# §19.20 Special Cases

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

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

The first lemniscate constant is given by

 19.20.2 $\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}=\mathop{R_{F}\/}\nolimits\!% \left(0,1,2\right)=\frac{\left(\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{4}% \right)\right)^{2}}{4(2\pi)^{1/2}}=1.31102\;87771\;46059\;90523\dots.$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: 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)

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

 19.20.3 $\mathop{R_{F}\/}\nolimits\!\left(x,a,y\right)=\mathop{R_{-\frac{1}{4}}\/}% \nolimits\!\left(\tfrac{3}{4},\tfrac{1}{2};a^{2},xy\right),$ $a=\frac{1}{2}(x+y)$.

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

 19.20.4 $\displaystyle\mathop{R_{G}\/}\nolimits\!\left(x,x,x\right)$ $\displaystyle=x^{1/2},$ $\displaystyle\mathop{R_{G}\/}\nolimits\!\left(\lambda x,\lambda y,\lambda z\right)$ $\displaystyle=\lambda^{1/2}\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right),$ $\displaystyle\mathop{R_{G}\/}\nolimits\!\left(0,y,y\right)$ $\displaystyle=\tfrac{1}{4}\pi y^{1/2},$ $\displaystyle\mathop{R_{G}\/}\nolimits\!\left(0,0,z\right)$ $\displaystyle=\tfrac{1}{2}z^{1/2},$ Symbols: $\mathop{R_{G}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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)
 19.20.5 $2\!\mathop{R_{G}\/}\nolimits\!\left(x,y,y\right)=y\mathop{R_{C}\/}\nolimits\!% \left(x,y\right)+\sqrt{x}.$

## §19.20(iii) $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$

 19.20.6 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,x,x,x\right)$ $\displaystyle=x^{-3/2},$ $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(\lambda x,\lambda y,\lambda z,% \lambda p\right)$ $\displaystyle=\lambda^{-3/2}\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right),$ $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,z,z\right)$ $\displaystyle=\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right),$ $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(0,0,z,p\right)$ $\displaystyle=\infty,$ $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,x,x,p\right)$ $\displaystyle=\mathop{R_{D}\/}\nolimits\!\left(p,p,x\right)=\frac{3}{x-p}\left% (\mathop{R_{C}\/}\nolimits\!\left(x,p\right)-\frac{1}{\sqrt{x}}\right),$ $x\neq p$, $xp\neq 0$.
 19.20.7 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)\to+\infty,$ $p\to 0+$ or $0-$; $x,y,z>0$. Symbols: $\mathop{R_{J}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: 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)
 19.20.8 $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(0,y,y,p\right)$ $\displaystyle=\frac{3\pi}{2(y\sqrt{p}+p\sqrt{y})},$ $p>0$, $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(0,y,y,-q\right)$ $\displaystyle=\frac{-3\pi}{2\sqrt{y}(y+q)},$ $q>0$, $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,y,p\right)$ $\displaystyle=\frac{3}{p-y}(\mathop{R_{C}\/}\nolimits\!\left(x,y\right)-% \mathop{R_{C}\/}\nolimits\!\left(x,p\right)),$ $p\neq y$, $\displaystyle\mathop{R_{J}\/}\nolimits\!\left(x,y,y,y\right)$ $\displaystyle=\mathop{R_{D}\/}\nolimits\!\left(x,y,y\right).$
 19.20.9 $\mathop{R_{J}\/}\nolimits\!\left(0,y,z,\pm\sqrt{yz}\right)=\pm\frac{3}{2\sqrt{% yz}}\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right).$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $\mathop{R_{J}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: 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)
 19.20.10 $\displaystyle\lim_{p\to 0+}\sqrt{p}\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)$ $\displaystyle=\frac{3\pi}{2\sqrt{yz}},$ $\displaystyle\lim_{p\to 0-}\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)$ $\displaystyle=-\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)-\mathop{R_{D}\/}% \nolimits\!\left(0,z,y\right)=\frac{-6}{yz}\mathop{R_{G}\/}\nolimits\!\left(0,% y,z\right).$
 19.20.11 $\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)\sim\frac{3}{2p\sqrt{z}}\mathop% {\ln\/}\nolimits\left(\frac{16z}{y}\right),$ $y\to 0+$; $p$ ($\neq 0$) real.
 19.20.12 $\lim_{p\to\pm\infty}p\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=3\!% \mathop{R_{F}\/}\nolimits\!\left(x,y,z\right).$
 19.20.13 $2(p-x)\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=3\!\mathop{R_{F}\/}% \nolimits\!\left(x,y,z\right)-3\sqrt{x}\mathop{R_{C}\/}\nolimits\!\left(yz,p^{% 2}\right),$ $p=x\pm\sqrt{(y-x)(z-x)}$,

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

When the variables are real and distinct, the various cases of $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ 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 $\mathop{R_{J}\/}\nolimits$ is given by

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

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)

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)

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

 19.20.17 $(q+z)\mathop{R_{J}\/}\nolimits\!\left(0,y,z,-q\right)=(p-z)\mathop{R_{J}\/}% \nolimits\!\left(0,y,z,p\right)-3\!\mathop{R_{F}\/}\nolimits\!\left(0,y,z% \right),$ $p=z(y+q)/(z+q)$, $w=z/(z+q)$.

## §19.20(iv) $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$

 19.20.18 $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(x,x,x\right)$ $\displaystyle=x^{-3/2},$ $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(\lambda x,\lambda y,\lambda z\right)$ $\displaystyle=\lambda^{-3/2}\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right),$ $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(0,y,y\right)$ $\displaystyle=\tfrac{3}{4}\pi\,y^{-3/2},$ $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(0,0,z\right)$ $\displaystyle=\infty.$ Symbols: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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)
 19.20.19 $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)\sim 3(xyz)^{-1/2},$ $z/\sqrt{xy}\to 0$. Symbols: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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)
 19.20.20 $\mathop{R_{D}\/}\nolimits\!\left(x,y,y\right)=\frac{3}{2(y-x)}\left(\mathop{R_% {C}\/}\nolimits\!\left(x,y\right)-\frac{\sqrt{x}}{y}\right),$ $x\neq y$, $y\neq 0$,
 19.20.21 $\mathop{R_{D}\/}\nolimits\!\left(x,x,z\right)=\frac{3}{z-x}\left(\mathop{R_{C}% \/}\nolimits\!\left(z,x\right)-\frac{1}{\sqrt{z}}\right),$ $x\neq z$, $xz\neq 0$.

The second lemniscate constant is given by

 19.20.22 $\int_{0}^{1}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{4}}}=\tfrac{1}{3}\mathop{R_{D}% \/}\nolimits\!\left(0,2,1\right)=\frac{\left(\mathop{\Gamma\/}\nolimits\!\left% (\frac{3}{4}\right)\right)^{2}}{(2\pi)^{1/2}}=0.59907\;01173\;67796\;10371\dots.$ Symbols: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: 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, 19.3.6 Permalink: http://dlmf.nist.gov/19.20.E22 Encodings: TeX, pMML, png See also: Annotations for 19.20(iv)

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 $\mathop{R_{D}\/}\nolimits\!\left(x,y,a\right)=\mathop{R_{-\frac{3}{4}}\/}% \nolimits\!\left(\tfrac{5}{4},\tfrac{1}{2};a^{2},xy\right),$ $a=\tfrac{1}{2}x+\tfrac{1}{2}y$.

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

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

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

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

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