# §19.28 Integrals of Elliptic Integrals

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

 19.28.1 $\displaystyle\int_{0}^{1}t^{\sigma-1}\mathop{R_{F}\/}\nolimits\!\left(0,t,1% \right)\mathrm{d}t$ $\displaystyle=\tfrac{1}{2}\left(\mathop{\mathrm{B}\/}\nolimits\!\left(\sigma,% \tfrac{1}{2}\right)\right)^{2},$ 19.28.2 $\displaystyle\int_{0}^{1}t^{\sigma-1}\mathop{R_{G}\/}\nolimits\!\left(0,t,1% \right)\mathrm{d}t$ $\displaystyle=\frac{\sigma}{4\sigma+2}\left(\mathop{\mathrm{B}\/}\nolimits\!% \left(\sigma,\tfrac{1}{2}\right)\right)^{2},$
 19.28.3 $\int_{0}^{1}t^{\sigma-1}(1-t)\mathop{R_{D}\/}\nolimits\!\left(0,t,1\right)% \mathrm{d}t=\frac{3}{4\sigma+2}\left(\mathop{\mathrm{B}\/}\nolimits\!\left(% \sigma,\tfrac{1}{2}\right)\right)^{2}.$
 19.28.4 $\int_{0}^{1}t^{\sigma-1}(1-t)^{c-1}\mathop{R_{-a}\/}\nolimits\!\left(b_{1},b_{% 2};t,1\right)\mathrm{d}t=\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)% \mathop{\Gamma\/}\nolimits\!\left(\sigma\right)\mathop{\Gamma\/}\nolimits\!% \left(\sigma+b_{2}-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(\sigma+c-a% \right)\mathop{\Gamma\/}\nolimits\!\left(\sigma+b_{2}\right)},$ $c=b_{1}+b_{2}>0$, $\Re{\sigma}>\max(0,a-b_{2})$.

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

 19.28.5 $\int_{z}^{\infty}\mathop{R_{D}\/}\nolimits\!\left(x,y,t\right)\mathrm{d}t=6\!% \mathop{R_{F}\/}\nolimits\!\left(x,y,z\right),$
 19.28.6 $\int_{0}^{1}\mathop{R_{D}\/}\nolimits\!\left(x,y,v^{2}z+(1-v^{2})p\right)% \mathrm{d}v=\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right).$
 19.28.7 $\int_{0}^{\infty}\mathop{R_{J}\/}\nolimits\!\left(x,y,z,r^{2}\right)\mathrm{d}% r=\tfrac{3}{2}\pi\mathop{R_{F}\/}\nolimits\!\left(xy,xz,yz\right),$
 19.28.8 $\int_{0}^{\infty}\mathop{R_{J}\/}\nolimits\!\left(tx,y,z,tp\right)\mathrm{d}t=% \frac{6}{\sqrt{p}}\mathop{R_{C}\/}\nolimits\!\left(p,x\right)\mathop{R_{F}\/}% \nolimits\!\left(0,y,z\right).$
 19.28.9 $\int_{0}^{\pi/2}\mathop{R_{F}\/}\nolimits\!\left({\mathop{\sin\/}\nolimits^{2}% }\theta{\mathop{\cos\/}\nolimits^{2}}\!\left(x+y\right),{\mathop{\sin\/}% \nolimits^{2}}\theta{\mathop{\cos\/}\nolimits^{2}}\!\left(x-y\right),1\right)% \mathrm{d}\theta=\mathop{R_{F}\/}\nolimits\!\left(0,{\mathop{\cos\/}\nolimits^% {2}}x,1\right)\mathop{R_{F}\/}\nolimits\!\left(0,{\mathop{\cos\/}\nolimits^{2}% }y,1\right),$
 19.28.10 $\int_{0}^{\infty}\mathop{R_{F}\/}\nolimits\!\left((ac+bd)^{2},(ad+bc)^{2},4% abcd{\mathop{\cosh\/}\nolimits^{2}}z\right)\mathrm{d}z=\tfrac{1}{2}\mathop{R_{% F}\/}\nolimits\!\left(0,a^{2},b^{2}\right)\mathop{R_{F}\/}\nolimits\!\left(0,c% ^{2},d^{2}\right),$ $a,b,c,d>0$.

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)).