isolated singularity
Let $\u2102\cup \{\mathrm{\infty}\}$ denote the Riemann sphere, and let $U\subset \u2102$ be open. Let $f:U\to \u2102\cup \{\mathrm{\infty}\}$ be a function. We say that $z$ is an isolated singularity^{} of $f$ if there exists an open set $V\subset U$ containing $z$ and such that $f$ is analytic on $V\setminus \{z\}$.
In other , if we take the set $S$ of points in $U$ where $f$ is not analytic, the isolated singularities are exactly the isolated points of $S$ in the usual topological sense.
There are three kinds of isolated singularities:

•
removable singularities^{} $\left(\text{e.g.}z=0\text{for the function}{\displaystyle \frac{\mathrm{sin}z}{z}}\right)$

•
poles $\left(\text{e.g.}z=0\text{for the function}{\displaystyle \frac{1}{{z}^{2}}}\right)$

•
essential singularities^{} $\left(\text{e.g.}z=0\text{for the function}\mathrm{exp}{\displaystyle \frac{1}{z}}\right)$
Title  isolated singularity 

Canonical name  IsolatedSingularity 
Date of creation  20130322 14:01:04 
Last modified on  20130322 14:01:04 
Owner  bwebste (988) 
Last modified by  bwebste (988) 
Numerical id  10 
Author  bwebste (988) 
Entry type  Definition 
Classification  msc 3000 