Riemann mapping theorem RiemannMappingTheorem 2002-12-11 08:10:41 2002-12-11 08:31:39 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $U$ be a simply connected open proper subset of $\mathbb{C}$, and let $a\in U$. There is a unique analytic function $f:U\rightarrow\mathbb{C}$ such that \begin{enumerate} \item $f(a)=0$, and $f'(a)$ is real and positive; \item $f$ is injective; \item $f(U)=\{z\in \mathbb{C}:|z|<1\}$. \end{enumerate} \textbf{Remark.} As a consequence of this theorem, any two simply connected regions, none of which is the whole plane, are conformally equivalent.