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Barnes G-function ：ウィキペディア英語版

Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a 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.〔E. W. Barnes, "The theory of the G-function", ''Quarterly Journ. Pure and Appl. Math.'' 31 (1900), 264–314.〕 Up to elementary factors, it is a special case of the double gamma function.
Formally, the Barnes ''G''-function is defined in the following Weierstrass product form:
:

G(1+z)=(2\pi)^ \text\left(- \frac \right) \, \prod_^ \left\\right)^k \text\left(\frac-z\right) \right\}

where

\, \gamma \,

is the Euler–Mascheroni constant, exp(''x'') = ''e''^''x'', and ∏ is capital pi notation.
==Functional equation and integer arguments==

The Barnes ''G''-function satisfies the functional equation
:

G(z+1)=\Gamma(z)\, G(z)

with normalisation ''G''(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler Gamma function:
:

\Gamma(z+1)=z \, \Gamma(z) .

The functional equation implies that ''G'' takes the following values at integer arguments:
:

G(n)=\begin 0&\textn=-1,-2,\dots\\ \prod_^ i!&\textn=0,1,2,\dots\end

(in particular,

\,G(0)=G(1)=1\,

)
and thus
:

G(n)=\frac

where

\,\Gamma(x)\,

denotes the Gamma function and ''K'' denotes the K-function. The functional equation uniquely defines the G function if the convexity condition:

\, \fracG(x)\geq 0\,

is added.〔M. F. Vignéras, ''L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL

(2,\mathbb)

'', Astérisque 61, 235–249 (1979).〕

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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