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31: 17.13 Integrals

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17.13.2 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-x{q}^{\alpha }/c;q)}_{\mathrm{\infty }}{(x{q}^{\beta }/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}\frac{cd}{c+d}\frac{{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(-q^{\beta }c/d;q)}_{\mathrm{\infty }}{(-q^{\alpha }d/c;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

Erratum (V1.0.1) for Equation (17.13.3)

Permalink:

http://dlmf.nist.gov/17.13.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the differential was identified incorrectly as a $q$-differential; the correct differential is $dt$.

See also:

Annotations for §17.13, §17.13 and Ch.17

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17.13.4 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-ct{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-ct;q)}_{\mathrm{\infty }}}{d}_{q}t=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right){(-c{q}^{\alpha };q)}_{\mathrm{\infty }}{(-q^{1-\alpha }/c;q)}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right){(-c;q)}_{\mathrm{\infty }}{(-q/c;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13, §17.13 and Ch.17

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32: 15.14 Integrals

… ►Integrals of the form $\int x^{\alpha }{(x+t)}^{\beta }F(a,b;c;x)dx$ and more complicated forms are given in Apelblat (1983, pp. 370–387), Prudnikov et al. (1990, §§1.15 and 2.21), Gradshteyn and Ryzhik (2000, §7.5) and Koornwinder (2015). …

33: 19.16 Definitions

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19.16.9 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a^{\prime }-1}\prod _{j=1}^n{(t+z_j)}^{-b_j}dt=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a-1}\prod _{j=1}^n{(1+t{z}_j)}^{-b_j}dt,$$ $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$, $z_j\in ℂ\setminus (-\mathrm{\infty },0]$,

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Defines:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function

Symbols:

$\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $ℝ$: real line, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.16(ii), §19.16(ii), §19.19, §19.20(iv), Erratum (V1.0.18) for Equation (19.16.9)

Permalink:

http://dlmf.nist.gov/19.16.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.18):

The constraint $a,a^{\prime }>0$, was replaced with $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$. It therefore follows from (19.16.10) that $a^{\prime }>0$.

Suggested 2017-07-25 by Bastien Roucariès

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

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19.16.12 $$R_{-a}(b_1,\mathrm{\dots },b_4;c-1,c-k^2,c,c-{\alpha }^2)=\frac{2{({\sin }^2\varphi )}^{1-a^{\prime }}}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\varphi }{(\sin \theta )}^{2a-1}{({\sin }^2\varphi -{\sin }^2\theta )}^{a^{\prime }-1}{(\cos \theta )}^{1-2b_1}{(1-k^2{\sin }^2\theta )}^{-b_2}{(1-{\alpha }^2{\sin }^2\theta )}^{-b_4}d\theta ,$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.16(ii), §19.25(i)

Permalink:

http://dlmf.nist.gov/19.16.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

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19.16.19 $$R_{-a}(b_1,\mathrm{\dots },b_n;0,z_2,\mathrm{\dots },z_n)=\frac{\mathrm{B}(a,a^{\prime }-b_1)}{\mathrm{B}(a,a^{\prime })}{R}_{-a}(b_2,\mathrm{\dots },b_n;z_2,\mathrm{\dots },z_n),$$ $a+a^{\prime }>0$, $a^{\prime }>b_1$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function and $n$: nonnegative integer

Referenced by:

§19.16(iii), §19.20(v)

Permalink:

http://dlmf.nist.gov/19.16.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(iii), §19.16 and Ch.19

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19.16.24 $$R_{-a}(𝐛;𝐳)=\frac{z_1^{a^{\prime }-b_1}}{\mathrm{B}(b_1,a^{\prime }-b_1)}{\int }_0^{\mathrm{\infty }}{t}^{b_1-1}{(t+z_1)}^{-a^{\prime }}{R}_{-a}(𝐛;0,t+z_2,\mathrm{\dots },t+z_n)dt,$$ $a^{\prime }>b_1$, $a+a^{\prime }>b_1>0$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

§19.16(iii), §19.28

Permalink:

http://dlmf.nist.gov/19.16.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(iii), §19.16 and Ch.19

34: 18.18 Sums

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18.18.1 $$a_n=\frac{n!(2n+\alpha +\beta +1)\mathrm{\Gamma }\left(n+\alpha +\beta +1\right)}{2^{\alpha +\beta +1}\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{\int }_{-1}^{1}f(x)P_n^{(\alpha ,\beta )}\left(x\right){(1-x)}^{\alpha }{(1+x)}^{\beta }dx.$$

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Defines:

$a_n$ (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable

Referenced by:

§18.18(i), §18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

… ►Assume also the integrals ${\int }_{-1}^1{(f(x))}^2{(1-x)}^{\alpha }{(1+x)}^{\beta }dx$ and ${\int }_{-1}^1{(f^{\prime }(x))}^2{(1-x)}^{\alpha +1}{(1+x)}^{\beta +1}dx$ converge. …

35: 19.23 Integral Representations

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19.23.8 $$R_{-a}(𝐛;𝐳)=\frac{2}{\mathrm{B}(b_1,b_2)}{\int }_0^{\pi /2}{(z_1{\cos }^2\theta +z_2{\sin }^2\theta )}^{-a}{(\cos \theta )}^{2b_1-1}{(\sin \theta )}^{2b_2-1}d\theta ,$$ $b_1,b_2>0$; $\mathrm{\Re }{z}_1,\mathrm{\Re }{z}_2>0$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Referenced by:

§19.2(iv), §19.23, §19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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19.23.10 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^1{u}^{a-1}{(1-u)}^{a^{\prime }-1}\prod _{j=1}^n{(1-u+u{z}_j)}^{-b_j}du,$$ $a,a^{\prime }>0$; $a+a^{\prime }={\sum }_{j=1}^n{b}_j$; $z_j\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.19, §19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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36: 32.9 Other Elementary Solutions

… ► ${\text{P}}_{\text{III}}$ with $\beta =\delta =0$ has a first integral …

37: 3.10 Continued Fractions

… ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). …

38: 22.19 Physical Applications

… ►There is bounded oscillatory motion near $x=0$, with period $4K\left(k\right)/\sqrt{1-\eta }$, and modulus $k=1/\sqrt{{\eta }^{-1}-1}$, for initial displacements with $|a|\le \sqrt{1/\beta }$. …

39: 1.5 Calculus of Two or More Variables

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1.5.22 $$\frac{d}{dx}{\int }_{\alpha (x)}^{\beta (x)}f(x,y)dy=f(x,\beta (x)){\beta }^{\prime }(x)-f(x,\alpha (x)){\alpha }^{\prime }(x)+{\int }_{\alpha (x)}^{\beta (x)}\frac{\partial f}{\partial x}dy.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

A&S Ref:

3.3.7

Permalink:

http://dlmf.nist.gov/1.5.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(iv), §1.5(iv), §1.5 and Ch.1

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40: 28.11 Expansions in Series of Mathieu Functions

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$${\beta }_n=\frac{1}{\pi }{\int }_0^{2\pi }f(x){\mathrm{se}}_n(x,q)dx.$$

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