Isolated singularity

Definition

Suppose $U \subset C$ is an open subset and $f : U \to m a t h b b C$ is a holomorphic function. An isolated singularity for $f$ is a point $z_{0} \in C ∖ U$ such that there exists a neighborhood $V ∋ z_{0}$ such that $V ∖ z_{0} \subset U$ . 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.