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

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

In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. (The terminology comes from the Ancient Greek “meros” (), meaning part, as opposed to “holos” (), meaning whole.) Such functions are sometimes said to be regular functions or regular on D.