Casorati-Weierstrass theorem CasoratiWeierstrassTheorem 2003-04-03 11:33:27 2008-06-18 20:01:08 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 Given a domain $U\subset\mathbb{C}$, $a\in U$, and $f:U\setminus\{a\}\to\mathbb{C}$ being holomorphic, then $a$ is an essential singularity of $f$ if and only if the image of any punctured neighborhood of $a$ under $f$ is dense in $\mathbb{C}$. Put another way, a holomorphic function can come in an arbitrarily small neighborhood of its essential singularity arbitrarily close to any complex value.