

Problem 3

Let $f$ be an entire function with finitely many zeros, and suppose there exists constants $0<\rho<1$ and $A,B>0$ so that

$$|f(z)|\le Ae^{B|z|^\rho}$$

for all $z\in\mathbb C$. Show that $f$ is a polynomial.

Problem 4

Let $(f_n)_{n\in\mathbb N}$ be a sequence of entire functions which is uniformly bounded on compact subsets. Suppose that there exists a polynomial $p(z)$ of degree $d$ such that for each $z\in\mathbb C$, $(f_n(z))_{n\in\mathbb N}$ converges to to $p(z)$. Show that there exists $N\in\mathbb N$ so that for all $n\ge N$, $f_n(z)$ has at least $d$ zeros (counting multiplicities).

Problem 5

Let $\Omega\subset\mathbb C$ be a simply connected domain. Suppose that $f:\Omega\to\Omega$ is analytic and that for some $z_0\in\Omega$ we have $f(z_0)=z_0$ and $f'(z_0)=1$. Show that $f(z)=z$.