Meromorphic Function - Meromorphic Functions On Riemann Surfaces

Meromorphic Functions On Riemann Surfaces

On a Riemann surface every point admits an open neighborhood which is homeomorphic to an open subset of the complex plane. Thereby the notion of a meromorphic function can be defined for every Riemann surface.

When D is the entire Riemann sphere, the field of meromorphic functions is simply the field of rational functions in one variable over the complex field, since one can prove that any meromorphic function on the sphere is rational. (This is a special case of the so-called GAGA principle.)

For every Riemann surface, 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 ∞.

On a non-compact Riemann surface every meromorphic function can be realized as a quotient of two (globally defined) holomorphic functions. In contrast, on a compact Riemann surface every holomorphic function is constant, while there always exist non-constant meromorphic functions.

Meromorphic functions on an elliptic curve are also known as elliptic functions.