# Equivalence of Definitions of Meromorphic Function

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## 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 $1$.

By Meromorphic Function is Quotient of Holomorphic Functions:

- $f$ 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 $2$.

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

$\Box$

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## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**meromorphic function**