§19.28 Integrals of Elliptic Integrals

In (19.28.1)–(19.28.3) we assume $\mathrm{\Re }\sigma >0$. Also, $\mathrm{B}$ again denotes the beta function (§5.12).

19.28.1		${\displaystyle {\int }_0^1}{t}^{\sigma -1}{R}_F(0,t,1)dt$	$=\frac{1}{2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$
ⓘ 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: TeX, pMML, png See also: Annotations for §19.28 and Ch.19
19.28.2		${\displaystyle {\int }_0^1}{t}^{\sigma -1}{R}_G(0,t,1)dt$	$={\displaystyle \frac{\sigma }{4\sigma +2}}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$
ⓘ 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: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

19.28.3		$${\int }_0^1{t}^{\sigma -1}(1-t)R_D(0,t,1)dt=\frac{3}{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2.$$
ⓘ 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: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

19.28.4		$${\int }_0^1{t}^{\sigma -1}{(1-t)}^{c-1}{R}_{-a}(b_1,b_2;t,1)dt=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(\sigma \right)\mathrm{\Gamma }\left(\sigma +b_2-a\right)}{\mathrm{\Gamma }\left(\sigma +c-a\right)\mathrm{\Gamma }\left(\sigma +b_2\right)},$$
$c=b_1+b_2>0$, $\mathrm{\Re }\sigma >\max (0,a-b_2)$.
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part and $\sigma$: parameter Referenced by: §19.28 Permalink: http://dlmf.nist.gov/19.28.E4 Encodings: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

In (19.28.5)–(19.28.9) we assume $x,y,z$, and $p$ are real and positive.

19.28.5		$${\int }_z^{\mathrm{\infty }}{R}_D(x,y,t)dt=6R_F(x,y,z),$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ and $\int$: integral Referenced by: §19.28, §19.28 Permalink: http://dlmf.nist.gov/19.28.E5 Encodings: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

19.28.6		$${\int }_0^1{R}_D(x,y,v^{2}z+(1-v^2)p)dv=R_J(x,y,z,p).$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ and $\int$: integral Referenced by: §19.28, §19.28 Permalink: http://dlmf.nist.gov/19.28.E6 Encodings: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

19.28.7		$${\int }_0^{\mathrm{\infty }}{R}_J(x,y,z,r^2)dr=\frac{3}{2}\pi R_F(xy,xz,yz),$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral Referenced by: §19.28 Permalink: http://dlmf.nist.gov/19.28.E7 Encodings: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

19.28.8		$${\int }_0^{\mathrm{\infty }}{R}_J(tx,y,z,tp)dt=\frac{6}{\sqrt{p}}{R}_C(p,x)R_F(0,y,z).$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ and $\int$: integral Referenced by: §19.28 Permalink: http://dlmf.nist.gov/19.28.E8 Encodings: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

19.28.9		$$\begin{array}{l}{\int }_0^{\pi /2}{R}_F({\sin }^2\theta {\cos }^2\left(x+y\right),{\sin }^2\theta {\cos }^2\left(x-y\right),1)d\theta \\ =R_F(0,{\cos }^{2}x,1)R_F(0,{\cos }^{2}y,1),\end{array}$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral and $\sin z$: sine function Referenced by: §19.28, §19.28 Permalink: http://dlmf.nist.gov/19.28.E9 Encodings: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

19.28.10		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{R}_F({(ac+bd)}^2,{(ad+bc)}^2,4abcd{\cosh }^{2}z)dz\\ =\frac{1}{2}{R}_F(0,a^2,b^2)R_F(0,c^2,d^2),\end{array}$$
$a,b,c,d>0$.
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function and $\int$: integral Referenced by: §19.28 Permalink: http://dlmf.nist.gov/19.28.E10 Encodings: TeX, pMML, png See also: Annotations for §19.28 and Ch.19

See also (19.16.24). To replace a single component of $\mathbf{z}$ in $R_{-a}(\mathbf{b};\mathbf{z})$ by several different variables (as in (19.28.6)), see Carlson (1963, (7.9)).