Runge's theorem RungesTheorem 2002-12-11 09:39:30 2004-06-07 19:24:19 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 $K$ be a compact subset of $\mathbb{C}$, and let $E$ be a subset of $\mathbb{C}_\infty=\mathbb{C}\cup\{\infty\}$ (the extended complex plane) which intersects every connected component of $\mathbb{C}_\infty-K$. If $f$ is an analytic function in an open set containing $K$, given $\varepsilon>0$, there is a rational function $R(z)$ whose only poles are in $E$, such that $|f(z)-R(z)|<\varepsilon$ for all $z\in K$.