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!

  • $\begingroup$ A proof of this is outlined in, I think, the exercises in Gamelin's book. Not sure if you have access to those though. $\endgroup$ – Future Jan 1 '16 at 21:04
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    $\begingroup$ See Is a meromorphic function always a ratio of two holomorphic functions?. $\endgroup$ – Martin R Jan 1 '16 at 21:05
  • $\begingroup$ $f$ should be defined on $A\setminus E,$ where $E$ is a discrete subset of $A.$ $\endgroup$ – zhw. Jan 1 '16 at 21:08

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

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