Isolated singularity

Definition

Suppose $U\subset \mathbb{C}$ is an open subset and $f:U\to \mathbb{C}$ is a holomorphic function. An isolated singularity for $f$ is a point $z_0\in \mathbb{C}\setminus U$ such that there exists a neighborhood $V\ni z_0$ such that $V\setminus z_0\subset U$. In other words, it is a point outside $U$, such that a small disc about the point, excluding the point itself, lies completely inside $U$.

Classification

There are three types of isolated singularities:

Removable singularity

Further information: removable singularity

$z_0$ is a removable singularity if we can extend $f$ to a holomorphic function on the open subset $U\cup \{z_0\}$.

Pole

Further information: pole

$z_0$ is a pole of order $n$ if the function $z\mapsto (z-z_0{)}^{n}f(z)$ has a removable singularity at $z_0$. The minimum such $n$ is termed the order of the pole at $z_0$.

Essential singularity

Further information: essential singularity

$z_0$ is an essential singularity if it is a singularity that is neither removable nor a pole.