§19.20 Special Cases

In this subsection, and also §§19.20(ii)–19.20(v), the variables of all $R$-functions satisfy the constraints specified in §19.16(i) unless other conditions are stated.

19.20.1		$R_F(x,x,x)$	$=x^{-1/2},$
		$R_F(\lambda x,\lambda y,\lambda z)$	$={\lambda }^{-1/2}{R}_F(x,y,z),$
		$R_F(x,y,y)$	$=R_C(x,y),$
		$R_F(0,y,y)$	$=\frac{1}{2}\pi y^{-1/2},$
		$R_F(0,0,z)$	$=\mathrm{\infty }.$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $\pi$: the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/19.20.E1 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.20(i), §19.20 and Ch.19

The first lemniscate constant is given by

19.20.2		$${\int }_0^1\frac{dt}{\sqrt{1-t^4}}=R_F(0,1,2)=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{4{(2\pi )}^{1/2}}=\mathrm{1.31102\hspace{0.33em}87771\hspace{0.33em}46059\hspace{0.33em}90523}\mathrm{\dots }.$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral Notes: For more digits see OEIS Sequence A085565; see also Sloane (2003). Referenced by: §19.20(i), §19.20(iv), §19.21(i) Permalink: http://dlmf.nist.gov/19.20.E2 Encodings: TeX, pMML, png See also: Annotations for §19.20(i), §19.20 and Ch.19

Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. The general lemniscatic case is

19.20.3		$$R_F(x,a,y)=R_{-\frac{1}{4}}(\frac{3}{4},\frac{1}{2};a^2,xy),$$
$a=\frac{1}{2}(x+y)$.
ⓘ 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 $R_F(x,y,z)$: symmetric elliptic integral of first kind Referenced by: §19.20(i) Permalink: http://dlmf.nist.gov/19.20.E3 Encodings: TeX, pMML, png See also: Annotations for §19.20(i), §19.20 and Ch.19

19.20.4		$R_G(x,x,x)$	$=x^{1/2},$
		$R_G(\lambda x,\lambda y,\lambda z)$	$={\lambda }^{1/2}{R}_G(x,y,z),$
		$R_G(0,y,y)$	$=\frac{1}{4}\pi y^{1/2},$
		$R_G(0,0,z)$	$=\frac{1}{2}{z}^{1/2},$
ⓘ Symbols: $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.20(ii) Permalink: http://dlmf.nist.gov/19.20.E4 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.20(ii), §19.20 and Ch.19

19.20.5		$$2R_G(x,y,y)=y{R}_C(x,y)+\sqrt{x}.$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_G(x,y,z)$: symmetric elliptic integral of second kind Referenced by: §19.20(ii) Permalink: http://dlmf.nist.gov/19.20.E5 Encodings: TeX, pMML, png See also: Annotations for §19.20(ii), §19.20 and Ch.19

