Meromorphic Functions

Date: 09/18/98 at 12:33:37
From: Duke Desabrais
Subject: Modular Forms

What is a meromorphic function?

Date: 09/18/98 at 12:57:29
From: Doctor Nick
Subject: Re: Modular Forms

Hi Duke -

I don't know what your background is so I have to guess a bit. A 
meromorphic function is a complex function that is analytic everywhere 
except for possibly a discrete subset of its domain, that is, except 
at some, possibly infinitely many, points. At these points, the 
function goes to infinity in a polynomial way. Another way of saying 
this is that the function is analytic, except for some non-essential 
isolated singularies.

As plainly as I can say it: a meromorphic function is one that is 
really nice, except at some isolated points where it doesn't behave 
too badly.

Here is the definition given by the CRC Concise Encyclopedia of 
Mathematics: 

   A meromorphic function is complex analytic in all but a discrete 
   subset of its domain, and at those singularities it must go to   
   infinity like a polynomial (i.e., have no essential singularities). 
   An equivalent definition of a meromorphic function is a complex 
   analytic map to the Riemann Sphere. 

- Doctor Nick, The Math Forum
  http://mathforum.org/dr.math/