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