19 Elliptic Integrals Symmetric Integrals 19.27 Asymptotic Approximations and Expansions 19.29 Reduction of General Elliptic Integrals

§19.28 Integrals of 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 $t=(ab/cd)\mathop{\exp\/}\nolimits\!\left(2z\right)$ . To prove (19.28.9) from (19.28.10), put $a=\mathop{\exp\/}\nolimits(ix)=1/b$ , $c=\mathop{\exp\/}\nolimits(iy)=1/d$ , $\mathop{\cosh\/}\nolimits z=1/\mathop{\sin\/}\nolimits\theta$ , and on the right-hand side use (19.22.1).
Keywords:: symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.28

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

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

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : 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

19.28.2 $\int_{0}^{1}t^{{\sigma-1}}\mathop{R_{G}\/}\nolimits\!\left(0,t,1\right)dt=% \frac{\sigma}{4\sigma+2}\left(\mathop{\mathrm{B}\/}\nolimits\!\left(\sigma,% \tfrac{1}{2}\right)\right)^{2},$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , $\int$ : integral and $\sigma$ : parameter
Permalink:: http://dlmf.nist.gov/19.28.E2
Encodings:: TeX, pMML, png

19.28.3 $\int_{0}^{1}t^{{\sigma-1}}(1-t)\mathop{R_{D}\/}\nolimits\!\left(0,t,1\right)dt% =\frac{3}{4\sigma+2}\left(\mathop{\mathrm{B}\/}\nolimits\!\left(\sigma,\tfrac{% 1}{2}\right)\right)^{2}.$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : 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

19.28.4 $\int_{0}^{1}t^{{\sigma-1}}(1-t)^{{c-1}}\mathop{R_{{-a}}\/}\nolimits\!\left(b_{% 1},b_{2};t,1\right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{% \Gamma\/}\nolimits\!\left(\sigma\right)\mathop{\Gamma\/}\nolimits\!\left(% \sigma+b_{2}-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(\sigma+c-a\right)% \mathop{\Gamma\/}\nolimits\!\left(\sigma+b_{2}\right)},$ $c=b_{1}+b_{2}>0$ , $\realpart{\sigma}>\max(0,a-b_{2})$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, : open interval, $\realpart{}$ : real part and $\sigma$ : parameter
Referenced by:: §19.28
Permalink:: http://dlmf.nist.gov/19.28.E4
Encodings:: TeX, pMML, png

In (19.28.5)–(19.28.9) we assume , and are real and positive.

19.28.5 $\int_{z}^{{\infty}}\mathop{R_{D}\/}\nolimits\!\left(x,y,t\right)dt=6\!\mathop{% R_{F}\/}\nolimits\!\left(x,y,z\right),$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : differential of and $\int$ : integral
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E5
Encodings:: TeX, pMML, png

19.28.6 $\int_{0}^{1}\mathop{R_{D}\/}\nolimits\!\left(x,y,v^{2}z+(1-v^{2})p\right)dv=% \mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, : differential of and $\int$ : integral
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E6
Encodings:: TeX, pMML, png

19.28.7 $\int_{0}^{{\infty}}\mathop{R_{J}\/}\nolimits\!\left(x,y,z,r^{2}\right)dr=% \tfrac{3}{2}\pi\mathop{R_{F}\/}\nolimits\!\left(xy,xz,yz\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, : differential of and $\int$ : integral
Referenced by:: §19.28
Permalink:: http://dlmf.nist.gov/19.28.E7
Encodings:: TeX, pMML, png

19.28.8 $\int_{0}^{{\infty}}\mathop{R_{J}\/}\nolimits\!\left(tx,y,z,tp\right)dt=\frac{6% }{\sqrt{p}}\mathop{R_{C}\/}\nolimits\!\left(p,x\right)\mathop{R_{F}\/}% \nolimits\!\left(0,y,z\right).$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, : differential of and $\int$ : integral
Referenced by:: §19.28
Permalink:: http://dlmf.nist.gov/19.28.E8
Encodings:: TeX, pMML, png

19.28.9 $\int_{0}^{{\pi/2}}\mathop{R_{F}\/}\nolimits\!\left({\mathop{\sin\/}\nolimits^{% {2}}}\theta{\mathop{\cos\/}\nolimits^{{2}}}\!\left(x+y\right),{\mathop{\sin\/}% \nolimits^{{2}}}\theta{\mathop{\cos\/}\nolimits^{{2}}}\!\left(x-y\right),1% \right)d\theta=\mathop{R_{F}\/}\nolimits\!\left(0,{\mathop{\cos\/}\nolimits^{{% 2}}}x,1\right)\mathop{R_{F}\/}\nolimits\!\left(0,{\mathop{\cos\/}\nolimits^{{2% }}}y,1\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral and $\mathop{\sin\/}\nolimits z$ : sine function
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E9
Encodings:: TeX, pMML, png

19.28.10 $\int_{0}^{{\infty}}\mathop{R_{F}\/}\nolimits\!\left((ac+bd)^{2},(ad+bc)^{2},4% abcd{\mathop{\cosh\/}\nolimits^{{2}}}z\right)dz=\tfrac{1}{2}\mathop{R_{F}\/}% \nolimits\!\left(0,a^{2},b^{2}\right)\mathop{R_{F}\/}\nolimits\!\left(0,c^{2},% d^{2}\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\int$ : integral
Referenced by:: §19.28
Permalink:: http://dlmf.nist.gov/19.28.E10
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.27 Asymptotic Approximations and Expansions 19.29 Reduction of General Elliptic Integrals