On Isomorphisms of Meromorphic Function Fields

1968 ◽  
Vol 20 ◽  
pp. 1230-1241 ◽  
Author(s):  
James J. Kelleher
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Algebraic Structure ◽  
Riemann Surfaces ◽  
Function Fields ◽  
Plane Region ◽  
Algebraic Properties ◽  
Compact Riemann Surfaces ◽  
Conformal Equivalence ◽  
Ring Of Functions

In this work we deal with algebraic properties of some fields of functions meromorphic in the complex plane with a view to determining the possible isomorphisms between two such fields. Interest in problems of this type began with a paper by Bers (2), in which it was shown that the algebraic structure of the ring of functions analytic on a plane region determines the conformai structure of the region to within conformai or anti-conformal equivalence, and this result was later extended to arbitrary non-compact Riemann surfaces by Nakai (7).

Rings of Meromorphic Functions on Non-Compact Riemann Surfaces

1969 ◽  
Vol 21 ◽  
pp. 284-300 ◽  
Author(s):  
James Kelleher
Keyword(s):  
Riemann Surface ◽  
Algebraic Structure ◽  
Riemann Surfaces ◽  
Meromorphic Functions ◽  
Doctoral Dissertation ◽  
University Of Illinois ◽  
Compact Riemann Surfaces ◽  
The University ◽  
Complex Valued

In this paper we shall be concerned with the algebraic structure of certain rings of functions meromorphic on a non-compact (connected) Riemann surface Ω. In this setting, A = A(Ω) and K= K(Ω) denote (respectively) the ring of all complex-valued functions analytic on Ω and its field of quotients, the field of functions meromorphic on Ω. The rings considered here are those subrings of K containing A,which we term A-rings of K. Most of the results given here were previously announced without proof (15) and are contained in the author's doctoral dissertation (16), completed at the University of Illinois under the direction of Professor M. Heins, whose encouragement and advice are gratefully acknowledged.

scholarly journals Remarks on the Elliptic Case of the Mapping Theorem for Simply-Connected Riemann Surfaces

1955 ◽  
Vol 9 ◽  
pp. 17-20 ◽  
Author(s):  
Maurice Heins
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Riemann Surfaces ◽  
Simple Pole ◽  
Conformal Map ◽  
Elliptic Case ◽  
Extended Plane ◽  
Simply Connected ◽  
Conformal Equivalence

It is well-known that the conformal equivalence of a compact simply-connected Riemann surface to the extended plane is readily established once it is shown that given a local uniformizer t(p) which carries a given point p0 of the surface into 0, there exists a function u harmonic on the surface save at p0 which admits near p0 a representation of the form(α complex 0; h harmonic at p0). For the monodromy theorem then implies the existence of a meromorphic function on the surface whose real part is u. Such a meromorphic function has a simple pole at p0 and elsewhere is analytic. It defines a univalent conformal map of the surface onto the extended plane.

scholarly journals Functions meromorphic on some Riemann surfaces

1978 ◽  
Vol 70 ◽  
pp. 41-45
Author(s):  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
A Value ◽  
Extended Complex ◽  
The Ideal ◽  
Extended Complex Plane

Consider an open Riemann surface R and a single-valued meromorphic function w = f(p) defined on R. A value w0 in the extended complex plane is said to be a cluster value for w = f(p) if there exists a sequence {pn } in R accumulating at the ideal boundary of R such that

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