DEVELOPMENT OP THE THEORY OF CONFORMAL MAPPING AND RIEMANN SURFACES THROUGH A CENTURY

1953 ◽  
pp. 1-14
Author(s):  
Lars V. Ahlfors
Keyword(s):  
Conformal Mapping ◽  
Riemann Surfaces

scholarly journals Dirichlet Problems on Riemann Surfaces and Conformal Mappings

1951 ◽  
Vol 3 ◽  
pp. 91-137 ◽  
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Riemann Surface ◽  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Conformal Mappings ◽  
Dirichlet Problems ◽  
Boundary Correspondence ◽  
Abstract Riemann Surface

The object of this paper is an investigation of existence problems and Dirichlet problems on an abstract Riemann surface in the sense of Weyl-Radó or on a covering surface over it, and of boundary correspondence in the conformal mapping of the surface.

scholarly journals Boundary Components of Riemann Surfaces

1954 ◽  
Vol 7 ◽  
pp. 65-83
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Dirichlet Problem ◽  
Riemann Surface ◽  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Boundary Values ◽  
Abstract Riemann Surface

The boundary components of an abstract Riemann surface were defined by B. v. Kérékjértó [7] and utilized in the book [14] written by S. Stoïlow. It is the purpose of the present paper to investigate their images under conformal mapping and to solve the Dirichlet problem with boundary values distributed on them.

scholarly journals Fundamental Domains of Gamma and Zeta Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Cabiria Andreian Cazacu ◽  
Dorin Ghisa
Keyword(s):  
Conformal Mapping ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Riemann Surfaces ◽  
Zeta Functions ◽  
Branched Covering ◽  
Fundamental Domains ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Color Visualization

Branched covering Riemann surfaces(ℂ,f)are studied, wherefis the Euler Gamma function and the Riemann Zeta function. For both of them fundamental domains are found and the group of cover transformations is revealed. In order to find fundamental domains, preimages of the real axis are taken and a thorough study of their geometry is performed. The technique of simultaneous continuation, introduced by the authors in previous papers, is used for this purpose. Color visualization of the conformal mapping of the complex plane by these functions is used for a better understanding of the theory. A version of this paper containing colored images can be found in arXiv at Andrian Cazacu and Ghisa.

scholarly journals The Conformal Mapping of Simply Connected Riemann Surfaces II

1957 ◽  
Vol 12 ◽  
pp. 139-143
Author(s):  
Maurice Heins
Keyword(s):  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Present Note ◽  
Universal Covering ◽  
Riemann Mapping ◽  
Classical Theorem ◽  
Countable Base ◽  
Riemann Mapping Theorem ◽  
Technical Interest ◽  
Simply Connected

On reviewing recently the proof which I gave for the Riemann mapping theorem for simply-connected Riemann surfaces several years ago [2], I observed that the argument which I used could be so modified that the assumption of a countable base could be completely eliminated. The problem of treating the Riemann mapping theorem without this assumption has been current for some time. The object of the present note is to give an account of a solution of this question. Of course, the classical theorem of Radó permits us to dispense with an attack on the Riemann mapping theorem which does not appeal to the countable base assumption. In this connection, we recall that Nevanlinna [4] has given a straightforward potential-theoretic treatment of the Radó theorem in which neither the Riemann mapping theorem (nor the notion of a universal covering) enters as they do in Radó’s proof. Nevertheless, a certain technical interest attaches to a direct treatment of the Riemann mapping theorem without the countable base assumption. An immediate byproduct of such a treatment is a simple proof of the Radó theorem which invokes the notion of a universal covering but in a manner different from that of Radó’s proof. Indeed, it suffices to note that a manifold has a countable base if the domain of a universal covering does.

Conformal Mapping on Riemann Surfaces.

1968 ◽  
Vol 75 (10) ◽  
pp. 1132
Author(s):  
Emil Grosswald ◽  
Harvey Cohn
Keyword(s):  
Conformal Mapping ◽  
Riemann Surfaces

scholarly journals On a Ring Isomorphism Induced by Quasiconformal Mappings

1959 ◽  
Vol 14 ◽  
pp. 201-221 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Rings Of Continuous Functions ◽  
Ring Isomorphism ◽  
Function Ring ◽  
Conformal Equivalence ◽  
The Relationship

The purpose of this paper is to study the relationship between a certain isomorphism of some rings of functions on Riemann surfaces and a quasi-conformal mapping.It is well known that two compact Hausdorff spaces are topologically equivalent if and only if their rings of continuous functions are isomorphic. We shall establish an analougous result concerning a function ring on a Riemann surface and the quasi-conformal equivalence.

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