An inverse function of an Abelian integral. Abelian functions have two variables and four periods, and can be defined
by
(Baker 1907, p. 21). Abelian functions are a generalization of elliptic
functions, and are also called hyperelliptic functions.
Any Abelian function can be expressed as a ratio of homogeneous polynomials of the Riemann theta function (Igusa 1972, Deconinck
et al. 2004).
See also
Abelian Integral,
Elliptic
Function,
Riemann Theta Function
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References
Baker, H. F. An Introduction to the Theory of Multiply Periodic Functions. London: Cambridge
University Press, 1907.Baker, H. F. Abelian
Functions: Abel's Theorem and the Allied Theory, Including the Theory of the Theta
Functions. New York: Cambridge University Press, 1995.Deconinck,
B.; Heil, M.; Bobenko, A.; van Hoeij, M.; and Schmies, M. "Computing Riemann
Theta Functions." Math. Comput. 73, 1417-1442, 2004.Igusa,
J.-I. Theta
Functions. New York: Springer-Verlag, 1972.Weisstein, E. W.
"Books about Abelian Functions." http://www.ericweisstein.com/encyclopedias/books/AbelianFunctions.html.Referenced
on Wolfram|Alpha
Abelian Function
Cite this as:
Weisstein, Eric W. "Abelian Function."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AbelianFunction.html
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