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.
Read more about this topic: Meromorphic Function
Famous quotes containing the words functions and/or surfaces:
“In today’s world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
—Urie Bronfenbrenner (b. 1917)
“Footnotes are the finer-suckered surfaces that allow tentacular paragraphs to hold fast to the wider reality of the library.”
—Nicholson Baker (b. 1957)