Complex Analysis - May 2022

Problem 1

Prove that for any $z_1,z_2\in\{z\in\mathbb C:\operatorname{Re}z<0\}$, $|e^{z_1}-e^{z_2}|\le|z_1-z_2|$.

Problem 2

Let $f$ be a non-constant meromorphic function on $\mathbb C$ with $P$ being the set of its poles. Prove that $f(\mathbb C\setminus P)$ is dense in $\mathbb C$.

Problem 3

Let $(f_n)$ be a sequence of conformal maps defined on a complex domain $D$. Suppose $f_n\to f$ uniformly on each compact subset of $D$. Prove that $f$ is either constant or a conformal map.

Remark. A conformal map is a holomorphic and injective map. A complex domain is a nonempty connected open subset of $\mathbb C$.

Problem 4

Let $z_0$ be a pole of $f$. Prove that $z_0$ is an essential singularity of $e^f$.

Problem 5

Compute

$$\int_{-\infty}^{\infty}\frac{\cos x}{(x^2+1)(x^2+4)}\,dx.$$