Picard's theorem PicardsTheorem 2002-12-11 10:24:50 2008-05-01 10:19:39 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 $f$ be an holomorphic function with an essential singularity at $w\in \mathbb{C}$. Then there is a number $z_0\in \mathbb{C}$ such that the image of any neighborhood of $w$ by $f$ contains $\mathbb{C}-\{z_0\}$. In other words, $f$ assumes every complex value, with the possible exception of $z_0$, in any neighborhood of $w$. \emph{Remark.} Little Picard theorem follows as a corollary: Given a nonconstant entire function $f$, if it is a polynomial, it assumes every value in $\mathbb{C}$ as a consequence of the fundamental theorem of algebra. If $f$ is not a polynomial, then $g(z)=f(1/z)$ has an essential singularity at $0$; Picard's theorem implies that $g$ (and thus $f$) assumes every complex value, with one possible exception.