# §19.20 Special Cases

## §19.20(i) $R_{F}\left(x,y,z\right)$

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 $\displaystyle R_{F}\left(x,x,x\right)$ $\displaystyle=x^{-1/2},$ $\displaystyle R_{F}\left(\lambda x,\lambda y,\lambda z\right)$ $\displaystyle=\lambda^{-1/2}R_{F}\left(x,y,z\right),$ $\displaystyle R_{F}\left(x,y,y\right)$ $\displaystyle=R_{C}\left(x,y\right),$ $\displaystyle R_{F}\left(0,y,y\right)$ $\displaystyle=\tfrac{1}{2}\pi y^{-1/2},$ $\displaystyle R_{F}\left(0,0,z\right)$ $\displaystyle=\infty.$

The first lemniscate constant is given by

 19.20.2 $\int_{0}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{F}\left(0,1,2\right)=\frac% {\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}=1.31102\;8777% 1\;46059\;90523\;\dots.$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential 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 $R_{F}\left(x,a,y\right)=R_{-\frac{1}{4}}\left(\tfrac{3}{4},\tfrac{1}{2};a^{2},% xy\right),$ $a=\frac{1}{2}(x+y)$.

## §19.20(ii) $R_{G}\left(x,y,z\right)$

 19.20.4 $\displaystyle R_{G}\left(x,x,x\right)$ $\displaystyle=x^{1/2},$ $\displaystyle R_{G}\left(\lambda x,\lambda y,\lambda z\right)$ $\displaystyle=\lambda^{1/2}R_{G}\left(x,y,z\right),$ $\displaystyle R_{G}\left(0,y,y\right)$ $\displaystyle=\tfrac{1}{4}\pi y^{1/2},$ $\displaystyle R_{G}\left(0,0,z\right)$ $\displaystyle=\tfrac{1}{2}z^{1/2},$ ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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 $2R_{G}\left(x,y,y\right)=yR_{C}\left(x,y\right)+\sqrt{x}.$ ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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(iii) $R_{J}\left(x,y,z,p\right)$

 19.20.6 $\displaystyle R_{J}\left(x,x,x,x\right)$ $\displaystyle=x^{-3/2},$ $\displaystyle R_{J}\left(\lambda x,\lambda y,\lambda z,\lambda p\right)$ $\displaystyle=\lambda^{-3/2}R_{J}\left(x,y,z,p\right),$ $\displaystyle R_{J}\left(x,y,z,z\right)$ $\displaystyle=R_{D}\left(x,y,z\right),$ $\displaystyle R_{J}\left(0,0,z,p\right)$ $\displaystyle=\infty,$ $\displaystyle R_{J}\left(x,x,x,p\right)$ $\displaystyle=R_{D}\left(p,p,x\right)=\frac{3}{x-p}\left(R_{C}\left(x,p\right)% -\frac{1}{\sqrt{x}}\right),$ $x\neq p$, $xp\neq 0$.
 19.20.7 $R_{J}\left(x,y,z,p\right)\to+\infty,$ $p\to 0+$ or $0-$; $x,y,z>0$. ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: 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 $\displaystyle R_{J}\left(0,y,y,p\right)$ $\displaystyle=\frac{3\pi}{2(y\sqrt{p}+p\sqrt{y})},$ $p>0$, $\displaystyle R_{J}\left(0,y,y,-q\right)$ $\displaystyle=\frac{-3\pi}{2\sqrt{y}(y+q)},$ $q>0$, $\displaystyle R_{J}\left(x,y,y,p\right)$ $\displaystyle=\frac{3}{p-y}(R_{C}\left(x,y\right)-R_{C}\left(x,p\right)),$ $p\neq y$, $\displaystyle R_{J}\left(x,y,y,y\right)$ $\displaystyle=R_{D}\left(x,y,y\right).$
 19.20.9 $R_{J}\left(0,y,z,\pm\sqrt{yz}\right)=\pm\frac{3}{2\sqrt{yz}}R_{F}\left(0,y,z% \right).$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: 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 $\displaystyle\lim_{p\to 0+}\sqrt{p}R_{J}\left(0,y,z,p\right)$ $\displaystyle=\frac{3\pi}{2\sqrt{y}\sqrt{z}},$ $\displaystyle\lim_{p\to 0-}R_{J}\left(0,y,z,p\right)$ $\displaystyle=-R_{D}\left(0,y,z\right)-R_{D}\left(0,z,y\right)=\frac{-6}{yz}R_% {G}\left(0,y,z\right).$ ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: 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), Erratum (V1.1.3) for Chapter 19 Permalink: http://dlmf.nist.gov/19.20.E10 Encodings: TeX, TeX, pMML, pMML, png, png Correction (effective with 1.1.3): The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior. Suggested 2021-06-07 by Luc Maisonobe See also: Annotations for §19.20(iii), §19.20 and Ch.19
 19.20.11 $R_{J}\left(0,y,z,p\right)=\frac{3}{2p\sqrt{z}}\ln\left(\frac{16z}{y}\right)-% \frac{3}{p}R_{C}\left(z,p\right)+O\left(y\ln y\right),$ $y\to 0+$; $p$ ($\neq 0$) real. ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: symmetric elliptic integral of third kind, $\sim$: Poincaré asymptotic expansion and $\ln\NVar{z}$: 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 $\frac{-3}{p}R_{C}\left(z,p\right)$ 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 $\lim_{p\to\pm\infty}pR_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right).$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: 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 $2(p-x)R_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right)-3\sqrt{x}R_{C}\left(% yz,p^{2}\right),$ $p=x\pm\sqrt{(y-x)(z-x)}$,

