beta integral

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21: 20.10 Integrals

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

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See also:

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

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

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

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22: Bibliography M

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Algorithm AS 63. The incomplete beta integral. Appl. Statist. 22 (3), pp. 409–411.

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Notes:

Double-precision Fortran. For remark see same journal 26 (1977), pp. 111-114

External Links:

FullText, Code

Referenced by:

4th item.

Encodings:

BibTeX

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23: 19.33 Triaxial Ellipsoids

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19.33.11 $$U=\frac{1}{2}{(\alpha \beta \gamma )}^2{R}_F({\alpha }^2,{\beta }^2,{\gamma }^2){\int }_0^{\mathrm{\infty }}{(g(r))}^{2}dr,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ , $\int$: integral, $\alpha$: positive constant, $\beta$: positive constant, $\gamma$: positive constant, $U$: energy and $g(r)$: function

Permalink:

http://dlmf.nist.gov/19.33.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(iv), §19.33 and Ch.19

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24: 36.6 Scaling Relations

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${\mathrm{\Psi }}_K(𝐱;k)=k^{{\beta }_K}{\mathrm{\Psi }}_K\left(𝐲(k)\right),$

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${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)=k^{{\beta }^{(\mathrm{U})}}{\mathrm{\Psi }}^{(\mathrm{U})}\left({𝐲}^{(\mathrm{U})}(k)\right),$

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25: 19.28 Integrals of Elliptic Integrals

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19.28.1 $${\int }_0^1{t}^{\sigma -1}{R}_F(0,t,1)dt=\frac{1}{2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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19.28.2 $${\int }_0^1{t}^{\sigma -1}{R}_G(0,t,1)dt=\frac{\sigma }{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Permalink:

http://dlmf.nist.gov/19.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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19.28.3 $${\int }_0^1{t}^{\sigma -1}(1-t)R_D(0,t,1)dt=\frac{3}{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2.$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E3

Encodings:

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See also:

Annotations for §19.28 and Ch.19

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26: 1.15 Summability Methods

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1.15.48 $${𝐼}^{\alpha }{𝐼}^{\beta }={𝐼}^{\alpha +\beta },$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\beta >0$.

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Symbols:

$\mathrm{\Re }$: real part and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vi)

Permalink:

http://dlmf.nist.gov/1.15.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vi), §1.15 and Ch.1

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27: 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.

28: 21.7 Riemann Surfaces

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21.7.9 $$E(P_1,P_2)=\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left({\int }_{P_1}^{P_2}𝝎\right|𝛀)/\left(\sqrt{\zeta (P_1)}\sqrt{\zeta (P_2)}\right),$$

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Defines:

$E(P_1,P_2)$: prime form (locally)

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\int$: integral, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $P_1,P_2$: points and $𝝎$: vector of differentials

Permalink:

http://dlmf.nist.gov/21.7.E9

Encodings:

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See also:

Annotations for §21.7(ii), §21.7 and Ch.21

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29: 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:

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See also:

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

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30: 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:

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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:

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See also:

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

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19.29.15 $$b_{j}I({𝐞}_l-{𝐞}_j)=d_{lj}I(-{𝐞}_j)+b_{l}I(\mathrm{𝟎}),$$ $j,l=1,2,\mathrm{\dots },n$,

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Symbols:

$l$: nonnegative integer, $n$: nonnegative integer, $I(𝐦)$: integral and $d_{\alpha \beta }$

Referenced by:

§19.15, §19.29(ii), §19.34

Permalink:

http://dlmf.nist.gov/19.29.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(ii), §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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19.29.18 $$b_j^{q}I(q{𝐞}_l)=\sum _{r=0}^q\left(\genfrac{}{}{0.0pt}{}{q}{r}\right)b_l^r{d}_{lj}^{q-r}I(r{𝐞}_j),$$ $j,l=1,2,\mathrm{\dots },n$;

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $l$: nonnegative integer, $n$: nonnegative integer, $I(𝐦)$: integral and $d_{\alpha \beta }$

Referenced by:

§19.29(ii), §19.34

Permalink:

http://dlmf.nist.gov/19.29.E18

Encodings:

pMML, png, TeX

See also:

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

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