# §19.16 Definitions

## §19.16(i) Symmetric Integrals

 19.16.1 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\frac{1}{2}\int_{0}^{\infty}% \frac{\mathrm{d}t}{s(t)},$ Defines: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $s(t)$: function Referenced by: §19.16(i), §19.16(i), §19.18(i), §19.25(vi) Permalink: http://dlmf.nist.gov/19.16.E1 Encodings: TeX, pMML, png See also: Annotations for 19.16(i)
 19.16.2 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=\frac{3}{2}\int_{0}^{\infty}% \frac{\mathrm{d}t}{s(t)(t+p)},$ Defines: $\mathop{R_{J}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: symmetric elliptic integral of third kind Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $s(t)$: function Referenced by: §19.16(i), §19.19, §19.20(iii), §19.28 Permalink: http://dlmf.nist.gov/19.16.E2 Encodings: TeX, pMML, png See also: Annotations for 19.16(i)
 19.16.3 $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=\frac{1}{4\pi}\int_{0}^{2\pi}\!% \!\!\!\int_{0}^{\pi}\left(x{\mathop{\sin\/}\nolimits^{2}}\theta{\mathop{\cos\/% }\nolimits^{2}}\phi+y{\mathop{\sin\/}\nolimits^{2}}\theta{\mathop{\sin\/}% \nolimits^{2}}\phi+z{\mathop{\cos\/}\nolimits^{2}}\theta\right)^{\frac{1}{2}}% \mathop{\sin\/}\nolimits\theta\mathrm{d}\theta\mathrm{d}\phi,$ Defines: $\mathop{R_{G}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function and $\phi$: real or complex argument Referenced by: §19.16(i), §19.16(ii), §19.20(ii), §19.23 Permalink: http://dlmf.nist.gov/19.16.E3 Encodings: TeX, pMML, png See also: Annotations for 19.16(i)

where $p$ ($\neq 0$) is a real or complex constant, and

 19.16.4 $s(t)=\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}.$ Symbols: $s(t)$: function Referenced by: §19.16(i) Permalink: http://dlmf.nist.gov/19.16.E4 Encodings: TeX, pMML, png See also: Annotations for 19.16(i)

In (19.16.1) and (19.16.2), $x,y,z\in\mathbb{C}\setminus(-\infty,0]$ except that one or more of $x,y,z$ may be 0 when the corresponding integral converges. In (19.16.2) the Cauchy principal value is taken when $p$ is real and negative. See also (19.20.14). In (19.16.3) $\Re{x}$, $\Re{y}$, $\Re{z}\geq 0$. It should be noted that the integrals (19.16.1)–(19.16.3) have been normalized so that $\mathop{R_{F}\/}\nolimits\!\left(1,1,1\right)=\mathop{R_{J}\/}\nolimits\!\left% (1,1,1,1\right)=\mathop{R_{G}\/}\nolimits\!\left(1,1,1\right)=1$.

A fourth integral that is symmetric in only two variables is defined by

 19.16.5 $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=\mathop{R_{J}\/}\nolimits\!\left% (x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)(t+z)},$ Defines: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables Symbols: $\mathop{R_{J}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: symmetric elliptic integral of third kind, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $s(t)$: function Referenced by: §19.18(i), §19.19, §19.20(iii), §19.20(iv), §19.24(ii), §19.6(iv) Permalink: http://dlmf.nist.gov/19.16.E5 Encodings: TeX, pMML, png See also: Annotations for 19.16(i)

with the same conditions on $x$, $y$, $z$ as for (19.16.1), but now $z\neq 0$.

Just as the elementary function $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$19.2(iv)) is the degenerate case

 19.16.6 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\mathop{R_{F}\/}\nolimits\!\left(x% ,y,y\right),$

and $\mathop{R_{D}\/}\nolimits$ is a degenerate case of $\mathop{R_{J}\/}\nolimits$, so is $\mathop{R_{J}\/}\nolimits$ a degenerate case of the hyperelliptic integral,

 19.16.7 $\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\prod_{j=1}^{5}\sqrt{t+x_{j}}}.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/19.16.E7 Encodings: TeX, pMML, png See also: Annotations for 19.16(i)

## §19.16(ii) $\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$

All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.16.3), are special cases of a multivariate hypergeometric function

 19.16.8 $\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\mathop{R_{-a}% \/}\nolimits\!\left(b_{1},\dots,b_{n};z_{1},\dots,z_{n}\right),$

which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\dots,n$. Thus $\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ is symmetric in the variables $z_{j}$ and $z_{\ell}$ if the parameters $b_{j}$ and $b_{\ell}$ are equal. The $R$-function is often used to make a unified statement of a property of several elliptic integrals. Before 1969 $\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ was denoted by $R(a;\mathbf{b};\mathbf{z})$.

