Definitions for "Meromorphic function"
An function that is an analytic function except at a discrete set of points where it has singularity called "poles". At such a point, the power series expansion of the function has a finite number of terms with negative powers of z.
a function that is holomorphic almost everywhere on the complex plane , except for a set of isolated poles , which are certain well-behaved singularities
a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface ) except at points in a set of isolated poles , which are certain well-behaved singularities