A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These
sheets can have very complicated structures and interconnections (Knopp 1996, pp. 98-99).
Riemann surfaces are one way of representing multiple-valued
functions; another is branch cuts. The above plot
shows Riemann surfaces for solutions of the equation
Schwarz proved at the end of nineteenth century that the automorphism group of a compact Riemann surface of genus is finite, and Hurwitz (1893)
subsequently showed that its order is at most (Arbarello et al. 1985, pp. 45-47; Karcher
and Weber 1999, p. 9). This bound is attained for infinitely many , with the smallest of such an extremal surface being 3 (corresponding to the
Klein quartic). However, it is also known that there
are infinitely many genera for which the bound is not attained (Belolipetsky 1997, Belolipetsky and
Jones).
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G. Introduction
to Riemann Surfaces, 2nd ed. New York: Chelsea, 1981.Trott,
M. "Visualization of Riemann Surfaces of Algebraic Functions." Mathematica
J.6, 15-36, 1997.Trott, M. "Visualization of Riemann
Surfaces IIa." Mathematica J.7, 465-496, 2000. Trott, M. "Visualization of Riemann Surfaces." http://library.wolfram.com/examples/riemannsurface/.