Meromorphic function

A meromorphic function is 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. Every meromorphic function can be expressed as the ratio between two holomorphic 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.

By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient f/g can be formed unless g(z) = 0 for all z. Thus, the meromorphic functions form a field, in fact a field extension of the complex numbers.

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