 19.16.9 $\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{% \mathop{\mathrm{B}\/}\nolimits\!\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{% a^{\prime}-1}\prod^{n}_{j=1}(t+z_{j})^{-b_{j}}\mathrm{d}t=\frac{1}{\mathop{% \mathrm{B}\/}\nolimits\!\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{a-1}% \prod^{n}_{j=1}(1+tz_{j})^{-b_{j}}\mathrm{d}t,$ $a,a^{\prime}>0$, $z_{j}\in\mathbb{C}\setminus(-\infty,0]$, Defines: $\mathop{R_{\NVar{-a}}\/}\nolimits\!\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar% {z_{1}},\dots,\NVar{z_{n}}\right)$ or $\mathop{R_{\NVar{-a}}\/}\nolimits\!\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$: multivariate hypergeometric function Symbols: $\mathop{\mathrm{B}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: beta function, $\mathbb{C}$: complex plane, $\mathrm{d}\NVar{x}$: differential of $x$, $\in$: element of, $\int$: integral, $(\NVar{a},\NVar{b}]$: half-closed interval, $\setminus$: set subtraction and $n$: nonnegative integer Referenced by: §19.16(ii), §19.16(ii), §19.19, §19.20(iv) Permalink: http://dlmf.nist.gov/19.16.E9 Encodings: TeX, pMML, png See also: Annotations for 19.16(ii)

where $\mathop{\mathrm{B}\/}\nolimits\!\left(x,y\right)$ is the beta function (§5.12) and

 19.16.10 $a^{\prime}=-a+\sum_{j=1}^{n}b_{j}.$ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/19.16.E10 Encodings: TeX, pMML, png See also: Annotations for 19.16(ii)
 19.16.11 $\displaystyle\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\lambda\mathbf{z}\right)$ $\displaystyle=\lambda^{-a}\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf% {z}\right),$ $\displaystyle\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};x\boldsymbol{{1}}\right)$ $\displaystyle=x^{-a}$, $\boldsymbol{{1}}=(1,\dots,1)$.

When $n=4$ a useful version of (19.16.9) is given by

 19.16.12 $\mathop{R_{-a}\/}\nolimits\!\left(b_{1},\dots,b_{4};c-1,c-k^{2},c,c-\alpha^{2}% \right)=\frac{2({\mathop{\sin\/}\nolimits^{2}}\phi)^{1-a^{\prime}}}{\mathop{% \mathrm{B}\/}\nolimits\!\left(a,a^{\prime}\right)}\int_{0}^{\phi}(\mathop{\sin% \/}\nolimits\theta)^{2a-1}{({\mathop{\sin\/}\nolimits^{2}}\phi-{\mathop{\sin\/% }\nolimits^{2}}\theta)}^{a^{\prime}-1}\*(\mathop{\cos\/}\nolimits\theta)^{1-2b% _{1}}{(1-k^{2}{\mathop{\sin\/}\nolimits^{2}}\theta)}^{-b_{2}}{(1-\alpha^{2}{% \mathop{\sin\/}\nolimits^{2}}\theta)}^{-b_{4}}\mathrm{d}\theta,$

where

 19.16.13 $\displaystyle c$ $\displaystyle={\mathop{\csc\/}\nolimits^{2}}\phi;$ $\displaystyle a,a^{\prime}$ $\displaystyle>0;$ $\displaystyle b_{3}$ $\displaystyle=a+a^{\prime}-b_{1}-b_{2}-b_{4}.$ Symbols: $\mathop{\csc\/}\nolimits\NVar{z}$: cosecant function and $\phi$: real or complex argument Permalink: http://dlmf.nist.gov/19.16.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 19.16(ii)

For further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b).

