Isolated singularities: removable, poles, and essential singularities

Let $U\subseteq \mathbb{C}$ be open, let $a\in U$, and let $f$ be holomorphic on a punctured neighbourhood of $a$, meaning that for some $r>0$ the set $\{z\in \mathbb{C}:0<\mathrm{\mid }z-a\mathrm{\mid }<r\}$ is contained in $U$ and $f$ is holomorphic there. Then $a$ is an isolated singularity of $f$.

Such an isolated singularity is:

• removable when there is a holomorphic function $F$ on a neighbourhood of $a$ with $F(z)=f(z)$ for all $z\mathrm{\ne }a$ near $a$;
• a pole of order $m\ge 1$ when $(z-a{)}^{m}f(z)$ extends holomorphically across $a$ and the extended value at $a$ is nonzero;
• essential when it is neither removable nor a pole.

This definition does not assume that every isolated singularity falls into exactly one of the three classes. That trichotomy is a theorem later on this page.

The order of a pole is part of the definition, not an afterthought: the smallest $m$ for which $(z-a{)}^{m}f(z)$ extends holomorphically and nonvanishingly at $a$ is the pole order.