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).
Arbarello, E.; Cornalba, M.; Griffiths, P. A.; and Harris, J. Geometry
of Algebraic Curves, I. New York: Springer-Verlag, 1985.Belolipetsky,
M. "On the Number of Automorphisms of a Nonarithmetic Riemann Surface."
Siberian Math. J.38, 860-867, 1997.Belolipetsky, M. and
Jones, G. "A Bound for the Number of Automorphisms of an Arithmetic Riemann
Surface." Math. Proc. Camb. Phil. Soc.138, 289-299, 2005.Borwein,
J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics."
Amer. Math. Monthly106, 899-909, 1999.Corless, R. M.
and Jeffrey, D. J. "Graphing Elementary Riemann Surfaces." ACM
Sigsam Bulletin: Commun. Comput. Algebra32, 11-17, 1998.Derbyshire,
J. Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.
New York: Penguin, pp. 209-210, 2004.Fischer, G. (Ed.). Plates
123-126 in Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig,
Germany: Vieweg, pp. 120-123, 1986.Hurwitz, A. "Über
algebraische Gebilde mit eindeutigen Transformationen in sich." Math. Ann.41,
403-442, 1893.Karcher, H. and Weber, M. "The Geometry of Klein's
Riemann Surface." In The
Eightfold Way: The Beauty of the Klein Quartic (Ed. S. Levy). New York:
Cambridge University Press, pp. 9-49, 1999.Knopp, K. Theory
of Functions Parts I and II, Two Volumes Bound as One, Part II. New York:
Dover, pp. 99-118, 1996.Krantz, S. G. "The Idea of a
Riemann Surface." §10.4 in Handbook
of Complex Variables. Boston, MA: Birkhäuser, pp. 135-139, 1999.Kulkarni,
R. S. "Pseudofree Actions and Hurwitz's Theorem." Math. Ann.261, 209-226,
1982.Lehner, J. and Newman, M. "On Riemann Surfaces with Maximal
Automorphism Groups." Glasgow Math. J.8, 102-112, 1967.Macbeath,
A. M. "On a Curve of Genus 7." Proc. Amer. Math. Soc.15,
527-542, 1965.Mathews, J. H. and Howell, R. W. Complex
Analysis for Mathematics and Engineering, 4th ed. Boston, MA: Jones and Bartlett,
2000.Monna, A. F. Dirichlet's
Principle: A Mathematical Comedy of Errors and Its Influence on the Development of
Analysis. Utrecht, Netherlands: Osothoek, Scheltema, and Holkema, 1975.Springer,
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/.
![[w(z)]^d+w(z)+z^(d-1)=0](/images/equations/RiemannSurface/NumberedEquation1.svg)
,
3, 4, and 5, where 
is the Lambert W-function (M. Trott).
of the function field 
is the set of nontrivial discrete valuations on 
. Here, the set 
corresponds to the ideals of the
ring 
of integers of 
over 
. (
consists of the elements of 
that are roots of monic
polynomials over ![C[z]](/images/equations/RiemannSurface/Inline12.svg)
.)
Riemann surfaces provide a geometric visualization of functions
elements and their analytic continuations.
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).
Theorem." Math. Ann. 261, 209-226,
1982.Lehner, J. and Newman, M. "On Riemann Surfaces with Maximal
Automorphism Groups." Glasgow Math. J. 8, 102-112, 1967.Macbeath,
A. M. "On a Curve of Genus 7." Proc. Amer. Math. Soc. 15,
527-542, 1965.Mathews, J. H. and Howell, R. W. 
Trott, M. "Visualization of Riemann Surfaces."