## §19.16(iii) Various Cases of $\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$

$\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ is an elliptic integral iff the $z$’s are distinct and exactly four of the parameters $a,a^{\prime},b_{1},\dots,b_{n}$ are half-odd-integers, the rest are integers, and none of $a$, $a^{\prime}$, $a+a^{\prime}$ is zero or a negative integer. The only cases that are integrals of the first kind are the four in which each of $a$ and $a^{\prime}$ is either $\frac{1}{2}$ or 1 and each $b_{j}$ is $\frac{1}{2}$. The only cases that are integrals of the third kind are those in which at least one $b_{j}$ is a positive integer. All other elliptic cases are integrals of the second kind.

 19.16.14 $\displaystyle\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=\mathop{R_{-\frac{1}{2}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{% 1}{2},\tfrac{1}{2};x,y,z\right),$ 19.16.15 $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=\mathop{R_{-\frac{3}{2}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{% 1}{2},\tfrac{3}{2};x,y,z\right),$ 19.16.16 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ $\displaystyle=\mathop{R_{-\frac{3}{2}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{% 1}{2},\tfrac{1}{2},1;x,y,z,p\right),$ 19.16.17 $\displaystyle\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=\mathop{R_{\frac{1}{2}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1% }{2},\tfrac{1}{2};x,y,z\right),$ 19.16.18 $\displaystyle\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ $\displaystyle=\mathop{R_{-\frac{1}{2}}\/}\nolimits\!\left(\tfrac{1}{2},1;x,y% \right).$

(Note that $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ is not an elliptic integral.)

When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an $R$-function with one less variable:

 19.16.19 $\mathop{R_{-a}\/}\nolimits\!\left(b_{1},\dots,b_{n};0,z_{2},\dots,z_{n}\right)% =\frac{\mathop{\mathrm{B}\/}\nolimits\!\left(a,a^{\prime}-b_{1}\right)}{% \mathop{\mathrm{B}\/}\nolimits\!\left(a,a^{\prime}\right)}\mathop{R_{-a}\/}% \nolimits\!\left(b_{2},\dots,b_{n};z_{2},\dots,z_{n}\right),$ $a+a^{\prime}>0$, $a^{\prime}>b_{1}$.

Thus

 19.16.20 $\displaystyle\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)$ $\displaystyle=\tfrac{1}{2}\pi\mathop{R_{-\frac{1}{2}}\/}\nolimits\!\left(% \tfrac{1}{2},\tfrac{1}{2};y,z\right),$ 19.16.21 $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)$ $\displaystyle=\tfrac{3}{4}\pi\mathop{R_{-\frac{3}{2}}\/}\nolimits\!\left(% \tfrac{1}{2},\tfrac{3}{2};y,z\right),$ 19.16.22 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)$ $\displaystyle=\tfrac{3}{4}\pi\mathop{R_{-\frac{3}{2}}\/}\nolimits\!\left(% \tfrac{1}{2},\tfrac{1}{2},1;y,z,p\right),$ 19.16.23 $\displaystyle\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)$ $\displaystyle=\tfrac{1}{4}\pi\mathop{R_{\frac{1}{2}}\/}\nolimits\!\left(\tfrac% {1}{2},\tfrac{1}{2};y,z\right)=\tfrac{1}{4}\pi z\mathop{R_{-\frac{1}{2}}\/}% \nolimits\!\left(-\tfrac{1}{2},\tfrac{3}{2};y,z\right).$

The last $R$-function has $a=a^{\prime}=\frac{1}{2}$.

Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral:

 19.16.24 $\mathop{R_{-a}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\frac{z_{1}^{a^% {\prime}-b_{1}}}{\mathop{\mathrm{B}\/}\nolimits\!\left(b_{1},a^{\prime}-b_{1}% \right)}\int_{0}^{\infty}t^{b_{1}-1}(t+z_{1})^{-a^{\prime}}\*\mathop{R_{-a}\/}% \nolimits\!\left(\mathbf{b};0,t+z_{2},\dots,t+z_{n}\right)\mathrm{d}t,$ $a^{\prime}>b_{1}$, $a+a^{\prime}>b_{1}>0$.