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Homework 14: Complex Analysis

Due May 22nd

Problem 1

Let $latex a\in\mathbb C$ and $latex f,g\in\mathscr O_a$. Then $latex f = g$ if and only if $latex f^{(k)}(a) = g^{(k)}(a)$ for all $latex k\in\mathbb N$.

Problem 2

Let $latex \gamma_0, \gamma_1$ paths from $latex a$ to $latex b$ in $latex V$. Then $latex \gamma_0$ and $latex \gamma_1$ are path-homotopic in $latex V$ if and only if $latex \gamma_0-\gamma_1$ is homotopic to a constant in $latex V$, where $latex \gamma_0-\gamma_1:[0,1]\to\mathbb C$ is defined by
$latex (\gamma_0-\gamma_1)(t) = \begin{cases}\gamma_0(2t) & 0\le t\le 1/2\\ \gamma_1(2-2t) & 1/2\le t\le 1. \end{cases}$

Problem 3

Let $latex V\subset W\subset\mathbb C$ open connected, $latex u:W\to\mathbb R$ harmonic and $latex f\in H(V)$ such that $latex \Re f = u$ in $latex V$. Then $latex (f,V)$ admits unrestricted continuation in $latex W$.