§19.16 Definitions

19.16.1		$$R_F(x,y,z)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{s(t)},$$
ⓘ Defines: $R_F(x,y,z)$: symmetric elliptic integral of first kind Symbols: $dx$: 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		$$R_J(x,y,z,p)=\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{s(t)(t+p)},$$
ⓘ Defines: $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Symbols: $dx$: 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		$$R_G(x,y,z)=\frac{1}{4}{\int }_0^{\mathrm{\infty }}\frac{1}{s(t)}\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)tdt.$$
ⓘ Defines: $R_G(x,y,z)$: symmetric elliptic integral of second kind Symbols: $dx$: 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 $R_G(x,y,z)$. 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, §19.25(vi), 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 $p$ ($\ne 0$) 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), $x,y,z\in ℂ\setminus (-\mathrm{\infty },0]$ 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 $R_F(1,1,1)=R_J(1,1,1,1)=R_G(1,1,1)=1$.

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

19.16.5		$$R_D(x,y,z)=R_J(x,y,z,z)=\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{s(t)(t+z)},$$
ⓘ Defines: $R_D(x,y,z)$: elliptic integral symmetric in only two variables Symbols: $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: 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 $z\ne 0$.

Just as the elementary function $R_C(x,y)$ (§19.2(iv)) is the degenerate case

19.16.6		$$R_C(x,y)=R_F(x,y,y),$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_F(x,y,z)$: 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		$$\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{{\prod }_{j=1}^5\sqrt{t+x_j}}.$$
ⓘ Symbols: $dx$: 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

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		$$R_{-a}(\mathbf{b};\mathbf{z})=R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n),$$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function and $n$: nonnegative integer Referenced by: §19.16(ii) Permalink: http://dlmf.nist.gov/19.16.E8 Encodings: TeX, pMML, png See also: Annotations for §19.16(ii), §19.16 and Ch.19

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

19.16.9		$$\begin{array}{ll}{R}_{-a}(\mathbf{b};\mathbf{z})& =\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a^{\prime }-1}\prod _{j=1}^n{(t+z_j)}^{-b_j}dt\\ & =\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a-1}\prod _{j=1}^n{(1+t{z}_j)}^{-b_j}dt,\end{array}$$
$b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$, $z_j\in ℂ\setminus (-\mathrm{\infty },0]$,
ⓘ Defines: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function Symbols: $\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$, $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $ℝ$: 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 $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$. 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 $\mathrm{B}(x,y)$ 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		$R_{-a}(\mathbf{b};\lambda \mathbf{z})$	$={\lambda }^{-a}{R}_{-a}(\mathbf{b};\mathbf{z}),$
		$R_{-a}(\mathbf{b};x\mathbf{1})$	$=x^{-a}$,
	$\mathbf{1}=(1,\mathrm{\dots },1)$.
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: 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		$$\begin{array}{l}{R}_{-a}(b_1,\mathrm{\dots },b_4;c-1,c-k^2,c,c-{\alpha }^2)\\ =\frac{2{({\sin }^2\varphi )}^{1-a^{\prime }}}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\varphi }\begin{array}{l}{(\sin \theta )}^{2a-1}{({\sin }^2\varphi -{\sin }^2\theta )}^{a^{\prime }-1}{(\cos \theta )}^{1-2b_1}\\ \times {(1-k^2{\sin }^2\theta )}^{-b_2}{(1-{\alpha }^2{\sin }^2\theta )}^{-b_4}d\theta ,\end{array}\end{array}$$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter Referenced by: §19.16(ii), §19.25(i) Permalink: http://dlmf.nist.gov/19.16.E12 Encodings: TeX, pMML, png See also: Annotations for §19.16(ii), §19.16 and Ch.19

where

19.16.13		$c$	$={\csc }^2\varphi ;$
		$a,a^{\prime }$	$>0;$
		$b_3$	$=a+a^{\prime }-b_1-b_2-b_4.$
ⓘ Symbols: $\csc z$: cosecant function and $\varphi$: 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).

