Isolated singularity

Definition

Let $c \in \mathbb{C}$ and let $f$ be a complex function. The point $c$ is an isolated singularity of $f$ if there is $r > 0$ such that $f$ is defined and holomorphic on the punctured disc $\{z : 0 < |z - c| < r\}$. The value of $f$ at $c$, if there is one, plays no role.

Remarks

Each isolated singularity is of exactly one of three types: removable, a pole, or essential. For example, $0$ is an isolated singularity of $1/z$, of $\frac{\sin z}{z}$ and of $e^{1/z}$. The point $0$ is not an isolated singularity of $\frac{1}{\sin(1/z)}$, because the poles $\frac{1}{k\pi}$ accumulate at $0$. It is not an isolated singularity of $\operatorname{Log} z$ either, because $\operatorname{Log}$ is not holomorphic on any punctured disc around $0$.

Explanations

L1 human

An isolated singularity is a single bad point of a function, with only good points around it. The function is holomorphic on a small disc around the point, with only the center removed.