Mellin transform

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1: 2.5 Mellin Transform Methods

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2.5.4 $$ℳI\left(z\right)=ℳf\left(1-z\right)ℳh\left(z\right).$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $f(x)$: locally integrable function, $I(x)$: convolution integral and $h(x)$: function

Keywords:

Mellintransform

Permalink:

http://dlmf.nist.gov/2.5.E4

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellintransform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5 and Ch.2

… ► … ►To apply the Mellin transform method outlined in §2.5(i), we require the transforms $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ to have a common strip of analyticity. … ►

2.5.26 $$ℳf\left(z\right)=ℳf_1\left(z\right)+ℳf_2\left(z\right)$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $f(x)$: locally integrable function and $f_j(t)$: truncated functions

Keywords:

Mellintransform

Referenced by:

§2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E26

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellintransform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

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2.5.28 $$ℳh\left(z\right)=ℳh_1\left(z\right)+ℳh_2\left(z\right)$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $f(x)$: locally integrable function and $h(x)$: locally integrable function

Referenced by:

§2.5(iii), §2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E28

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellintransform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

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2: 1.14 Integral Transforms

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§1.14(iv) Mellin Transform

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1.14.32 $$ℳ\left(f\right)\left(s\right)=ℳf\left(s\right)={\int }_0^{\mathrm{\infty }}{x}^{s-1}f(x)dx.$$

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Defines:

$ℳ\left(f\right)\left(s\right)$: Mellin transform

Symbols:

$dx$: differential of $x$ and $\int$: integral

Keywords:

Mellintransform

Referenced by:

§1.14(iv), §1.14(iv)

Permalink:

http://dlmf.nist.gov/1.14.E32

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

Annotations for §1.14(iv), §1.14 and Ch.1

… ►

Inversion

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Convolution

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Table 1.14.5: Mellin transforms.

► ►►

$f(x)$	${\displaystyle {\int }_0^{\mathrm{\infty }}}{x}^{s-1}f(x)dx$
…

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3: 2.6 Distributional Methods

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2.6.18 $$c_s=\frac{{(-1)}^s}{(s-1)!}ℳf\left(s\right),$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $!$: factorial (as in $n!$), $c\ne 0$: real and $f(t)$: locally integrable function

Keywords:

Mellintransform

Referenced by:

§2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E18

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

► $ℳf\left(z\right)$ being the Mellin transform of $f(t)$ or its analytic continuation (§2.5(ii)). … ►

2.6.60 $$ℳf\left(z\right)=ℳf_n\left(z\right),$$

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $f(t)$: locally integrable function, $n$: positive integer and $f_n(t)$: remainder

Keywords:

Mellintransform

Permalink:

http://dlmf.nist.gov/2.6.E60

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

Annotations for §2.6(iv), §2.6 and Ch.2

►where $ℳf\left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. … ►

2.6.61 $$ℳh_x\left(j+\alpha \right)=x^{-j-\alpha }ℳh\left(j+\alpha \right),$$

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $f(t)$: locally integrable function and $h(x)$: function

Keywords:

Mellintransform

Referenced by:

§2.6(iv)

Permalink:

http://dlmf.nist.gov/2.6.E61

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

Annotations for §2.6(iv), §2.6 and Ch.2

…

4: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by ►

15.14.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}𝐅(\genfrac{}{}{0pt}{}{a,b}{c};-x)dx=\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(a-s\right)\mathrm{\Gamma }\left(b-s\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-s\right)},$$ $\min (\mathrm{\Re }a,\mathrm{\Re }b)>\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $s$: nonnegative integer, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellintransform

Referenced by:

§15.14

Permalink:

http://dlmf.nist.gov/15.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.14 and Ch.15

… ►Mellin transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §6.9), Oberhettinger (1974, §1.15), and Marichev (1983, pp. 288–299). Inverse Mellin transforms are given in Erdélyi et al. (1954a, §7.5). …

5: 2.3 Integrals of a Real Variable

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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 Mellintransform 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).

6: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellintransform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellintransform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_3\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_3\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.3 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}(1-{\theta }_4\left(0\right|\mathrm{i}{x}^2))dx=(1-2^{1-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellintransform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>0$ because $1-{\theta }_4\left(0\right|\mathrm{i}{x}^2)=1-x^{-1}{\theta }_2\left(0\right|\mathrm{i}{x}^{-2})\sim 1$ as $x\to 0+$, which follows from (20.7.33) and (20.4.3).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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7: 13.23 Integrals

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§13.23(i) Laplace and Mellin Transforms

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13.23.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)}{{\left(z+\frac{1}{2}\right)}^{\mu +\nu +\frac{1}{2}}}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}+\mu -\kappa ,\frac{1}{2}+\mu +\nu }{1+2\mu };\frac{1}{z+\frac{1}{2}}),$$ $\mathrm{\Re }\mu +\nu +\frac{1}{2}>0$, $\mathrm{\Re }z>\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellintransform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

►

13.23.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(2\mu +1\right){\left(z+\frac{1}{2}\right)}^{-\kappa -\mu -\frac{1}{2}}{\left(z-\frac{1}{2}\right)}^{\kappa -\mu -\frac{1}{2}},$$ $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }z>\frac{1}{2}$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellintransform

Permalink:

http://dlmf.nist.gov/13.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

►

13.23.3 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\kappa -\nu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\nu \right)},$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform, Mellintransform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

… ►For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), and Prudnikov et al. (1992a, §§3.34, 3.35). …

8: 13.10 Integrals

… ►

§13.10(iii) Mellin Transforms

►

13.10.10 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}𝐌(a,b,-t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-\lambda \right)},$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellintransform

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iii), §13.10 and Ch.13

►

13.10.11 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda -1}U(a,b,t)dt=\frac{\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(a-\lambda \right)\mathrm{\Gamma }\left(\lambda -b+1\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)},$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellintransform

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iii), §13.10 and Ch.13

►For additional Mellin transforms see Erdélyi et al. (1954a, §§6.9, 7.5), Marichev (1983, pp. 283–287), and Oberhettinger (1974, §§1.13, 2.8). …

9: 16.15 Integral Representations and Integrals

… ►

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

ⓘ

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:

Mellintransform

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

►

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

ⓘ

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:

Mellintransform

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §16.15 and Ch.16

►

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

ⓘ

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:

Mellintransform

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

… ►

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:

Mellintransform

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

…

10: 18.17 Integrals

… ►

§18.17(vii) Mellin Transforms

►

Jacobi

… ►

Ultraspherical

… ►

Legendre

… ►

Laguerre

…