19 Elliptic Integrals Symmetric Integrals 19.19 Taylor and Related Series 19.21 Connection Formulas

§19.20 Special Cases

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Keywords:: special cases, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.20
See also:: Annotations for Ch.19

§19.20(i) $R_{F}\left(x,y,z\right)$
§19.20(ii) $R_{G}\left(x,y,z\right)$
§19.20(iii) $R_{J}\left(x,y,z,p\right)$
§19.20(iv) $R_{D}\left(x,y,z\right)$
§19.20(v) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

§19.20(i) $R_{F}\left(x,y,z\right)$

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Keywords:: general lemniscatic case, lemniscate constants, symmetric elliptic integrals
Notes:: In (19.20.2) put $t=1/\sqrt{s+1}$ ; alternatively use (19.29.19). For the second equality replace $t^{4}$ by $t$ and apply (5.12.1). For (19.20.3) use Carlson (1977b, Ex. 6.9-5 and p. 309) and (19.25.42).
Permalink:: http://dlmf.nist.gov/19.20.i
See also:: Annotations for §19.20 and Ch.19

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	$\displaystyle R_{F}\left(x,x,x\right)$	$\displaystyle=x^{-1/2},$
	$\displaystyle R_{F}\left(\lambda x,\lambda y,\lambda z\right)$	$\displaystyle=\lambda^{-1/2}R_{F}\left(x,y,z\right),$
	$\displaystyle R_{F}\left(x,y,y\right)$	$\displaystyle=R_{C}\left(x,y\right),$
	$\displaystyle R_{F}\left(0,y,y\right)$	$\displaystyle=\tfrac{1}{2}\pi y^{-1/2},$
	$\displaystyle R_{F}\left(0,0,z\right)$	$\displaystyle=\infty.$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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		$\int_{0}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{F}\left(0,1,2\right)=\frac% {\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}=1.31102\;8777% 1\;46059\;90523\;\dots.$
ⓘ 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 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		$R_{F}\left(x,a,y\right)=R_{-\frac{1}{4}}\left(\tfrac{3}{4},\tfrac{1}{2};a^{2},% xy\right),$
$a=\frac{1}{2}(x+y)$ .
ⓘ 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 and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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(ii) $R_{G}\left(x,y,z\right)$

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Notes:: For (19.20.4) use (19.20.5) and (19.23.6_5). For (19.20.5) put $z=y$ in (19.21.10).
Referenced by:: §19.20(i)
Permalink:: http://dlmf.nist.gov/19.20.ii
See also:: Annotations for §19.20 and Ch.19

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) $R_{J}\left(x,y,z,p\right)$

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Keywords:: circular cases, hyperbolic cases, symmetric elliptic integrals
Notes:: For (19.20.6) substitute in (19.16.2) and (19.16.5). In (19.20.7) see (19.27.12) for $p\rightarrow 0+$ ; for $p\rightarrow 0-$ use (19.20.17) and (19.6.15). In (19.20.8) the third equation is proved by partial fractions, and also implies the first two equations by (19.6.15). For (19.20.9) put $x=0$ in (19.20.13). For (19.20.10) interchange $x$ and $z$ in (19.27.14) and use (19.6.15). For (19.20.11) use (19.27.13), (19.20.17), and (19.27.2). For (19.20.12) see (19.27.11) and (19.21.12). For (19.20.13) let $q=p$ in (19.21.12). For (19.20.14) exchange $x$ and $z$ in (19.21.12) and use (19.2.20).
Referenced by:: §19.21(iii)
Permalink:: http://dlmf.nist.gov/19.20.iii
See also:: Annotations for §19.20 and Ch.19

19.20.6		$\displaystyle R_{J}\left(x,x,x,x\right)$	$\displaystyle=x^{-3/2},$
		$\displaystyle R_{J}\left(\lambda x,\lambda y,\lambda z,\lambda p\right)$	$\displaystyle=\lambda^{-3/2}R_{J}\left(x,y,z,p\right),$
		$\displaystyle R_{J}\left(x,y,z,z\right)$	$\displaystyle=R_{D}\left(x,y,z\right),$
		$\displaystyle R_{J}\left(0,0,z,p\right)$	$\displaystyle=\infty,$
		$\displaystyle R_{J}\left(x,x,x,p\right)$	$\displaystyle=R_{D}\left(p,p,x\right)=\frac{3}{x-p}\left(R_{C}\left(x,p\right)% -\frac{1}{\sqrt{x}}\right),$
	$x\neq p$ , $xp\neq 0$ .
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables 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.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		$R_{J}\left(x,y,z,p\right)\to+\infty,$
$p\to 0+$ 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		$\displaystyle R_{J}\left(0,y,y,p\right)$	$\displaystyle=\frac{3\pi}{2(y\sqrt{p}+p\sqrt{y})},$
	$p>0$ ,
		$\displaystyle R_{J}\left(0,y,y,-q\right)$	$\displaystyle=\frac{-3\pi}{2\sqrt{y}(y+q)},$
	$q>0$ ,
		$\displaystyle R_{J}\left(x,y,y,p\right)$	$\displaystyle=\frac{3}{p-y}(R_{C}\left(x,y\right)-R_{C}\left(x,p\right)),$
	$p\neq y$ ,
		$\displaystyle R_{J}\left(x,y,y,y\right)$	$\displaystyle=R_{D}\left(x,y,y\right).$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind and $\pi$ : the ratio of the circumference of a circle to its diameter Referenced by: Figure 19.17.8, 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		$R_{J}\left(0,y,z,\pm\sqrt{yz}\right)=\pm\frac{3}{2\sqrt{yz}}R_{F}\left(0,y,z% \right).$
ⓘ 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	$\displaystyle\lim_{p\to 0+}\sqrt{p}R_{J}\left(0,y,z,p\right)$	$\displaystyle=\frac{3\pi}{2\sqrt{y}\sqrt{z}},$
19.20.10	$\displaystyle\lim_{p\to 0-}R_{J}\left(0,y,z,p\right)$	$\displaystyle=-R_{D}\left(0,y,z\right)-R_{D}\left(0,z,y\right)=\frac{-6}{yz}R_% {G}\left(0,y,z\right).$
ⓘ 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		$2(p-x)R_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right)-3\sqrt{x}R_{C}\left(% yz,p^{2}\right),$
$p=x\pm\sqrt{(y-x)(z-x)}$ ,
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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(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 $R_{J}\left(x,y,z,p\right)$ 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 $0\leq x<y<z$ , then the Cauchy principal value of $R_{J}$ is given by

