# Riemann Surface

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## Riemann Surface

one of the basic concepts of the theory of functions of a complex variable. The Riemann surface was introduced by B. Riemann in 1851 for the purpose of replacing the study of multiple-valued analytic functions by the study of single-valued analytic functions of a point on corresponding Riemann surfaces (seeANALYTIC FUNCTIONS).

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On a potential-theoretically parabolic Riemann surface, there exists a so-called Evans-Selberg potential, whose existence is equivalent to the parabolicity condition (see [E, K, Na62, S]).
Along the lines of other important contributions to the development of topology--and in particular that of Felix Klein--Hermann Weyl has pointed out this feature of Riemann's work, by arguing "that it is always the Riemann surface, not the analytic form, which is regarded as the given object" (2010: 157).
He also uses Quillen's holomorphic theorem to evaluate the dependence of the determinants of the Laplace operations on the boundary conditions on the Riemann surface.
At the first glance, the idea of a Riemann surface looks as a cheap trick: we replace the domain of a given multi-valued function by a new more complex domain on which the function becomes single-valued.
Hence if, conversely, a map A from a simply connected Riemann surface [summation] into ([h.
Another interesting instance is that of a Riemann surface endowed with the Poincare metric.
lambda]] and its natural domain of definition is the two-sheeted Riemann surface [R.
For any Riemann surface S of genus g [less than or equal to] 2,
s]/[partial]z are discontinuous on crossing the screen, and is taken into account in Sommerfeld's theory by "wrapping" the diffracting half plane in a semi-infinite, two-sided Riemann surface so that its positive and negative sides are distinguished by the values 2[pi] and 0 of the polar angle [PHI] in Fig.
The set [gamma]\[gamma] is a Riemann surface, which can be seen to have infinite genus because of the presence of infinitely many branch points.
We numerically solved the equations, which involved path integrals of meromorphic functions on a genus-one Riemann surface.
They discuss topics related to Kleinian groups, classical Riemann surface theory, mapping class groups, geometric group theory, and statistical mechanics.

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