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G-function (other)
*Barnes G-function, related to the Gamma function *Meijer G-function In mathematics, the G-function was introduced by 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 ..., a generalization of the hypergeometric function * Siegel G-function, a class of functions in transcendence theory {{Mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a function (mathematics), function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function. Formally, the Barnes ''G''-function is defined in the following Weierstrass product form: : G(1+z)=(2\pi)^ \exp\left(- \frac \right) \, \prod_^\infty \left\ where \, \gamma is the Euler–Mascheroni constant, exponential function, exp(''x'') = ''e''''x'' is the exponential function, and Π denotes multiplication (capital pi notation). The integral representation, which may be deduced from the relation to the double gamma function, is : \log G(1+z) = \frac\log(2\pi) +\int_0^\infty\frac\left[\frac +\frace^ -\frac\right] As an entire function, ''G'' is of order two, and of infinite type. This can be deduced from the asymptoti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meijer G-function
In mathematics, the G-function was introduced by 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |