Homework 12: Complex Analysis

Due May 8th

Problem 1

If $P(z)$ is a polynomial, then there exists $z_0$ such that $|z_0|=1$ and $|P(z_0) - \dfrac{1}{z_0}| \ge 1$.

Problem 2

There exists a sequence $P_n(z)$ of polynomials such that $P_n(0) =1$ for all $n$ and $P_n(z)\to 0$ for all $z\in\mathbb C$, $z\not=0$.

Problem 3

Assume Runge's Theorem and prove the following theorem: Let $V\subset\mathbb C$ open and $\Gamma$ a cycle in $V$. If  $\displaystyle \int_\Gamma f(z) dz = 0$ for all $f\in H(V)$ of the form $f(z) = 1/(z-a)$, then $\displaystyle \int_\Gamma f(z) dz = 0$ for all $f\in H(V)$.

Problem 4

Let $V\subset\mathbb C$ be bounded, connected and open. There exists $f\in H(V)$ which cannot be extended to a function holomorphic in a strictly larger open set.