Meijer G-Function

The Meijer -function is a very general function which reduces to simpler special functions in many common cases. The Meijer -function is defined by

G_(p,q)^(m,n)(x|a_1,...,a_p; b_1,...,b_q)=1/(2pii)int_(gamma_L)(product_(j=1)^(m)Gamma(b_j-s)product_(j=1)^(n)Gamma(1-a_j+s))/(product_(j=n+1)^(p)Gamma(a_j-s)product_(j=m+1)^(q)Gamma(1-b_j+s))x^sds,

(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

G_(p,q)^(m,n)(x,r|a_1,...,a_p; b_1,...,b_q) =1/(2pii)int_(gamma_L)(product_(j=1)^(m)Gamma(b_j+s)product_(j=1)^(n)Gamma(1-a_j-s))/(product_(j=n+1)^(p)Gamma(a_j+s)product_(j=m+1)^(q)Gamma(1-b_j-s))x^(-s/r)ds,

(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 G_(1,2)^(2,1)(2z|1/2; 3,-3) 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.

Special cases include

			(4)
			(5)
			(6)
			(7)

A special case of the 2-argument form is

(8)

Related Wolfram sites

http://functions.wolfram.com/HypergeometricFunctions/MeijerG/, http://functions.wolfram.com/HypergeometricFunctions/MeijerG1/

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References

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. Analiz 27, 3-146, 1989.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Meijer