Abelian Function

An inverse function of an Abelian integral. Abelian functions have two variables and four periods, and can be defined by

 Theta(v,tau;q^'; q)=sum_(lambda=-infty)^inftye^(2piiv(lambda+q^')+piitau(lambda+q^')^2+2piiq(lambda+q^'))

(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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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."

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Abelian Function

Cite this as:

Weisstein, Eric W. "Abelian Function." From MathWorld--A Wolfram Web Resource.

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