19.20.14		$(q+z)R_{J}\left(x,y,z,-q\right)=(p-z)R_{J}\left(x,y,z,p\right)-3R_{F}\left(x,y% ,z\right)+3\left(\frac{xyz}{xy+pq}\right)^{1/2}R_{C}\left(xy+pq,pq\right),$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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: §19.16(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	$\displaystyle q$	$\displaystyle>0,$
19.20.15	$\displaystyle p$	$\displaystyle=\frac{z(x+y+q)-xy}{z+q},$
ⓘ 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

19.20.16	$\displaystyle p$	$\displaystyle=wy+(1-w)z,$
	$\displaystyle w$	$\displaystyle=\frac{z-x}{z+q},$
	$\displaystyle 0$	$\displaystyle<w<1.$
ⓘ 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 $x<y<p<z$ , $p$ 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) $R_{D}\left(x,y,z\right)$

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Keywords:: general lemniscatic case, lemniscate constants, symmetric elliptic integrals
Notes:: For the third equation in (19.20.18) put $t=y{\tan}^{2}\theta$ in (19.16.5); for the fourth equation see (19.27.7). For (19.20.19) see (19.27.8). For (19.20.20) and (19.20.21) use (19.16.15), (19.16.9), and Carlson (1977b, Table 8.5-1). In (19.20.22) put $t=1/\sqrt{s+1}$ ; alternatively use (19.29.20). For the second equality replace $t^{4}$ by $t$ and apply (5.12.1). For (19.20.23) use Carlson (1977b, Ex. 6.9-5 and p. 309) and (19.25.42).
Permalink:: http://dlmf.nist.gov/19.20.iv
See also:: Annotations for §19.20 and Ch.19

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 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		$R_{D}\left(x,y,a\right)=R_{-\frac{3}{4}}\left(\tfrac{5}{4},\tfrac{1}{2};a^{2},% xy\right),$
$a=\tfrac{1}{2}x+\tfrac{1}{2}y$ .
ⓘ 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 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.E23 Encodings: TeX, pMML, png See also: Annotations for §19.20(iv), §19.20 and Ch.19

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

ⓘ

Keywords:: special cases, symmetric elliptic integrals
Notes:: See Carlson (1977b, (6.2-1), (6.8-15)).
Referenced by:: §19.20(i)
Permalink:: http://dlmf.nist.gov/19.20.v
See also:: Annotations for §19.20 and Ch.19

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

19.20.24		$\displaystyle R_{0}\left(\mathbf{b};\mathbf{z}\right)$	$\displaystyle=1,$
		$\displaystyle R_{N}\left(\mathbf{b};\mathbf{z}\right)$	$\displaystyle=\frac{N!}{{\left(c\right)_{N}}}T_{N}(\mathbf{b},\mathbf{z}),$
	$N=0,1,2,\dots$ ,
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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		$R_{-c}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}},$
ⓘ 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 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		$R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}}R_{-a^{% \prime}}\left(\mathbf{b};\boldsymbol{{z^{-1}}}\right),$
$a+a^{\prime}=c$ , $\boldsymbol{{z^{-1}}}=(z_{1}^{-1},\dots,z_{n}^{-1})$ .
ⓘ 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 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

§19.20 Special Cases

Contents

§19.20(i) RF⁡(x,y,z)

§19.20(ii) RG⁡(x,y,z)

§19.20(iii) RJ⁡(x,y,z,p)

§19.20(iv) RD⁡(x,y,z)

§19.20(v) R−a⁡(𝐛;𝐳)

§19.20(i) $R_{F}\left(x,y,z\right)$

§19.20(ii) $R_{G}\left(x,y,z\right)$

§19.20(iii) $R_{J}\left(x,y,z,p\right)$

§19.20(iv) $R_{D}\left(x,y,z\right)$

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