# §19.23 Integral Representations

In (19.23.1)–(19.23.3) we assume $\Re y>0$ and $\Re z>0$.

 19.23.1 $R_{F}\left(0,y,z\right)=\int_{0}^{\pi/2}(y{\cos^{2}}\theta+z{\sin^{2}}\theta)^% {-1/2}\mathrm{d}\theta,$
 19.23.2 $R_{G}\left(0,y,z\right)=\frac{1}{2}\int_{0}^{\pi/2}(y{\cos^{2}}\theta+z{\sin^{% 2}}\theta)^{1/2}\mathrm{d}\theta,$
 19.23.3 $R_{D}\left(0,y,z\right)=3\int_{0}^{\pi/2}(y{\cos^{2}}\theta+z{\sin^{2}}\theta)% ^{-3/2}{\sin^{2}}\theta\mathrm{d}\theta.$
 19.23.4 $R_{F}\left(0,y,z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(y,z{\cos^{2}}% \theta\right)\mathrm{d}\theta=\frac{2}{\pi}\int_{0}^{\infty}R_{C}\left(y{\cosh% ^{2}}t,z\right)\mathrm{d}t.$
 19.23.5 $R_{F}\left(x,y,z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(x,y{\cos^{2}}% \theta+z{\sin^{2}}\theta\right)\mathrm{d}\theta,$ $\Re y>0$, $\Re z>0$,
 19.23.6 $4\pi R_{F}\left(x,y,z\right)=\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\frac{\sin% \theta\mathrm{d}\theta\mathrm{d}\phi}{(x{\sin^{2}}\theta{\cos^{2}}\phi+y{\sin^% {2}}\theta{\sin^{2}}\phi+z{\cos^{2}}\theta)^{1/2}},$

where $x$, $y$, and $z$ have positive real parts—except that at most one of them may be 0.

In (19.23.7)–(19.23.10) one or more of the variables may be 0 if the integral converges. In (19.23.8) $n=2$, and in (19.23.9) $n=3$. Also, in (19.23.8) and (19.23.10) $\mathrm{B}$ denotes the beta function (§5.12).

 19.23.7 $R_{G}\left(x,y,z\right)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{\sqrt{t+x}\sqrt{t% +y}\sqrt{t+z}}\*\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm% {d}t,$ $x,y,z\in\mathbb{C}\setminus(-\infty,0]$.
 19.23.8 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{2}{\mathrm{B}\left(b_{1},b_{2}% \right)}\int_{0}^{\pi/2}{(z_{1}{\cos^{2}}\theta+z_{2}{\sin^{2}}\theta)}^{-a}\*% (\cos\theta)^{2b_{1}-1}(\sin\theta)^{2b_{2}-1}\mathrm{d}\theta,$ $b_{1},b_{2}>0$; $\Re z_{1},\Re z_{2}>0$.

With $l_{1},l_{2},l_{3}$ denoting any permutation of $\sin\theta\cos\phi$, $\sin\theta\sin\phi$, $\cos\theta$,

 19.23.9 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{4\Gamma\left(b_{1}+b_{2}+b_{3}% \right)}{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\Gamma\left(b_{3}% \right)}\int_{0}^{\pi/2}\!\!\!\!\int_{0}^{\pi/2}\left(\sum_{j=1}^{3}z_{j}l_{j}% ^{2}\right)^{-a}\*\prod_{j=1}^{3}l_{j}^{2b_{j}-1}\sin\theta\mathrm{d}\theta% \mathrm{d}\phi,$ $b_{j}>0$, $\Re z_{j}>0$.
 19.23.10 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{1}u^{a-1}(1-u)^{a^{\prime}-1}\*\prod_{j=1}^{n}(1-u+uz_{j})^{% -b_{j}}\mathrm{d}u,$ $a,a^{\prime}>0$; $a+a^{\prime}=\sum_{j=1}^{n}b_{j}$; $z_{j}\in\mathbb{C}\setminus(-\infty,0]$.

For generalizations of (19.16.3) and (19.23.8) see Carlson (1964, (6.2), (6.12), and (6.1)).