Euler beta integral

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11: 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

12: 29.18 Mathematical Applications

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$\beta =K+\mathrm{i}{\beta }^{\prime },$ $0\le {\beta }^{\prime }\le 2K^{\prime },0\le \gamma \le 4K$,

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13: 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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14: 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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15: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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17.6.28 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{\alpha },q^{\beta }}{q^{\gamma }};q,z)=\frac{{\mathrm{\Gamma }}_q\left(\gamma \right)}{{\mathrm{\Gamma }}_q\left(\beta \right){\mathrm{\Gamma }}_q\left(\gamma -\beta \right)}{\int }_0^1\frac{t^{\beta -1}{(tq;q)}_{\gamma -\beta -1}}{{(xt;q)}_{\alpha }}{d}_{q}t.$$

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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, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $z$: complex variable and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.6.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(v), §17.6 and Ch.17

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16: 19.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. ►The first set of main functions treated in this chapter are Legendre’s complete integrals …of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals … ►However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. … ►The first three functions are incomplete integrals of the first, second, and third kinds, and the $\mathrm{cel}$ function includes complete integrals of all three kinds.

17: 19.29 Reduction of General Elliptic Integrals

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19.29.7 $${\int }_y^x\frac{a_{\alpha }+b_{\alpha }t}{a_{\delta }+b_{\delta }t}\frac{dt}{s(t)}=\frac{2}{3}{d}_{\alpha \beta }{d}_{\alpha \gamma }{R}_D(U_{\alpha \beta }^2,U_{\alpha \gamma }^2,U_{\alpha \delta }^2)+\frac{2X_{\alpha }{Y}_{\alpha }}{X_{\delta }{Y}_{\delta }{U}_{\alpha \delta }},$$ $U_{\alpha \delta }\ne 0$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $dx$: differential of $x$ , $\int$: integral, $X_{\alpha }$ , $U_n$ , $Y_{\alpha }$ , $d_{\alpha \beta }$ , $s(t)$: product, $\alpha$: parameter, $\beta$: parameter, $\gamma$: parameter and $\delta$: parameter

Referenced by:

§19.29(i), §19.29(i), §19.29(ii), §19.29(ii), §19.29(iii), §19.30(ii), §19.34

Permalink:

http://dlmf.nist.gov/19.29.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(i), §19.29 and Ch.19

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19.29.8 $${\int }_y^x\frac{a_{\alpha }+b_{\alpha }t}{a_5+b_{5}t}\frac{dt}{s(t)}=\frac{2}{3}\frac{d_{\alpha \beta }{d}_{\alpha \gamma }{d}_{\alpha \delta }}{d_{\alpha 5}}{R}_J(U_{12}^2,U_{13}^2,U_{23}^2,U_{\alpha 5}^2)+2R_C(S_{\alpha 5}^2,Q_{\alpha 5}^2),$$ $S_{\alpha 5}^2\in ℂ\setminus (-\mathrm{\infty },0)$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b)$: open interval, $\setminus$: set subtraction, $U_n$ , $S_n$ , $Q_{\alpha \beta }$ , $d_{\alpha \beta }$ , $s(t)$: product, $\alpha$: parameter, $\beta$: parameter, $\gamma$: parameter and $\delta$: parameter

Referenced by:

§19.29(i), §19.29(ii), §19.29(ii), §19.29(ii)

Permalink:

http://dlmf.nist.gov/19.29.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(i), §19.29 and Ch.19

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19.29.16 $$b_{\beta }{b}_{\gamma }I({𝐞}_{\alpha })=d_{\alpha \beta }{d}_{\alpha \gamma }I(-{𝐞}_{\alpha })+2b_{\alpha }\left(\frac{s(x)}{a_{\alpha }+b_{\alpha }x}-\frac{s(y)}{a_{\alpha }+b_{\alpha }y}\right),$$

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Symbols:

$I(𝐦)$: integral, $s(t)$: product, $d_{\alpha \beta }$ , $\alpha$: parameter, $\beta$: parameter and $\gamma$: parameter

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.29.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(ii), §19.29 and Ch.19

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18: 18.10 Integral Representations

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18.10.1 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{2^{\alpha +\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)}{(\sin \theta )}^{-2\alpha }{\int }_0^{\theta }\frac{\cos \left((n+\alpha +\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{-\alpha +\frac{1}{2}}}d\varphi ,$$ , $\alpha >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer

A&S Ref:

22.10.11

Referenced by:

§18.10(i), §18.10(i)

Permalink:

http://dlmf.nist.gov/18.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

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18.10.3 $$\frac{P_n^{(\alpha ,\beta )}\left(\cos \theta \right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\frac{2\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha -\beta \right)\mathrm{\Gamma }\left(\beta +\frac{1}{2}\right)}{\int }_0^1{\int }_0^{\pi }{\left({(\cos \frac{1}{2}\theta )}^2-r^2{(\sin \frac{1}{2}\theta )}^2+ir\sin \theta \cos \varphi \right)}^n{(1-r^2)}^{\alpha -\beta -1}{r}^{2\beta +1}{(\sin \varphi )}^{2\beta }d\varphi dr,$$ $\alpha >\beta >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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18.10.4 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^n{(\sin \varphi )}^{2\alpha }d\varphi ,$$ $\alpha >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer

A&S Ref:

22.10.10

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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19: 19.14 Reduction of General Elliptic Integrals

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19.14.4 $${\int }_y^x\frac{dt}{\sqrt{(a_1+b_1{t}^2)(a_2+b_2{t}^2)}}=\frac{1}{\sqrt{\gamma -\alpha }}F(\varphi ,k),$$ $k^2=(\gamma -\beta )/(\gamma -\alpha )$.

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Symbols:

$dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i), §19.14(i), §19.14(i), §19.15, §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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20: 32.9 Other Elementary Solutions

… ►Similar results hold when $\delta =0$ and $\beta \gamma \ne 0$. ► ${\text{P}}_{\text{III}}$ with $\beta =\delta =0$ has a first integral …${\text{P}}_{\text{III}}$ with $\alpha =\beta =\gamma =\delta =0$, has the general solution $w(z)=C{z}^{\mu }$, with $C$ and $\mu$ arbitrary constants. … ► ${\text{P}}_{\text{V}}$, with $\gamma =\delta =0$, has a first integral … ►Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of ${\text{P}}_{\text{VI}}$ with $\beta =\gamma =0$, $\delta =\frac{1}{2}$. …