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'essential singularity'
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| Title of object: |
essential singularity |
| Canonical Name: |
EssentialSingularity |
| Type: |
Definition |
| Created on: |
2003-03-28 14:57:15.842505-05 |
| Modified on: |
2003-03-31 09:59:33.7469-05 |
| Classification: |
msc:30D30 |
Preamble:
% 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}
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%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
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% define commands here |
Content:
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 Weierstrass-Casorati 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. |
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