A Riemann surface is a connected one-dimensional complex manifold . When used to represent a multiple-valued function , it can be pictured
as several, and in general infinitely many, "sheets" over the complex
plane , with the sheets joined by analytic
continuation (Knopp 1996, pp. 98-99). A single-valued branch can instead be selected by introducing branch
cuts . The above plot shows Riemann surfaces for solutions of the equation
with algebraic function (M. Trott).
The Riemann surface function field ideals of the
ring integers of roots of monic
polynomials over functions
elements and their analytic continuations .
Schwarz proved at the end of nineteenth century that the automorphism group of a compact Riemann surface of genus finite
group . Its group order is at most the Hurwitz
bound et al. 1985, pp. 45-47; Karcher and Weber 1999,
p. 9). This bound is attained for infinitely many Klein quartic ). However, it is also known that there
are infinitely many genera for which the bound
See also Analytic Continuation ,
Branch Cut ,
Branch Point ,
Complex Manifold ,
Complex
Structure ,
Function Field ,
Ideal ,
Ring Explore
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References Arbarello, E.; Cornalba, M.; Griffiths, P. A.; and Harris, J. Geometry
of Algebraic Curves, I. 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. Monthly 106 , 899-909, 1999. Corless, R. M.
and Jeffrey, D. J. "Graphing Elementary Riemann Surfaces." ACM
Sigsam Bulletin: Commun. Comput. Algebra 32 , 11-17, 1998. Derbyshire,
J. Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Fischer, G. (Ed.). Plates
123-126 in Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Bildband. 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 Knopp, K. Theory
of Functions Parts I and II, Two Volumes Bound as One, Part II. Krantz, S. G. "The Idea of a
Riemann Surface." §10.4 in Handbook
of Complex Variables. Kulkarni,
R. S. "Pseudofree Actions and Hurwitz's 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. McMullen, C. T. "Riemann Surfaces." Math 213b course
notes. Cambridge, MA: Harvard University, Apr. 28, 2026. https://abel.math.harvard.edu/~ctm/math213b/home/course/course.pdf . Monna,
A. F. Dirichlet's
Principle: A Mathematical Comedy of Errors and Its Influence on the Development of
Analysis. Springer,
G. Introduction
to Riemann Surfaces, 2nd ed. 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. "Trott's Corner: Visualization of Riemann Surfaces IIb: Compositions of Elementary
Transcendental Functions." Mathematica J. 8 , 50-62, 2001. https://library.wolfram.com/infocenter/Articles/1987/ . Trott,
M. "Trott's Corner: Visualization of Riemann Surfaces IIc: Compositions of Elementary
Transcendental Functions." Mathematica J. 8 , 409-432, 2002. https://library.wolfram.com/infocenter/Articles/3900/ . Trott,
M. The
Mathematica GuideBook for Graphics. https://doi.org/10.1007/978-1-4419-8576-7 . Trott,
M. The
Mathematica GuideBook for Numerics. https://www.mathematicaguidebooks.org/ . https://library.wolfram.com/infocenter/Demos/15/ .Referenced
on Wolfram|Alpha Riemann Surface
Cite this as:
Weisstein, Eric W. "Riemann Surface."
From MathWorld https://mathworld.wolfram.com/RiemannSurface.html
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![[w(z)]^d+w(z)+z^(d-1)=0](/images/equations/RiemannSurface/NumberedEquation1.svg)
,
3, 4, and 5, where 
is an algebraic 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 a finite
group. Its group order is at most the Hurwitz
bound 
(Hurwitz 1893; 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."