A meromorphic function is a single-valued function that is analytic in all but possibly a discrete
subset of its domain, and at those singularities
it must go to infinity like a polynomial (i.e., these
exceptional points must be poles and not essential
singularities). A simpler definition states that a meromorphic function is a
function 
of the form
 |
where 
and 
are entire functions with 
(Krantz 1999, p. 64).
A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities
in its domain. A meromorphic function with an infinite
number of poles is exemplified by 
on the punctured disk 
, where 
is the open unit disk.
An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere.
The word derives from the Greek 
(meros), meaning "part,"
and 
(morphe), meaning "form" or "appearance."
