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 words, 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 $\displaystyle \left( {e.g.\;\;} z = 0 {\; for the function\,} \frac{\sin{z}}{z} \right)$
• poles $\displaystyle \left( {e.g.\;\;} z = 0 {\; for the function\,} \frac{1}{z^2} \right)$
• essential singularities $\displaystyle \left( {e.g.\;\;} z = 0 {\; for the function\,} \exp{\frac{1}{z}} \right)$