Equivalent definitions of meromorphic function

Question

My complex analysis course gives the following definition of a meromorphic function:

"A function $f\colon A \rightarrow \mathbb{C}$ with $A\subset \mathbb{C}$ is meromorphic if it is holomorphic on $A$ except for isolated singularities, which should be poles. "

Searching through the web, I found that some authors or websites define a meromorphic function as a quotient of two holomorphic functions e.g. the wikipedia page mentions this: https://en.wikipedia.org/wiki/Meromorphic_function

Could anybody give me a brief outline of the proof why these two definitions are equivalent, or give me an (internet-accessible) reference?

Thanks in advance!

A proof of this is outlined in, I think, the exercises in Gamelin's book. Not sure if you have access to those though. — Future, Jan 01 '16 at 21:04
See Is a meromorphic function always a ratio of two holomorphic functions?. — Martin R, Jan 01 '16 at 21:05
$f$ should be defined on $A\setminus E,$ where $E$ is a discrete subset of $A.$ — zhw., Jan 01 '16 at 21:08

score 2 · Answer 1 · answered Jan 01 '16 at 21:08

2

The equivalence of the two "definitions" is a consequence of Weierstrass Factorisation Theorem, which has a relatively long proof.

answered Jan 01 '16 at 21:08

Yiorgos S. Smyrlis

83,933

Equivalent definitions of meromorphic function

1 Answers1