Meromorphic function: Difference between revisions

Revision as of 13:45, 13 June 2018

Revision as of 09:35, 29 June 2018

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==On Riemann surfaces==

On a [[Riemann surface]], every point admits an open neighborhood

which is [[homeomorphismbiholomorphism|homeomorphicbiholomorphic]] 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.)