Homework 12: Complex Analysis

Due May 8th

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

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