19.20.6		$R_J(x,x,x,x)$	$=x^{-3/2},$
		$R_J(\lambda x,\lambda y,\lambda z,\lambda p)$	$={\lambda }^{-3/2}{R}_J(x,y,z,p),$
		$R_J(x,y,z,z)$	$=R_D(x,y,z),$
		$R_J(0,0,z,p)$	$=\mathrm{\infty },$
		$R_J(x,x,x,p)$	$=R_D(p,p,x)={\displaystyle \frac{3}{x-p}}\left(R_C(x,p)-{\displaystyle \frac{1}{\sqrt{x}}}\right),$
	$x\ne p$, $xp\ne 0$.
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.20(iii) Permalink: http://dlmf.nist.gov/19.20.E6 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.7		$$R_J(x,y,z,p)\to +\mathrm{\infty },$$
$p\to 0+$ or $0-$; $x,y,z>0$.
ⓘ Symbols: $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.20(iii) Permalink: http://dlmf.nist.gov/19.20.E7 Encodings: TeX, pMML, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.8		$R_J(0,y,y,p)$	$={\displaystyle \frac{3\pi }{2(y\sqrt{p}+p\sqrt{y})}},$
	$p>0$,
		$R_J(0,y,y,-q)$	$={\displaystyle \frac{-3\pi }{2\sqrt{y}(y+q)}},$
	$q>0$,
		$R_J(x,y,y,p)$	$={\displaystyle \frac{3}{p-y}}(R_C(x,y)-R_C(x,p)),$
	$p\ne y$,
		$R_J(x,y,y,y)$	$=R_D(x,y,y).$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: Figure 19.17.8, Figure 19.17.8, Figure 19.17.8, §19.20(iii), §19.26(i), §19.6(iv) Permalink: http://dlmf.nist.gov/19.20.E8 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.9		$$R_J(0,y,z,\pm \sqrt{yz})=\pm \frac{3}{2\sqrt{yz}}{R}_F(0,y,z).$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.20(iii), §19.22(i) Permalink: http://dlmf.nist.gov/19.20.E9 Encodings: TeX, pMML, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.10		$\underset{p\to 0+}{lim}\sqrt{p}{R}_J(0,y,z,p)$	$={\displaystyle \frac{3\pi }{2\sqrt{yz}}},$
		$\underset{p\to 0-}{lim}{R}_J(0,y,z,p)$	$=-R_D(0,y,z)-R_D(0,z,y)={\displaystyle \frac{-6}{yz}}{R}_G(0,y,z).$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: Figure 19.17.7, Figure 19.17.7, Figure 19.17.8, Figure 19.17.8, Figure 19.17.8, §19.20(iii), §19.21(i) Permalink: http://dlmf.nist.gov/19.20.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.11		$$R_J(0,y,z,p)=\frac{3}{2p\sqrt{z}}\ln \left(\frac{16z}{y}\right)-\frac{3}{p}{R}_C(z,p)+O\left(y\ln y\right),$$
$y\to 0+$; $p$ ($\ne 0$) real.
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\sim$: Poincaré asymptotic expansion and $\ln z$: principal branch of logarithm function Referenced by: Figure 19.17.8, Figure 19.17.8, Figure 19.17.8, §19.20(iii), Erratum (V1.0.26) for Equation (19.20.11) Permalink: http://dlmf.nist.gov/19.20.E11 Encodings: TeX, pMML, png Clarification (effective with 1.0.26): The constant term $\frac{-3}{p}{R}_C(z,p)$ and the order term $O\left(y\ln y\right)$ were added, and hence $\sim$ was replaced by $=$. See also: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.12		$$\underset{p\to \pm \mathrm{\infty }}{lim}p{R}_J(x,y,z,p)=3R_F(x,y,z).$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.20(iii) Permalink: http://dlmf.nist.gov/19.20.E12 Encodings: TeX, pMML, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.13		$$2(p-x)R_J(x,y,z,p)=3R_F(x,y,z)-3\sqrt{x}{R}_C(yz,p^2),$$
$p=x\pm \sqrt{(y-x)(z-x)}$,
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.20(iii), §19.22(iii) Permalink: http://dlmf.nist.gov/19.20.E13 Encodings: TeX, pMML, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

where $x,y,z$ may be permuted.

When the variables are real and distinct, the various cases of $R_J(x,y,z,p)$ are called circular (hyperbolic) cases if $(p-x)(p-y)(p-z)$ is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. If $x,y,z$ are permuted so that , then the Cauchy principal value of $R_J$ is given by

19.20.14		$$(q+z)R_J(x,y,z,-q)=\begin{array}{l}(p-z)R_J(x,y,z,p)-3R_F(x,y,z)\\ +3{\left(\frac{xyz}{xy+pq}\right)}^{1/2}{R}_C(xy+pq,pq),\end{array}$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.16(i), §19.16(i), §19.20(iii), §19.20(iii), §19.21(iii), §19.36(i) Permalink: http://dlmf.nist.gov/19.20.E14 Encodings: TeX, pMML, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

valid when

19.20.15		$q$	$>0,$
		$p$	$={\displaystyle \frac{z(x+y+q)-xy}{z+q}},$
ⓘ Permalink: http://dlmf.nist.gov/19.20.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

or

19.20.16		$p$	$=wy+(1-w)z,$
		$w$	$={\displaystyle \frac{z-x}{z+q}},$
		$0$
ⓘ Permalink: http://dlmf.nist.gov/19.20.E16 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

Since , $p$ is in a hyperbolic region. In the complete case ($x=0$) (19.20.14) reduces to

19.20.17		$$(q+z)R_J(0,y,z,-q)=(p-z)R_J(0,y,z,p)-3R_F(0,y,z),$$
$p=z(y+q)/(z+q)$, $w=z/(z+q)$.
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.20(iii) Permalink: http://dlmf.nist.gov/19.20.E17 Encodings: TeX, pMML, png See also: Annotations for §19.20(iii), §19.20 and Ch.19

19.20.18		$R_D(x,x,x)$	$=x^{-3/2},$
		$R_D(\lambda x,\lambda y,\lambda z)$	$={\lambda }^{-3/2}{R}_D(x,y,z),$
		$R_D(0,y,y)$	$=\frac{3}{4}\pi y^{-3/2},$
		$R_D(0,0,z)$	$=\mathrm{\infty }.$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.20(iv) Permalink: http://dlmf.nist.gov/19.20.E18 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.20(iv), §19.20 and Ch.19

