# Equivalence of Definitions of Meromorphic Function

## Theorem

The following definitions of the concept of Meromorphic Function are equivalent:

### 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.

## Proof

### $(1)$ implies $(2)$

Let $f$ be a meromorphic function by definition $1$.

$f$ can be expressed as the ratio of two holomorphic functions.

Thus $f$ is a meromorphic function by definition $2$.

$\Box$

### $(2)$ implies $(1)$

Let $f$ be a meromorphic function by definition $2$.

Thus $f$ is a meromorphic function by definition $1$.

$\Box$