$R_{-a}(\mathbf{b};\mathbf{z})$ is an elliptic integral iff the $z$’s are distinct and exactly four of the parameters $a,a^{\prime },b_1,\mathrm{\dots },b_n$ 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		$R_F(x,y,z)$	$=R_{-\frac{1}{2}}(\frac{1}{2},\frac{1}{2},\frac{1}{2};x,y,z),$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function and $R_F(x,y,z)$: symmetric elliptic integral of first kind Referenced by: §19.16(iii), §19.18(i), §19.18(ii), §19.19, §19.19, §19.24(ii) Permalink: http://dlmf.nist.gov/19.16.E14 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19
19.16.15		$R_D(x,y,z)$	$=R_{-\frac{3}{2}}(\frac{1}{2},\frac{1}{2},\frac{3}{2};x,y,z),$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function and $R_D(x,y,z)$: elliptic integral symmetric in only two variables Referenced by: §19.20(iv) Permalink: http://dlmf.nist.gov/19.16.E15 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19
19.16.16		$R_J(x,y,z,p)$	$=R_{-\frac{3}{2}}(\frac{1}{2},\frac{1}{2},\frac{1}{2},1;x,y,z,p),$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Permalink: http://dlmf.nist.gov/19.16.E16 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19
19.16.17		$R_G(x,y,z)$	$=R_{\frac{1}{2}}(\frac{1}{2},\frac{1}{2},\frac{1}{2};x,y,z),$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function and $R_G(x,y,z)$: symmetric elliptic integral of second kind Referenced by: §19.16(iii), §19.24(ii) Permalink: http://dlmf.nist.gov/19.16.E17 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19
19.16.18		$R_C(x,y)$	$=R_{-\frac{1}{2}}(\frac{1}{2},1;x,y).$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function and $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions Referenced by: §19.18(i), §19.18(ii), §19.19, §19.19, §19.2(iv) Permalink: http://dlmf.nist.gov/19.16.E18 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19

(Note that $R_C(x,y)$ 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		$$R_{-a}(b_1,\mathrm{\dots },b_n;0,z_2,\mathrm{\dots },z_n)=\frac{\mathrm{B}(a,a^{\prime }-b_1)}{\mathrm{B}(a,a^{\prime })}{R}_{-a}(b_2,\mathrm{\dots },b_n;z_2,\mathrm{\dots },z_n),$$
$a+a^{\prime }>0$, $a^{\prime }>b_1$.
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function and $n$: nonnegative integer Referenced by: §19.16(iii), §19.20(v) Permalink: http://dlmf.nist.gov/19.16.E19 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19

Thus

19.16.20		$R_F(0,y,z)$	$=\frac{1}{2}\pi R_{-\frac{1}{2}}(\frac{1}{2},\frac{1}{2};y,z),$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.16(iii), §19.18(i), §19.18(ii), §19.18(ii), §19.19, §19.19, §19.22(ii), §19.24(ii), §19.24(i), §19.28 Permalink: http://dlmf.nist.gov/19.16.E20 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19
19.16.21		$R_D(0,y,z)$	$=\frac{3}{4}\pi R_{-\frac{3}{2}}(\frac{1}{2},\frac{3}{2};y,z),$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.9(i) Permalink: http://dlmf.nist.gov/19.16.E21 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19
19.16.22		$R_J(0,y,z,p)$	$=\frac{3}{4}\pi R_{-\frac{3}{2}}(\frac{1}{2},\frac{1}{2},1;y,z,p),$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.24(i) Permalink: http://dlmf.nist.gov/19.16.E22 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19
19.16.23		$R_G(0,y,z)$	$=\frac{1}{4}\pi R_{\frac{1}{2}}(\frac{1}{2},\frac{1}{2};y,z)=\frac{1}{4}\pi z{R}_{-\frac{1}{2}}(-\frac{1}{2},\frac{3}{2};y,z).$
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.16(iii), §19.16(iii), §19.18(i), §19.18(ii), §19.18(ii), §19.19, §19.19, §19.22(ii), §19.24(ii), §19.24(i), §19.33(i) Permalink: http://dlmf.nist.gov/19.16.E23 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19

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		$$R_{-a}(\mathbf{b};\mathbf{z})=\frac{z_1^{a^{\prime }-b_1}}{\mathrm{B}(b_1,a^{\prime }-b_1)}{\int }_0^{\mathrm{\infty }}{t}^{b_1-1}{(t+z_1)}^{-a^{\prime }}{R}_{-a}(\mathbf{b};0,t+z_2,\mathrm{\dots },t+z_n)dt,$$
$a^{\prime }>b_1$, $a+a^{\prime }>b_1>0$.
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$, $\int$: integral and $n$: nonnegative integer Referenced by: §19.16(iii), §19.28 Permalink: http://dlmf.nist.gov/19.16.E24 Encodings: TeX, pMML, png See also: Annotations for §19.16(iii), §19.16 and Ch.19