Mellin%E2%80%93Barnes%20type

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11: 12.5 Integral Representations

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§12.5(iii) Mellin–Barnes Integrals

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12.5.8 $$U(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}+a-2t\right)2^t{z}^{2t}dt,$$ $a\ne -\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\mathrm{\dots }$, ,

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter

Keywords:

Mellin transform

A&S Ref:

19.5.13 (modification and correction of)

Referenced by:

§12.5(iii)

Permalink:

http://dlmf.nist.gov/12.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(iii), §12.5 and Ch.12

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12.5.9 $$V(a,z)=\sqrt{\frac{2}{\pi }}\frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}-a-2t\right)2^t{z}^{2t}\cos \left(\pi t\right)dt,$$ $a\ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$, ,

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Symbols:

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

Keywords:

Mellin transform

A&S Ref:

19.5.13 (modification and correction of)

Referenced by:

§12.5(iii), §12.8(ii)

Permalink:

http://dlmf.nist.gov/12.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(iii), §12.5 and Ch.12

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12: 16.15 Integral Representations and Integrals

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16.15.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)}{\int }_0^1\frac{u^{\alpha -1}{(1-u)}^{\gamma -\alpha -1}}{{(1-ux)}^{\beta }{(1-uy)}^{{\beta }^{\prime }}}du,$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\left(\gamma -\alpha \right)>0$,

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/16.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

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16.15.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-{\beta }^{\prime }\right)}{\int }_0^1{\int }_0^1\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u)}^{\gamma -\beta -1}{(1-v)}^{{\gamma }^{\prime }-{\beta }^{\prime }-1}}{{(1-ux-vy)}^{\alpha }}dudv,$$ $\mathrm{\Re }\gamma >\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/16.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

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16.15.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,$$ $\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Notes:

Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.

Referenced by:

Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)

Permalink:

http://dlmf.nist.gov/16.15.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.14):

A stray “$f$” was removed; it appeared after the comma at the end of this equation.

See also:

Annotations for §16.15 and Ch.16

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16.15.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x(1-y),y(1-x))=\frac{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left({\gamma }^{\prime }\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left(\gamma -\alpha \right)\mathrm{\Gamma }\left({\gamma }^{\prime }-\beta \right)}{\int }_0^1{\int }_0^1\frac{u^{\alpha -1}{v}^{\beta -1}{(1-u)}^{\gamma -\alpha -1}{(1-v)}^{{\gamma }^{\prime }-\beta -1}}{{(1-ux)}^{\gamma +{\gamma }^{\prime }-\alpha -1}{(1-vy)}^{\gamma +{\gamma }^{\prime }-\beta -1}{(1-ux-vy)}^{\alpha +\beta -\gamma -{\gamma }^{\prime }+1}}dudv,$$ $\mathrm{\Re }\gamma >\mathrm{\Re }\alpha >0$, $\mathrm{\Re }{\gamma }^{\prime }>\mathrm{\Re }\beta >0$.

ⓘ

Symbols:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Source:

Burchnall and Chaundy (1940, (68))

Referenced by:

§16.15

Permalink:

http://dlmf.nist.gov/16.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

►For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). …

13: 7.7 Integral Representations

… ►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. … ►

Mellin–Barnes Integrals

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7.7.13 $$\mathrm{f}\left(z\right)=\frac{{(2\pi )}^{-3/2}}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\zeta }^{-s}\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(s+\frac{1}{2}\right)\mathrm{\Gamma }\left(s+\frac{3}{4}\right)\mathrm{\Gamma }\left(\frac{1}{4}-s\right)ds,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\zeta$: change of variable and $c$: constant

Keywords:

Mellin transform

Referenced by:

§7.7(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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7.7.14 $$\mathrm{g}\left(z\right)=\frac{{(2\pi )}^{-3/2}}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\zeta }^{-s}\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(s+\frac{1}{2}\right)\mathrm{\Gamma }\left(s+\frac{1}{4}\right)\mathrm{\Gamma }\left(\frac{3}{4}-s\right)ds.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\zeta$: change of variable and $c$: constant

Keywords:

Mellin transform

Referenced by:

§7.7(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

…

14: 2.3 Integrals of a Real Variable

… ►Other types of singular behavior in the integrand can be treated in an analogous manner. … ►

2.3.12 $${\int }_0^{\mathrm{\infty }}f(xt)q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}ℳf\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{x^{(s+\lambda )/\mu }},$$ $x\to +\mathrm{\infty }$,

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\int$: integral, $f(x)$: function, $a$: positive constant, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

§2.3(ii), §2.3(ii)

Permalink:

http://dlmf.nist.gov/2.3.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §2.3(ii), §2.3 and Ch.2

►where $ℳf\left(\alpha \right)$ is the Mellin transform of $f(t)$ (§2.5(i)). … ►

§2.3(vi) Asymptotics of Mellin Transforms

►For the asymptotics of the Mellin transform $ℳf\left(z\right)={\int }_0^{\mathrm{\infty }}{t}^{z-1}f(t)dt$ as $z\to \mathrm{\infty }$ see Frenzen (1987b), Sidi (1985, 2011).

