Euler beta integral

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21: 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

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22: 8.1 Special Notation

… ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

23: 32.2 Differential Equations

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32.2.13 $$z(1-z)𝐼\left({\int }_{\mathrm{\infty }}^w\frac{dt}{\sqrt{t(t-1)(t-z)}}\right)=\sqrt{w(w-1)(w-z)}\left(\alpha +\frac{\beta z}{w^2}+\frac{\gamma (z-1)}{{(w-1)}^2}+(\delta -\frac{1}{2})\frac{z(z-1)}{{(w-z)}^2}\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: real, $\alpha$: arbitrary constant, $𝐼$: operator, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(iv), §32.2 and Ch.32

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24: 15.14 Integrals

§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

►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). … ►For other integral transforms see Erdélyi et al. (1954b), Prudnikov et al. (1992b, §4.3.43), and also §15.9(ii).

25: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
5.24(v) $\mathrm{B}(a,b)$, $a,b\in ℝ$	✓	✓	✓	✓	✓	✓		✓	✓		✓	✓	✓	✓			✓		✓	✓	✓	✓	✓
5.24(vi) $\mathrm{B}(a,b)$, $a,b\in ℂ$	✓	✓		✓							✓	a	✓							✓	✓	✓	a
…
11.16(iii) Integrals of …														✓					✓		✓	✓	a
…
19 Elliptic Integrals
…
24 Bernoulli and Euler Polynomials
…

…

26: 18.17 Integrals

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18.17.9 $$\frac{{(1-x)}^{\alpha +\mu }{P}_n^{(\alpha +\mu ,\beta -\mu )}\left(x\right)}{\mathrm{\Gamma }\left(\alpha +\mu +n+1\right)}={\int }_x^1\frac{{(1-y)}^{\alpha }{P}_n^{(\alpha ,\beta )}\left(y\right)}{\mathrm{\Gamma }\left(\alpha +n+1\right)}\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$, ,

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Symbols:

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

Referenced by:

§1.15(vi), §18.17(iv), §18.17(iv), §18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E9

Encodings:

pMML, png, TeX

See also:

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

►

18.17.10 $$\frac{x^{\beta +\mu }{(x+1)}^n}{\mathrm{\Gamma }\left(\beta +\mu +n+1\right)}{P}_n^{(\alpha ,\beta +\mu )}\left(\frac{x-1}{x+1}\right)={\int }_0^x\frac{y^{\beta }{(y+1)}^n}{\mathrm{\Gamma }\left(\beta +n+1\right)}{P}_n^{(\alpha ,\beta )}\left(\frac{y-1}{y+1}\right)\frac{{(x-y)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$, $x>0$,

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Symbols:

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

Referenced by:

§18.17(iv), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E10

Encodings:

pMML, png, TeX

See also:

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

►

18.17.11 $$\frac{\mathrm{\Gamma }\left(n+\alpha +\beta -\mu +1\right)}{x^{n+\alpha +\beta -\mu +1}}{P}_n^{(\alpha ,\beta -\mu )}\left(1-2x^{-1}\right)={\int }_x^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n+\alpha +\beta +1\right)}{y^{n+\alpha +\beta +1}}{P}_n^{(\alpha ,\beta )}\left(1-2y^{-1}\right)\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\alpha +\beta +1>\mu >0$, $x>1$,

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Symbols:

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

Referenced by:

§1.15(vi), §18.17(iv), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E11

Encodings:

pMML, png, TeX

See also:

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

… ►

18.17.16 $${\int }_{-1}^1{(1-x)}^{\alpha }{(1+x)}^{\beta }{P}_n^{(\alpha ,\beta )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx=\frac{{(\mathrm{i}y)}^n{\mathrm{e}}^{\mathrm{i}y}}{n!}{2}^{n+\alpha +\beta +1}\mathrm{B}(n+\alpha +1,n+\beta +1){}_1{}^{}F_1^{}(n+\alpha +1;2n+\alpha +\beta +2;-2\mathrm{i}y).$$

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Symbols:

$\mathrm{B}(a,b)$: beta function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v), §18.17(vi)

Permalink:

http://dlmf.nist.gov/18.17.E16

Encodings:

pMML, png, TeX

See also:

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

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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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27: 19.5 Maclaurin and Related Expansions

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19.5.4 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{\pi }{2}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_n}{n!}\sum _{m=0}^n\frac{{\left(\frac{1}{2}\right)}_m}{m!}{k}^{2m}{\alpha }^{2n-2m}=\frac{\pi }{2}{F}_1(\frac{1}{2};\frac{1}{2},1;1;k^2,{\alpha }^2),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Source:

Carlson (1961b, (2.10))

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.4_1 $$F(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.8))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_2 $$E(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},-\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.9))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

►

19.5.4_3 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{F}_1(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};{\sin }^2\varphi ,{\alpha }^2{\sin }^2\varphi )k^{2m},$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Source:

Carlson (1961b, (2.11))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

…

28: 5.20 Physical Applications

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5.20.3 $${\psi }_n(\beta )={\int }_{{ℝ}^n}{\mathrm{e}}^{-\beta W}dx={(2\pi )}^{n/2}{\beta }^{-(n/2)-(\beta n(n-1)/4)}{\left(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right)\right)}^{-n}\prod _{j=1}^n\mathrm{\Gamma }\left(1+\frac{1}{2}j\beta \right).$$

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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, $\int$: integral, $ℝ$: real line, $j$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

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5.20.5 $${\psi }_n(\beta )=\frac{1}{{(2\pi )}^n}{\int }_{{[-\pi ,\pi ]}^n}{\mathrm{e}}^{-\beta W}d{\theta }_1\mathrm{\cdots }d{\theta }_n=\mathrm{\Gamma }\left(1+\frac{1}{2}n\beta \right){(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right))}^{-n}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $n$: nonnegative integer, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

…

29: 5.18 $q$-Gamma and $q$-Beta Functions

§5.18 $q$-Gamma and $q$-Beta Functions

… ►Also, $\ln {\mathrm{\Gamma }}_q\left(x\right)$ is convex for $x>0$, and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds. … ►For the $q$-digamma or $q$-psi function ${\psi }_q(z)={\mathrm{\Gamma }}_q^{\prime }\left(z\right)/{\mathrm{\Gamma }}_q\left(z\right)$ see Salem (2013). ►

§5.18(iii) $q$-Beta Function

… ►For $q$-integrals see §17.2(v).

30: 13.27 Mathematical Applications

… ►

13.27.1

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Permalink:

http://dlmf.nist.gov/13.27.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.27 and Ch.13

►where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma >0$. …The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. …