# §19.28 Integrals of Elliptic Integrals

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

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

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

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)).