# 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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The quotient space T \ H can be identified with a compact Riemann surface F of a genus g [greater than or equal to] 2.
In order to acquire the algebro-geometric solutions of systems (16), we first introduce the Riemann surface r of the hyperelliptic curve with genus N:
), and they found that the number of minimal reducing subspaces of [T.sub.B] equals the number of connected components of the Riemann surface of B(z) = B(w) when the order of B is 3,4,6.
Consider a compact connected Riemann surface [summation] of genus g with canonical homology cycle bases [a.sub.i], [b.sub.i] for i = 1, ..., g.
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).
Let [GAMMA]\H be a hyperbolic Riemann surface of finite area.
In the case of multi-valued analytic functions of a single complex variable, the proposed solution is called "The Riemann Surface'.
Hence if, conversely, a map A from a simply connected Riemann surface [summation] into ([h.sup.2n+1]) satisfying (3.5) is given, then a solution to [[phi].sup.-1]ld[phi] = Bdz + [bar.B]d[bar.z] exists and defines a harmonic map from [summation] into [H.sup.2n+1].
Another interesting instance is that of a Riemann surface endowed with the Poincare metric.
The function g[lambda] has branch points at [z.sup.+.sub.[lambda]] and [z.sup.-.sub.[lambda]] and its natural domain of definition is the two-sheeted Riemann surface [R.sub.[lambda]] defined by the relation
For any Riemann surface S of genus g [less than or equal to] 2,
This distinction is necessary because [u.sup.(p).sub.s] and [[partial]u.sup.(s).sub.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.

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