19.20.19		$$R_D(x,y,z)\sim 3{(xyz)}^{-1/2},$$
$z/\sqrt{xy}\to 0$.
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\sim$: asymptotic equality Referenced by: §19.20(iv) Permalink: http://dlmf.nist.gov/19.20.E19 Encodings: TeX, pMML, png See also: Annotations for §19.20(iv), §19.20 and Ch.19

19.20.20		$$R_D(x,y,y)=\frac{3}{2(y-x)}\left(R_C(x,y)-\frac{\sqrt{x}}{y}\right),$$
$x\ne y$, $y\ne 0$,
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables Referenced by: §19.20(iv), §19.22(iii) Permalink: http://dlmf.nist.gov/19.20.E20 Encodings: TeX, pMML, png See also: Annotations for §19.20(iv), §19.20 and Ch.19

19.20.21		$$R_D(x,x,z)=\frac{3}{z-x}\left(R_C(z,x)-\frac{1}{\sqrt{z}}\right),$$
$x\ne z$, $xz\ne 0$.
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables Referenced by: §19.20(iv) Permalink: http://dlmf.nist.gov/19.20.E21 Encodings: TeX, pMML, png See also: Annotations for §19.20(iv), §19.20 and Ch.19

The second lemniscate constant is given by

19.20.22		$${\int }_0^1\frac{t^{2}dt}{\sqrt{1-t^4}}=\frac{1}{3}{R}_D(0,2,1)=\frac{{\left(\mathrm{\Gamma }\left(\frac{3}{4}\right)\right)}^2}{{(2\pi )}^{1/2}}=\mathrm{0.59907\hspace{0.33em}01173\hspace{0.33em}67796\hspace{0.33em}10371}\mathrm{\dots }.$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral Notes: For more digits see OEIS Sequence A085566; see also Sloane (2003). Referenced by: §19.20(iv), §19.21(i), Figure 19.3.6, Figure 19.3.6, Figure 19.3.6 Permalink: http://dlmf.nist.gov/19.20.E22 Encodings: TeX, pMML, png See also: Annotations for §19.20(iv), §19.20 and Ch.19

Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. Compare (19.20.2). The general lemniscatic case is

19.20.23		$$R_D(x,y,a)=R_{-\frac{3}{4}}(\frac{5}{4},\frac{1}{2};a^2,xy),$$
$a=\frac{1}{2}x+\frac{1}{2}y$.
ⓘ 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 $R_D(x,y,z)$: elliptic integral symmetric in only two variables Referenced by: §19.20(iv) Permalink: http://dlmf.nist.gov/19.20.E23 Encodings: TeX, pMML, png See also: Annotations for §19.20(iv), §19.20 and Ch.19

Define $c={\sum }_{j=1}^n{b}_j$. Then

19.20.24		$R_0(\mathbf{b};\mathbf{z})$	$=1,$
		$R_N(\mathbf{b};\mathbf{z})$	$={\displaystyle \frac{N!}{{\left(c\right)}_N}}{T}_N(\mathbf{b},\mathbf{z}),$
	$N=0,1,2,\mathrm{\dots }$,
ⓘ 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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$) and $T_N(\mathbf{b},\mathbf{z})$: polynomial Permalink: http://dlmf.nist.gov/19.20.E24 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.20(v), §19.20 and Ch.19

where $T_N$ is defined by (19.19.1). Also,

19.20.25		$$R_{-c}(\mathbf{b};\mathbf{z})=\prod _{j=1}^n{z}_j^{-b_j},$$
ⓘ 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: §19.18(ii), §19.21(ii) Permalink: http://dlmf.nist.gov/19.20.E25 Encodings: TeX, pMML, png See also: Annotations for §19.20(v), §19.20 and Ch.19

19.20.26		$$R_{-a}(\mathbf{b};\mathbf{z})=\prod _{j=1}^n{z}_j^{-b_j}{R}_{-a^{\prime }}(\mathbf{b};{\mathit{z}}^{\mathbf{-}\mathbf{1}}),$$
$a+a^{\prime }=c$, ${\mathit{z}}^{\mathbf{-}\mathbf{1}}=(z_1^{-1},\mathrm{\dots },z_n^{-1})$.
ⓘ 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 and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/19.20.E26 Encodings: TeX, pMML, png See also: Annotations for §19.20(v), §19.20 and Ch.19

See also (19.16.11) and (19.16.19).