# isolated singularity

Let  $\mathbb{C}\cup\{\infty\}$  denote the Riemann sphere, and let  $U\subset\mathbb{C}$ be open.  Let  $f\colon U\to\mathbb{C}\cup\{\infty\}$  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  $V\!\smallsetminus\!\{z\}$.

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 $\displaystyle\left(\text{e.g.\;\;}z=0\text{\; for the function\,}\frac{\sin{z}% }{z}\right)$

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

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

Title isolated singularity IsolatedSingularity 2013-03-22 14:01:04 2013-03-22 14:01:04 bwebste (988) bwebste (988) 10 bwebste (988) Definition msc 30-00