§19.28 Integrals of Elliptic Integrals

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Keywords:: integrals of, symmetric elliptic integrals
Notes:: To prove (19.28.1)–(19.28.3) from (19.28.4) use §19.16(iii). To prove (19.28.4) expand the -function in powers of by (19.19.3), integrate term by term, and use Erdélyi et al. (1953a, 2.8(46)). (19.28.5) is equivalent to (19.18.1). In (19.28.6) let $v=\sqrt{u}$ and use Carlson (1963, (7.9)). In (19.28.7) substitute (19.16.2), change the order of integration, and use (19.29.4). Use Carlson (1963, (7.11)) and (19.16.20) to prove (19.28.8) and (19.28.10). In the first case Carlson (1977b, (5.9-21)) is needed; in the second case put $(z_{1},z_{2},\zeta_{1},\zeta_{2})=(a^{2},b^{2},c^{2},d^{2})$ , use Carlson (1977b, (9.8-4)), and substitute $t=(ab/cd)\exp\left(2z\right)$ . To prove (19.28.9) from (19.28.10), put $a=\exp\left(ix\right)=1/b$ , $c=\exp\left(iy\right)=1/d$ , $\cosh z=1/\sin\theta$ , and on the right-hand side use (19.22.1).
Permalink:: http://dlmf.nist.gov/19.28
See also:: Annotations for Ch.19

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

19.28.1 $\int_{0}^{1}t^{\sigma-1}R_{F}\left(0,t,1\right)\,\mathrm{d}t=\tfrac{1}{2}\left% (\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2},$

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : 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 $\int_{0}^{1}t^{\sigma-1}R_{G}\left(0,t,1\right)\,\mathrm{d}t=\frac{\sigma}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2},$

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Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : 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 $\int_{0}^{1}t^{\sigma-1}(1-t)R_{D}\left(0,t,1\right)\,\mathrm{d}t=\frac{3}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2}.$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : 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 $\int_{0}^{1}t^{\sigma-1}(1-t)^{c-1}R_{-a}\left(b_{1},b_{2};t,1\right)\,\mathrm% {d}t=\frac{\Gamma\left(c\right)\Gamma\left(\sigma\right)\Gamma\left(\sigma+b_{% 2}-a\right)}{\Gamma\left(\sigma+c-a\right)\Gamma\left(\sigma+b_{2}\right)},$ $c=b_{1}+b_{2}>0$ , $\Re\sigma>\max(0,a-b_{2})$ .

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : 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 , and are real and positive.

19.28.5 $\int_{z}^{\infty}R_{D}\left(x,y,t\right)\,\mathrm{d}t=6R_{F}\left(x,y,z\right),$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of 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 $\int_{0}^{1}R_{D}\left(x,y,v^{2}z+(1-v^{2})p\right)\,\mathrm{d}v=R_{J}\left(x,% y,z,p\right).$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\,\mathrm{d}\NVar{x}$ : differential of 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 $\int_{0}^{\infty}R_{J}\left(x,y,z,r^{2}\right)\,\mathrm{d}r=\tfrac{3}{2}\pi R_% {F}\left(xy,xz,yz\right),$

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of 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 $\int_{0}^{\infty}R_{J}\left(tx,y,z,tp\right)\,\mathrm{d}t=\frac{6}{\sqrt{p}}R_% {C}\left(p,x\right)R_{F}\left(0,y,z\right).$

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\,\mathrm{d}\NVar{x}$ : differential of 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 $\int_{0}^{\pi/2}R_{F}\left({\sin}^{2}\theta{\cos}^{2}\left(x+y\right),{\sin}^{% 2}\theta{\cos}^{2}\left(x-y\right),1\right)\,\mathrm{d}\theta=R_{F}\left(0,{% \cos}^{2}x,1\right)R_{F}\left(0,{\cos}^{2}y,1\right),$

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\sin\NVar{z}$ : 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 $\int_{0}^{\infty}R_{F}\left((ac+bd)^{2},(ad+bc)^{2},4abcd{\cosh}^{2}z\right)\,% \mathrm{d}z=\tfrac{1}{2}R_{F}\left(0,a^{2},b^{2}\right)R_{F}\left(0,c^{2},d^{2% }\right),$

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\cosh\NVar{z}$ : 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 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ by several different variables (as in (19.28.6)), see Carlson (1963, (7.9)).