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 $1$.

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