15: 13.16 Integral Representations

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§13.16(iii) Mellin–Barnes Integrals

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13.16.10 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left({\mathrm{e}}^{\pm \pi \mathrm{i}}z\right)=\frac{{\mathrm{e}}^{\frac{1}{2}z\pm (\frac{1}{2}+\mu )\pi \mathrm{i}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(t-\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu -t\right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)}{z}^{t}dt,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(iii), §13.16 and Ch.13

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13.16.11 $$W_{\kappa ,\mu }\left(z\right)=\frac{{\mathrm{e}}^{-\frac{1}{2}z}}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)\mathrm{\Gamma }\left(-\kappa -t\right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{z}^{-t}dt,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.16.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(iii), §13.16 and Ch.13

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13.16.12 $$W_{\kappa ,\mu }\left(z\right)=\frac{{\mathrm{e}}^{\frac{1}{2}z}}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)}{\mathrm{\Gamma }\left(1-\kappa +t\right)}{z}^{-t}dt,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.16.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(iii), §13.16 and Ch.13

…

16: 11.5 Integral Representations

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Mellin–Barnes Integrals

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11.5.8 $${(\frac{1}{2}x)}^{-\nu -1}{𝐇}_{\nu }\left(x\right)=-\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\pi \csc \left(\pi s\right)}{\mathrm{\Gamma }\left(\frac{3}{2}+s\right)\mathrm{\Gamma }\left(\frac{3}{2}+\nu +s\right)}{(\frac{1}{4}{x}^2)}^{s}ds,$$ $x>0$, $\mathrm{\Re }\nu >-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: real or complex order

Keywords:

Mellin transform

Referenced by:

§11.5(ii), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

►

11.5.9 $${(\frac{1}{2}z)}^{-\nu -1}{𝐋}_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }}^{(0+)}\frac{\pi \csc \left(\pi s\right)}{\mathrm{\Gamma }\left(\frac{3}{2}+s\right)\mathrm{\Gamma }\left(\frac{3}{2}+\nu +s\right)}{(-\frac{1}{4}{z}^2)}^{s}ds.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Keywords:

Mellin transform

Referenced by:

§11.5(ii), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

…

17: 13.4 Integral Representations

… ►

§13.4(iii) Mellin–Barnes Integrals

… ►

13.4.16 $$𝐌(a,b,-z)=\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(b+t\right)}{z}^{t}dt,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

A&S Ref:

13.2.9 (in different form)

Permalink:

http://dlmf.nist.gov/13.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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13.4.17 $$U(a,b,z)=\frac{z^{-a}}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(1+a-b+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(1+a-b\right)}{z}^{-t}dt,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

A&S Ref:

13.2.10

Referenced by:

§13.6(vi)

Permalink:

http://dlmf.nist.gov/13.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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13.4.18 $$U(a,b,z)=\frac{z^{1-b}{\mathrm{e}}^z}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(b-1+t\right)\mathrm{\Gamma }\left(t\right)}{\mathrm{\Gamma }\left(a+t\right)}{z}^{-t}dt,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric 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, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

…

18: 8.19 Generalized Exponential Integral

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8.19.2 $$E_p\left(z\right)=z^{p-1}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t^p}dt.$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§8.19(i), §8.19(viii)

Permalink:

http://dlmf.nist.gov/8.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19 and Ch.8

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8.19.3 $$E_p\left(z\right)={\int }_1^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^p}dt,$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

A&S Ref:

5.1.4

Referenced by:

§8.19(iii)

Permalink:

http://dlmf.nist.gov/8.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

►

8.19.4 $$E_p\left(z\right)=\frac{z^{p-1}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{t^{p-1}{\mathrm{e}}^{-zt}}{1+t}dt,$$ , $\mathrm{\Re }p>0$.

ⓘ

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, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/8.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

►Integral representations of Mellin–Barnes type for $E_p\left(z\right)$ follow immediately from (8.6.11), (8.6.12), and (8.19.1). …

19: 14.17 Integrals

… ►

§14.17(ii) Barnes’ Integral

… ►

§14.17(vi) Mellin Transforms

►For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).

20: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

§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 $m=1$. …