Ir al contenido principal

# Due May 22nd

## Problem 1

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

## Problem 2

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

## Problem 3

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