Meromorphic function

A meromorphic function is a function that is holomorphic (defined on the complex numbers, and that can be differentiated everywhere where it is defined) on all of an open set except for a set of isolated points, which are poles of the function.

All rational functions, for example$$f(z)={\frac {z^{3}-2z+10}{z^{5}+3z-1}},$$are meromorphic on the whole complex plane.