essential singularity EssentialSingularity 2003-03-28 14:57:15 2003-04-03 10:59:44 Definition % 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 $U\subset\mathbb{C}$ be a domain, $a\in U$, and let $f:U \setminus \{a\} \to \mathbb{C}$ be holomorphic. If the Laurent series expansion of $f(z)$ around $a$ contains infinitely many terms with negative powers of $z-a$, then $a$ is said to be an \emph{essential singularity} of $f$. Any singularity of $f$ is a removable singularity, a pole or an essential singularity. If $a$ is an essential singularity of $f$, then the image of any punctured neighborhood of $a$ under $f$ is dense in $\mathbb{C}$ (the Casorati-Weierstrass theorem). In fact, an even stronger statement is true: according to Picard's theorem, the image of any punctured neighborhood of $a$ is $\mathbb{C}$, with the possible exception of a single point.