Meromorphic

Analytic on a domain except for isolated poles -- no other kind of singularity is allowed.

A function is meromorphic on a domain $D$ if it is analytic at every point of $D$ except for a set of isolated singularities, every one of which is a pole -- never removable clutter left undealt with, and never essential. A meromorphic function is exactly what is left once every removable singularity has been filled in and every essential singularity has been ruled out; rational functions and $\tan z$ (meromorphic on $\mathbb C$ , with poles where $\cos z=0$ ) are typical examples.

Examples

Every rational function $p(z)/q(z)$ (in lowest terms) is meromorphic on $\mathbb C$ , with poles exactly at the roots of $q$ .
$f(z)=\dfrac{1}{\sin z}$ is meromorphic on $\mathbb C$ , with simple poles at every integer multiple of $\pi$ .