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.