Abelian Function

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).

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.