isolated singularity

Let  $ℂ\cup \{\mathrm{\infty }\}$  denote the Riemann sphere, and let  $U\subset ℂ$ be open.  Let  $f:U\to ℂ\cup \{\mathrm{\infty }\}$  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 $\left(\text{e.g. }z=0\text{ for the function }{\displaystyle \frac{\sin z}{z}}\right)$

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

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