Meijer G-function
In mathematics, the G-function was introduced by Cornelis Simon Meijer (1936) as a very general function intended to include most of the
known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert
E-function had the same aim, but Meijer's G-function was able to include those as particular cases as well. The first definition was made by Meijer
using a series; nowadays the accepted and more general definition is via a line integral in the complex plane, introduced in its full generality by
Arthur Erdélyi in 1953.
With the modern definition, the majority of the established special functions can be represented in terms of the Meijer G-function. A notable
property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a
functional equation that allows to liberate from a G-function G(z) any factor z
ρ
that is a constant power of its argument z, the closure implies that
whenever a function is expressible as a G-function of a constant multiple of some constant power of the function argument, f(x) = G(cx
γ
), the
derivative and the antiderivative of this function are expressible so too.
The wide coverage of special functions also lends power to uses of Meijer's G-function other than the representation and manipulation of
derivatives and antiderivatives. For example, the definite integral over the positive real axis of any function g(x) that can be written as a product
G
1
(cx
γ
)·G
2
(dx
δ
) of two G-functions with rational γ/δ equals just another G-function, and generalizations of integral transforms like the Hankel
transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels.
A still more general function, which introduces additional parameters into Meijer's G-function, is Fox's H-function.
Definition of the Meijer G-function
Differential equation
Relationship between the G-function and the generalized hypergeometric function
Polynomial cases
Basic properties of the G-function
Derivatives and antiderivatives
Recurrence relations
Multiplication theorems
Definite integrals involving the G-function
Laplace transform
Integral transforms based on the G-function
Narain transform
Wimp transform
Generalized Laplace transform
Meijer transform
Representation of other functions in terms of the G-function
See also
References
External links
A general definition of the Meijer G-function is given by the following line integral in the complex plane (Bateman & Erdélyi 1953, § 5.3-1):
where Γ denotes the gamma function. This integral is of the so-called Mellin–Barnes type, and may be viewed as an inverse Mellin transform. The
definition holds under the following assumptions:
Contents
Definition of the Meijer G-function
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