Appell Hypergeometric Function

A formal extension of the hypergeometric function to two variables, resulting in four kinds of functions (Appell 1925; Picard 1880ab, 1881; Goursat 1882; Whittaker and Watson 1990, Ex. 22, p. 300),


These double series are absolutely convergent for

 {F_1   for |x|<1,|y|<1; F_2   for |x|+|y|<1; F_3   for |x|<1,|y|<1; F_4   for |x|^(1/2)+|y|^(1/2)<1.

Appell defined the functions in 1880 and they were subsequently studied by Picard in 1881. The functions F_1, F_2, and F_3 can be expressed in terms of double integrals as


(Bailey 1934, pp. 76-77). There appears to be no simple integral representation of this type for the function F_4 (Bailey 1934, p. 77).

The function F_1 can also be expressed by the simple integral


(Bailey 1934, p. 77), for R[alpha]>0 and R[gamma-alpha]>0.

The Appell functions are special cases of the Kampé de Fériet function, and are the first four in the set of Horn functions. The F_1 function is implemented in the Wolfram Language as AppellF1[a, b1, b2, c, x, y], and the other 3 will be implemented in a future version of the Wolfram Language.

For general complex parameters, the F_1 function can be written as the contour integral


for |arg(-z_1)|,|arg(-z_2)|<pi, where Gamma(z) is a gamma function and L and L^* are complicated contours related to those used in the definition of the Meijer G-function. In fact, the four functions can also be expressed as double contour integrals taken along contours of the Barnes type (Bailey 1934).

In particular, the general integral




has a closed form in terms of F_1.

Integrals that result in particularly nice closed forms involving the F_1 function include


which arise in computing area and geometric centroid of the interior of the cranioid curve.

F_1(alpha;beta,beta^';gamma;x,y) reduces to the hypergeometric function in the cases


In addition,


where _2F_1(a,b;c;z) is a hypergeometric function.

See also

Elliptic Integral, Horn Function, Hypergeometric Function, Kampé de Fériet Function, Lauricella Functions

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Appell, P. "Sur les fonctions hypergéométriques de plusieurs variables." In Mémoir. Sci. Math. Paris: Gauthier-Villars, 1925.Appell, P. and Kampé de Fériet, J. Fonctions hypergéométriques et hypersphériques: polynomes d'Hermite. Paris: Gauthier-Villars, 1926.Bailey, W. N. "A Reducible Case of the Fourth Type of Appell's Hypergeometric Functions of Two Variables." Quart. J. Math. (Oxford) 4, 305-308, 1933.Bailey, W. N. "On the Reducibility of Appell's Function F_4." Quart. J. Math. (Oxford) 5, 291-292, 1934.Bailey, W. N. "Appell's Hypergeometric Functions of Two Variables." Ch. 9 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 73-83 and 99-101, 1935.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 222 and 224, 1981.Exton, H. Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. Chichester, England: Ellis Horwood, p. 27, 1978.Goursat, E. "Extension du problème de Riemann à des fonctions hypergéométriques de deux variables." Comptes Rendus Acad. Sci. Paris 95, 903 and 1044, 1882.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1461, 1980.Picard, E. "Sur une classe de fonctions de deux variables indépendantes." Comptes Rendus Acad. Sci. Paris 90, 1119-1121, 1880a.Picard, E. "Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques." Comptes Rendus Acad. Sci. Paris 90, 1267-1269, 1880b.Picard, E. "Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques." Ann. Ecole Norm. Sup. (2) 10, 305-322, 1881.Watson, G. N. "The Product of Two Hypergeometric Functions." Proc. London Math. Soc. 20, 189-195, 1922.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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Appell Hypergeometric Function

Cite this as:

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

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