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

(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.

Special cases include

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

A special case of the 2-argument form is

(8)