The Meijer -function
is a very general function which reduces to simpler special functions in many common
cases. The Meijer -function
is defined by
(1)
where
is the gamma function (Erdélyi et al.
1981, p. 1068; Gradshteyn and Ryzhik 2000). A different but equivalent form
is used by Prudnikov et al. (1990, p. 793),
(2)
This form provides more consistency with the definition of this function via an inverse
Mellin transform.
The Meijer -function
is implemented in the Wolfram Language
as MeijerG[a1, ..., an, a(n+1), ..., ap, b1, ..., bm, b(m+1), ..., bq, z]. A generalized form of the function defined by
(3)
is implemented in the Wolfram Language as MeijerG[a1, ..., an, a(n+1), ..., ap, b1, ..., bm, b(m+1), ..., bq, z, r].
In both (2) and (3), the contour lies between the poles
of
and the poles of . For example, the contour
for
is illustrated above, both in the complex plane
and superposed on the function itself (M. Trott).
Prudnikov et al. (1990) contains an extensive nearly 200-page listing of formulas for the Meijer -function.
Adamchik, V. "The Evaluation of Integrals of Bessel Functions via -Function
Identities." J. Comput. Appl. Math.64, 283-290, 1995.Erdélyi,
A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Definition of the
G-Function" et seq. §5.3-5.6 in Higher
Transcendental Functions, Vol. 1. New York: Krieger, pp. 206-222,
1981.Gradshteyn, I. S. and Ryzhik, I. M. "Meijer's and
MacRobert's Function ( and )" and "Meijer's -Function." §7.8 and 9.3 in Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
pp. 843-850 and 1022-1025, 2000.Luke, Y. L. The
Special Functions and Their Approximations, 2 vols. New York: Academic Press,
1969.Mathai, A. M. A
Handbook of Generalized Special Functions for Statistical and Physical Sciences.
New York: Oxford University Press, 1993.Meijer, C. S. "Multiplikationstheoreme
für di Funktion ."
Proc. Nederl. Akad. Wetensch.44, 1062-1070, 1941.Meijer,
C. S. "On the -Function. II." Proc. Nederl. Akad. Wetensch.49,
344-456, 1946.Meijer, C. S. "On the -Function. III." Proc. Nederl. Akad. Wetensch.49,
457-469, 1946.Meijer, C. S. "On the -Function. IV." Proc. Nederl. Akad. Wetensch.49,
632-641, 1946.Meijer, C. S. "On the -Function. V." Proc. Nederl. Akad. Wetensch.49,
765-772, 1946.Meijer, C. S. "On the -Function. VI." Proc. Nederl. Akad. Wetensch.49,
936-943, 1946.Meijer, C. S. "On the -Function. VII." Proc. Nederl. Akad. Wetensch.49,
1063-1072, 1946.Meijer, C. S. "On the -Function. VIII." Proc. Nederl. Akad. Wetensch.49,
1165-1175, 1946.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev,
O. I. "Evaluation of Integrals and the Mellin Transform." Itogi
Nauki i Tekhniki, Seriya Matemat. Analiz27, 3-146, 1989.Prudnikov,
A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Meijer -Function ." §8.2 in Integrals
and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach,
pp. 617-626, 1990.