Definition:Meromorphic Function

Definition

Let $U \subset \C$ be an open subset of the complex plane $\C$.

Definition 1

A meromorphic function on $U$ is a holomorphic function on all of $U$ except for a set of poles of $f$.

Definition 2

A meromorphic function on $U$ is a complex function that can be expressed as the ratio of two holomorphic functions.

That is:

$\map f z = \dfrac {\map g z} {\map h z}$

where:

$g: \C \to \C$ and $h: \C \to \C$ are holomorphic

$z \in \C$ such that $\map h z \ne 0$

Definition 3

A meromorphic function on $U$ is a complex function whose only singular points are poles.

Also see

• Equivalence of Definitions of Meromorphic Function

• Definition:Holomorphic Function

• Results about meromorphic functions can be found here.