# §19.23 Integral Representations

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

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

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) $\mathop{\mathrm{B}\/}\nolimits$ denotes the beta function (§5.12).

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

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

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

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