# §19.16 Definitions

## §19.16(i) Symmetric Integrals

 19.16.1 $R_{F}\left(x,y,z\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\,\mathrm{d}t}{s(t)},$ ⓘ Defines: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $s(t)$: scaling factor Referenced by: §19.16(i), §19.16(i), §19.18(i), (19.25.35), (19.25.40), §19.25(vi) Permalink: http://dlmf.nist.gov/19.16.E1 Encodings: TeX, pMML, png See also: Annotations for §19.16(i), §19.16 and Ch.19
 19.16.2 $R_{J}\left(x,y,z,p\right)=\frac{3}{2}\int_{0}^{\infty}\frac{\,\mathrm{d}t}{s(t% )(t+p)},$ ⓘ Defines: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: symmetric elliptic integral of third kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $s(t)$: scaling factor 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 and Ch.19
 19.16.2_5 $R_{G}\left(x,y,z\right)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)}\*\left(% \frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\,\mathrm{d}t.$ ⓘ Defines: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $s(t)$: scaling factor Source: Carlson (1977b, (9.1-9)) Referenced by: §19.16(i), §19.16(i), §19.21(ii), (19.23.7), (19.23.7), §19.23, Erratum (V1.1.0) for Rearrangement Permalink: http://dlmf.nist.gov/19.16.E2_5 Encodings: TeX, pMML, png Rearrangement (effective with 1.1.0): This integral representation was moved from (19.23.7) and is taken as the definition of $R_{G}\left(x,y,z\right)$. We have also employed the notation $s(t)$ for the factor of radicals. See also: Annotations for §19.16(i), §19.16 and Ch.19
 19.16.3 Moved to (19.23.6_5). ⓘ Referenced by: §19.16(i), (19.23.6_5), §19.23, Erratum (V1.1.0) for Rearrangement Permalink: http://dlmf.nist.gov/19.16.E3 Rearrangement (effective with 1.1.0): This equation has been moved to (19.23.6_5). See also: Annotations for §19.16(i), §19.16 and Ch.19

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

 19.16.4 $s(t)=\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}.$ ⓘ Defines: $s(t)$: scaling factor (locally) Referenced by: §19.16(i), (19.25.35), §19.25(vi) Permalink: http://dlmf.nist.gov/19.16.E4 Encodings: TeX, pMML, png See also: Annotations for §19.16(i), §19.16 and Ch.19

In (19.16.1)–(19.16.2_5), $x,y,z\in\mathbb{C}\setminus(-\infty,0]$ 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). It should be noted that the integrals (19.16.1)–(19.16.2_5) have been normalized so that $R_{F}\left(1,1,1\right)=R_{J}\left(1,1,1,1\right)=R_{G}\left(1,1,1\right)=1$.

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

 19.16.5 $R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}% \frac{\,\mathrm{d}t}{s(t)(t+z)},$ ⓘ Defines: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: symmetric elliptic integral of third kind, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $s(t)$: scaling factor 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), §19.16 and Ch.19

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

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

 19.16.6 $R_{C}\left(x,y\right)=R_{F}\left(x,y,y\right),$ ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Referenced by: §19.19, §19.24(ii), §19.6(ii) Permalink: http://dlmf.nist.gov/19.16.E6 Encodings: TeX, pMML, png See also: Annotations for §19.16(i), §19.16 and Ch.19

and $R_{D}$ is a degenerate case of $R_{J}$, so is $R_{J}$ 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 and Ch.19

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

All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function

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

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 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ 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 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ was denoted by $R(a;\mathbf{b};\mathbf{z})$.

 19.16.9 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{\infty}t^{a^{\prime}-1}\prod^{n}_{j=1}(t+z_{j})^{-b_{j}}\,% \mathrm{d}t=\frac{1}{\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{a% -1}\prod^{n}_{j=1}(1+tz_{j})^{-b_{j}}\,\mathrm{d}t,$ $b_{1}+\cdots+b_{n}>a>0$, $b_{j}\in\mathbb{R}$, $z_{j}\in\mathbb{C}\setminus(-\infty,0]$, ⓘ Defines: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$: multivariate hypergeometric function Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$: 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, $\mathbb{R}$: real line, $\setminus$: set subtraction and $n$: nonnegative integer Referenced by: §19.16(ii), §19.16(ii), §19.19, §19.20(iv), Erratum (V1.0.18) for Equation (19.16.9) Permalink: http://dlmf.nist.gov/19.16.E9 Encodings: TeX, pMML, png Clarification (effective with 1.0.18): The constraint $a,a^{\prime}>0$, was replaced with $b_{1}+\cdots+b_{n}>a>0$, $b_{j}\in\mathbb{R}$. It therefore follows from (19.16.10) that $a^{\prime}>0$. Suggested 2017-07-25 by Bastien Roucariès See also: Annotations for §19.16(ii), §19.16 and Ch.19

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

 19.16.10 $a^{\prime}=-a+\sum_{j=1}^{n}b_{j}.$ ⓘ Symbols: $n$: nonnegative integer Referenced by: (19.16.9), Erratum (V1.0.18) for Equation (19.16.9) Permalink: http://dlmf.nist.gov/19.16.E10 Encodings: TeX, pMML, png See also: Annotations for §19.16(ii), §19.16 and Ch.19
 19.16.11 $\displaystyle R_{-a}\left(\mathbf{b};\lambda\mathbf{z}\right)$ $\displaystyle=\lambda^{-a}R_{-a}\left(\mathbf{b};\mathbf{z}\right),$ $\displaystyle R_{-a}\left(\mathbf{b};x\boldsymbol{{1}}\right)$ $\displaystyle=x^{-a}$, $\boldsymbol{{1}}=(1,\dots,1)$. ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$: multivariate hypergeometric function Referenced by: §19.19, §19.20(v) Permalink: http://dlmf.nist.gov/19.16.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.16(ii), §19.16 and Ch.19

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

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

where

 19.16.13 $\displaystyle c$ $\displaystyle={\csc}^{2}\phi;$ $\displaystyle a,a^{\prime}$ $\displaystyle>0;$ $\displaystyle b_{3}$ $\displaystyle=a+a^{\prime}-b_{1}-b_{2}-b_{4}.$ ⓘ Symbols: $\csc\NVar{z}$: 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), §19.16 and Ch.19

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

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

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

(Note that $R_{C}\left(x,y\right)$ 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 $R_{-a}\left(b_{1},\dots,b_{n};0,z_{2},\dots,z_{n}\right)=\frac{\mathrm{B}\left% (a,a^{\prime}-b_{1}\right)}{\mathrm{B}\left(a,a^{\prime}\right)}R_{-a}\left(b_% {2},\dots,b_{n};z_{2},\dots,z_{n}\right),$ $a+a^{\prime}>0$, $a^{\prime}>b_{1}$.

Thus

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

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