The Appell hypergeometric functions are 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),
(1)
(2)
(3)
(4)
These double series are absolutely convergent for
(5)
Appell defined the functions in 1880 and they were subsequently studied by Picard in 1881. The functions , , and can be expressed in terms of double
integrals as
(6)
(7)
(8)
(Bailey 1934, pp. 76-77). There appears to be no simple integral representation of this type for the function (Bailey 1934, p. 77).
The function can also be expressed by the simple integral
For general complex parameters, the function can be written as the contour
integral
(10)
for ,
where
is a gamma function and and 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
(11)
where
(12)
has a closed form in terms of .
Integrals that result in particularly nice closed forms involving the function include
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 ." 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. Paris95, 903 and
1044, 1882.Ishkhanyan, T. "Hypergeometric Functions: From Euler
to Appell and Beyond." Jan. 25, 2024. https://blog.wolfram.com/2024/01/25/hypergeometric-functions-from-euler-to-appell-and-beyond/.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. Paris90, 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. Paris90, 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.
, 
, and 
can be expressed in terms of double
integrals as
(Bailey 1934, p. 77).
can also be expressed by the simple integral
![R[alpha]>0](/images/equations/AppellHypergeometricFunction/Inline27.svg)
and ![R[gamma-alpha]>0](/images/equations/AppellHypergeometricFunction/Inline28.svg)
.
function can be written as the contour
integral
,
where 
is a gamma function and 
and 
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).
^(n+1)[b(n+1)sqrt(1+(c^2)/(b^2))]^(-1)sqrt((b(sqrt(1+(c^2)/(b^2))-sinx)-ccosx)/(bsqrt(1+(c^2)/(b^2))+a))sqrt((b(sqrt(1+(c^2)/(b^2))+sinx)+ccosx)/(bsqrt(1+(c^2)/(b^2))-a)),](/images/equations/AppellHypergeometricFunction/NumberedEquation5.svg)
.
function include
reduces to the hypergeometric function
in the cases
is a hypergeometric function.
." Quart. J. Math. (Oxford) 5, 291-292,
1934.Bailey, W. N. "Appell's Hypergeometric Functions of Two
Variables." Ch. 9 in