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

When the variables are real and distinct, the various cases of $R_{J}\left(x,y,z,p\right)$ 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 $0\leq x, then the Cauchy principal value of $R_{J}$ is given by

 19.20.14 $(q+z)R_{J}\left(x,y,z,-q\right)=(p-z)R_{J}\left(x,y,z,p\right)-3R_{F}\left(x,y% ,z\right)+3\left(\frac{xyz}{xy+pq}\right)^{1/2}R_{C}\left(xy+pq,pq\right),$

valid when

 19.20.15 $\displaystyle q$ $\displaystyle>0,$ $\displaystyle p$ $\displaystyle=\frac{z(x+y+q)-xy}{z+q},$ ⓘ 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 $\displaystyle p$ $\displaystyle=wy+(1-w)z,$ $\displaystyle w$ $\displaystyle=\frac{z-x}{z+q},$ $\displaystyle 0$ $\displaystyle ⓘ 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 $x, $p$ is in a hyperbolic region. In the complete case ($x=0$) (19.20.14) reduces to

 19.20.17 $(q+z)R_{J}\left(0,y,z,-q\right)=(p-z)R_{J}\left(0,y,z,p\right)-3R_{F}\left(0,y% ,z\right),$ $p=z(y+q)/(z+q)$, $w=z/(z+q)$. ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: 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(iv) $R_{D}\left(x,y,z\right)$

 19.20.18 $\displaystyle R_{D}\left(x,x,x\right)$ $\displaystyle=x^{-3/2},$ $\displaystyle R_{D}\left(\lambda x,\lambda y,\lambda z\right)$ $\displaystyle=\lambda^{-3/2}R_{D}\left(x,y,z\right),$ $\displaystyle R_{D}\left(0,y,y\right)$ $\displaystyle=\tfrac{3}{4}\pi\,y^{-3/2},$ $\displaystyle R_{D}\left(0,0,z\right)$ $\displaystyle=\infty.$ ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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 $R_{D}\left(x,y,z\right)\sim 3x^{-1/2}y^{-1/2}z^{-1/2},$ $z/\sqrt{xy}\to 0$. ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables and $\sim$: asymptotic equality Referenced by: §19.20(iv), Erratum (V1.1.3) for Chapter 19 Permalink: http://dlmf.nist.gov/19.20.E19 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior. Suggested 2021-06-07 by Luc Maisonobe See also: Annotations for §19.20(iv), §19.20 and Ch.19
 19.20.20 $R_{D}\left(x,y,y\right)=\frac{3}{2(y-x)}\left(R_{C}\left(x,y\right)-\frac{% \sqrt{x}}{y}\right),$ $x\neq y$, $y\neq 0$, ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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 $R_{D}\left(x,x,z\right)=\frac{3}{z-x}\left(R_{C}\left(z,x\right)-\frac{1}{% \sqrt{z}}\right),$ $x\neq z$, $xz\neq 0$. ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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 $\int_{0}^{1}\frac{t^{2}\,\mathrm{d}t}{\sqrt{1-t^{4}}}=\tfrac{1}{3}R_{D}\left(0% ,2,1\right)=\frac{\left(\Gamma\left(\frac{3}{4}\right)\right)^{2}}{(2\pi)^{1/2% }}=0.59907\;01173\;67796\;10371\dots.$ ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables, $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential 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 $R_{D}\left(x,y,a\right)=R_{-\frac{3}{4}}\left(\tfrac{5}{4},\tfrac{1}{2};a^{2},% xy\right),$ $a=\tfrac{1}{2}x+\tfrac{1}{2}y$.

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

Define $c=\sum_{j=1}^{n}b_{j}$. Then

 19.20.24 $\displaystyle R_{0}\left(\mathbf{b};\mathbf{z}\right)$ $\displaystyle=1,$ $\displaystyle R_{N}\left(\mathbf{b};\mathbf{z}\right)$ $\displaystyle=\frac{N!}{{\left(c\right)_{N}}}T_{N}(\mathbf{b},\mathbf{z}),$ $N=0,1,2,\dots$,

where $T_{N}$ is defined by (19.19.1). Also,

 19.20.25 $R_{-c}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}},$
 19.20.26 $R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}}R_{-a^{% \prime}}\left(\mathbf{b};\boldsymbol{{z^{-1}}}\right),$ $a+a^{\prime}=c$, $\boldsymbol{{z^{-1}}}=(z_{1}^{-1},\dots,z_{n}^{-1})$.