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Meijer G-Function


The Meijer G-function is a very general function which reduces to simpler special functions in many common cases. The Meijer G-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 Gamma(s) 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),

 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^(-s)ds,
(2)

This form provides more consistency with the definition of this function via an inverse Mellin transform.

The Meijer G-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].

MeijerGContourPlane
MeijerGContour

In both (2) and (3), the contour gamma_L lies between the poles of Gamma(1-a_i-s) and the poles of Gamma(b_i+s). 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 G-function.

Special cases include

G_(22)^(12)(z|1,1; 1,0)=ln(z+1)
(4)
G_(22)^(12)(z|1,1; 1,1)=z/(z+1)
(5)
G_(02)^(10)(1/2z|-; 0,1/2)=(cos(sqrt(2z)))/(sqrt(pi))
(6)
G_(10)^(01)(z|1-a; -)=e^(-1/z)z^(-a).
(7)

A special case of the 2-argument form is

 G_(02)^(10)(1/2z,1/2|-; 0,1/2)=(cosz)/(sqrt(pi)).
(8)

See also

Barnes G-Function, Fox H-Function, G-Transform, Kampe de Feriet Function, MacRobert's E-Function, Ramanujan g- and G-Functions

Related Wolfram sites

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

Explore with Wolfram|Alpha

References

Adamchik, V. "The Evaluation of Integrals of Bessel Functions via G-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 (G and E)" and "Meijer's G-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 G_(p,q)^(m,n)(z)." Proc. Nederl. Akad. Wetensch. 44, 1062-1070, 1941.Meijer, C. S. "On the G-Function. II." Proc. Nederl. Akad. Wetensch. 49, 344-456, 1946.Meijer, C. S. "On the G-Function. III." Proc. Nederl. Akad. Wetensch. 49, 457-469, 1946.Meijer, C. S. "On the G-Function. IV." Proc. Nederl. Akad. Wetensch. 49, 632-641, 1946.Meijer, C. S. "On the G-Function. V." Proc. Nederl. Akad. Wetensch. 49, 765-772, 1946.Meijer, C. S. "On the G-Function. VI." Proc. Nederl. Akad. Wetensch. 49, 936-943, 1946.Meijer, C. S. "On the G-Function. VII." Proc. Nederl. Akad. Wetensch. 49, 1063-1072, 1946.Meijer, C. S. "On the G-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 G-Function G_(pq)^(mn)(z|(a_p); (b_p))." §8.2 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 617-626, 1990.

Referenced on Wolfram|Alpha

Meijer G-Function

Cite this as:

Weisstein, Eric W. "Meijer G-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MeijerG-Function.html

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