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41: 13.23 Integrals

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

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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, Mellin transform

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

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

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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, Mellin transform

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

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13.23.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{W}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\nu \right){}_2{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}-\mu +\nu ,\frac{1}{2}+\mu +\nu }{\nu -\kappa +1};\frac{1}{2}-z),$$ $\mathrm{\Re }\left(\nu +\frac{1}{2}\right)>|\mathrm{\Re }\mu |$, $\mathrm{\Re }z>-\frac{1}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E4

Encodings:

pMML, png, TeX

See also:

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

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13.23.6 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{M}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{I}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E6

Encodings:

pMML, png, TeX

See also:

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

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13.23.7 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{W}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{2z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{K}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E7

Encodings:

pMML, png, TeX

See also:

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

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42: 15.14 Integrals

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

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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:

Mellin transform

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

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43: 18.17 Integrals

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18.17.36 $${\int }_{-1}^1{(1-x)}^{z-1}{(1+x)}^{\beta }{P}_n^{(\alpha ,\beta )}\left(x\right)dx=\frac{2^{\beta +z}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1+\beta +n\right){\left(1+\alpha -z\right)}_n}{n!\mathrm{\Gamma }\left(1+\beta +z+n\right)},$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.37 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_n^{(\lambda )}\left(x\right)x^{z-1}dx=\frac{\pi {\mathrm{\hspace{0.17em}2}}^{1-2\lambda -z}\mathrm{\Gamma }\left(n+2\lambda \right)\mathrm{\Gamma }\left(z\right)}{n!\mathrm{\Gamma }\left(\lambda \right)\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}n+\lambda +\frac{1}{2}z\right)\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n\right)},$$ $\mathrm{\Re }z>0$.

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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 $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.38 $${\int }_0^1{P}_{2n}\left(x\right)x^{z-1}dx=\frac{{(-1)}^n{\left(\frac{1}{2}-\frac{1}{2}z\right)}_n}{2{\left(\frac{1}{2}z\right)}_{n+1}},$$ $\mathrm{\Re }z>0$,

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.39 $${\int }_0^1{P}_{2n+1}\left(x\right)x^{z-1}dx=\frac{{(-1)}^n{\left(1-\frac{1}{2}z\right)}_n}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)}_{n+1}},$$ $\mathrm{\Re }z>-1$.

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.40 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{L}_n^{(\alpha )}\left(bx\right)x^{z-1}dx=\frac{\mathrm{\Gamma }\left(z+n\right)}{n!}{(a-b)}^n{a}^{-n-z}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,1+\alpha -z}{1-n-z};\frac{a}{a-b}),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }z>0$.

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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, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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44: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

§17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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F. H. Jackson’s Transformations

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Transformations of ${}_3{}^{}\varphi _2^{}$-Series

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Sears–Carlitz Transformation

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Mixed-Base Heine-Type Transformations

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45: 18.7 Interrelations and Limit Relations

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§18.7(i) Linear Transformations

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18.7.7 $$T_n^{\ast }\left(x\right)=T_n\left(2x-1\right),$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $T_n^{\ast }\left(x\right)$: shifted Chebyshev polynomial of the first kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.14

Referenced by:

§18.7(i)

Permalink:

http://dlmf.nist.gov/18.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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18.7.11 $${\mathrm{𝐻𝑒}}_n\left(x\right)=2^{-\frac{1}{2}n}{H}_n\left(2^{-\frac{1}{2}}x\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.18

Referenced by:

§18.5(iii)

Permalink:

http://dlmf.nist.gov/18.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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18.7.12 $$H_n\left(x\right)=2^{\frac{1}{2}n}{\mathrm{𝐻𝑒}}_n\left(2^{\frac{1}{2}}x\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.19

Referenced by:

§18.7(i)

Permalink:

http://dlmf.nist.gov/18.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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§18.7(ii) Quadratic Transformations

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46: 12.17 Physical Applications

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12.17.2 $${\nabla }^2=\frac{{\partial }^2}{{\partial x}^2}+\frac{{\partial }^2}{{\partial y}^2}+\frac{{\partial }^2}{{\partial z}^2}$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

… ►Setting $w=U(\xi )V(\eta )W(\zeta )$ and separating variables, we obtain …The first two equations can be transformed into (12.2.2) or (12.2.3). ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. … ►Miller (1974) treats separation of variables by group theoretic methods. …

47: 2.3 Integrals of a Real Variable

… ►Assume that the Laplace transform …Then … ►where $ℳf\left(\alpha \right)$ is the Mellin transform of $f(t)$ (§2.5(i)). … ►The integral (2.3.24) transforms into … ►

§2.3(vi) Asymptotics of Mellin Transforms

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48: 29.18 Mathematical Applications

… ►when transformed to sphero-conal coordinates $r,\beta ,\gamma$: … ►The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha ,\beta ,\gamma$: …

49: 13.10 Integrals

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§13.10(ii) Laplace Transforms

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§13.10(iii) Mellin Transforms

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§13.10(iv) Fourier Transforms

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§13.10(v) Hankel Transforms

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50: 11.5 Integral Representations

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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

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

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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

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