Definition:Isolated Singularity/Pole

Definition
Let $U$ be an open subset of a Riemann surface.

Let $z_0 \in U$.

Let $f: U \setminus \left\{{z_0}\right\} \to \C$ be a holomorphic function.

Let $z_0$ be an isolated singularity of $f$.

Then $z_0$ is a pole iff:
 * $\displaystyle \lim_{z \to z_0} \left|{f \left({z}\right)}\right| \to \infty$