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Let $\mathrm{\Omega }$ be an open set in $ℂ$, and let $𝑢(𝑥,𝑦),𝑣(𝑥,𝑦)$ be real-valued functions with continuous partial derivatives in $\mathrm{\Omega }$ satisfying the Cauchy-Riemann equations. Assume also that

on $\mathrm{\Omega }$. Prove that

is harmonic in $\mathrm{\Omega }$ and find its harmonic conjugate on $\mathrm{\Omega }$ if it exists.

Let $𝑃(𝑧)$ be a polynomial of degree $𝑑$ with complex coefficients such that

for all $𝑥\in [-1,1]$. Show that

Use contour integration to show that for all integers $𝑛\ge 2$,

Suppose $𝑓:𝐷\to 𝐷$ is a holomorphic function with two distinct fixed points $𝑎,𝑏\in 𝐷$, so

Prove that

for all $𝑧\in 𝐷$.

Find an explicit conformal map between

and the unit disk $𝐷$.

Consider

Show that $𝑓$ is a bounded holomorphic function on the unit disk $𝐷$ that admits a continuous extension to , but cannot be holomorphically extended to any neighborhood $𝐷(𝑝,𝜀)$ of any point .

First, find an example of a pair of distinct entire functions $𝑓$ and $𝑔$ such that

for all $𝑛\in \mathbb{N}$.

Show that it is impossible to find a distinct pair with this property if we further assume

for all $𝑧\in ℂ$.

Prove or disprove that there exists a holomorphic function

such that

Let $𝑓$ be a holomorphic function on

such that $𝑓(0)=0$ and $𝑓(𝑧)\ne 0$ whenever $𝑧\ne 0$. Let $𝑛\in \mathbb{N}$. Prove that there exists a holomorphic function $𝑔$ on $𝔻$ such that

for every $𝑧\in 𝔻$.

Find all entire functions $𝑓$ such that

for every $𝑧\in ℂ$, and

for all $𝑚,𝑛\in \mathbb{N}$.