Barnes’ G-function

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1: 5.17 Barnes’ $G$-Function (Double Gamma Function)

§5.17 Barnes’ $G$-Function (Double Gamma Function)

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$G\left(z+1\right)=\mathrm{\Gamma }\left(z\right)G\left(z\right),$

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$G\left(1\right)=1,$

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5.17.2 $$G\left(n\right)=(n-2)!(n-3)!\mathrm{\cdots }1!,$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $!$: factorial (as in $n!$) and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

… ►When $z\to \mathrm{\infty }$ in , …

2: 15.12 Asymptotic Approximations

… ►where ►

15.12.13 $$G_0(\pm \beta )={\left(2+{\mathrm{e}}^{\pm \zeta }\right)}^{c-b-(1/2)}{\left(1+{\mathrm{e}}^{\pm \zeta }\right)}^{a-c+(1/2)}{\left(z-1-{\mathrm{e}}^{\pm \zeta }\right)}^{-a+(1/2)}\sqrt{\frac{\beta }{{\mathrm{e}}^{\zeta }-{\mathrm{e}}^{-\zeta }}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $G_0(\pm \zeta )$: function

Permalink:

http://dlmf.nist.gov/15.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

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3: 5.19 Mathematical Applications

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§5.19(ii) Mellin–Barnes Integrals

►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …

4: Richard B. Paris

… ► Wood), published by Longman Scientific and Technical in 1986, and Asymptotics and Mellin-Barnes Integrals (with D. …

5: 10.17 Asymptotic Expansions for Large Argument

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10.17.17 $$R_{\mathrm{\ell }}^{\pm }(\nu ,z)={(-1)}^{\mathrm{\ell }}2\cos \left(\nu \pi \right)\left(\sum _{k=0}^{m-1}{(\pm \mathrm{i})}^k\frac{a_k(\nu )}{z^k}{G}_{\mathrm{\ell }-k}\left(\mp 2\mathrm{i}z\right)+R_{m,\mathrm{\ell }}^{\pm }(\nu ,z)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $G_p\left(z\right)$: rescaled terminant function, $m$: integer, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder

Permalink:

http://dlmf.nist.gov/10.17.E17

Encodings:

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See also:

Annotations for §10.17(v), §10.17 and Ch.10

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6: 8.6 Integral Representations

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Mellin–Barnes Integrals

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8.6.10 $$\gamma (a,z)=\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s\right)}{a-s}{z}^{a-s}ds,$$ , $a\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter and $c$: real constant

Keywords:

Mellin transform

Referenced by:

§8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))

Permalink:

http://dlmf.nist.gov/8.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

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8.6.11 $$\mathrm{\Gamma }(a,z)=\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(s+a\right)\frac{z^{-s}}{s}ds,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter and $c$: real constant

Keywords:

Mellin transform

Referenced by:

§8.19(i), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))

Permalink:

http://dlmf.nist.gov/8.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

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8.6.12 $$\mathrm{\Gamma }(a,z)=-\frac{z^{a-1}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(1-a\right)}\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(s+1-a\right)\frac{\pi z^{-s}}{\sin \left(\pi s\right)}ds,$$ , $a\ne 1,2,3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $a$: parameter and $c$: real constant

Keywords:

Mellin transform

Referenced by:

§8.19(i), §8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))

Permalink:

http://dlmf.nist.gov/8.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

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7: 5.13 Integrals

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Barnes’ Beta Integral

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8: 5.1 Special Notation

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9: 12.5 Integral Representations

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§12.5(iii) Mellin–Barnes Integrals

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10: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(v) Mellin–Barnes Integrals

►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions ${}_p{}^{}F_q^{}$ and ${}_{p+1}{}^{}F_p^{}$ of matrix argument. …These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case $m=1$. …