Meromorphic function

A meromorphic function is a function that is holomorphic almost everywhere on the complex plane, except for a set of isolated poles, which are certain well-behaved singularities. Every meromorphic function can be expressed as the ratio between two entire functions (with the denominator not constant 0): the poles then occur at the zeroes of the denominator.

Examples of meromorphic functions are all rational functions such as f(z) = (z³-2z + 1)/(z⁵+3z-1), the functions f(z) = exp(z)/z and f(z) = sin(z)/(z-1)² as well as the Gamma function and the Riemann zeta function. The functions f(z) = ln(z) and f(z) = exp(1/z) are not meromorphic.

In the language of Riemann surfaces, a meromorphic function is the same as a holomorphic function from the complex plane to the Riemann sphere which is not constant ∞. The poles correspond to those complex numbers which are mapped to ∞.