§19.21 Connection Formulas

Legendre’s relation (19.7.1) can be written

19.21.1		$$R_F(0,z+1,z)R_D(0,z+1,1)+R_D(0,z+1,z)R_F(0,z+1,1)=3\pi /(2z),$$
$z\in ℂ\setminus (-\mathrm{\infty },0]$.
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval and $\setminus$: set subtraction Referenced by: §19.21(i), §19.21(i), §19.7(i) Permalink: http://dlmf.nist.gov/19.21.E1 Encodings: TeX, pMML, png See also: Annotations for §19.21(i), §19.21 and Ch.19

The case $z=1$ shows that the product of the two lemniscate constants, (19.20.2) and (19.20.22), is $\pi /4$.

19.21.2		$$3R_F(0,y,z)=z{R}_D(0,y,z)+y{R}_D(0,z,y).$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind Referenced by: §19.21(i) Permalink: http://dlmf.nist.gov/19.21.E2 Encodings: TeX, pMML, png See also: Annotations for §19.21(i), §19.21 and Ch.19

19.21.3		$$6R_G(0,y,z)=yz(R_D(0,y,z)+R_D(0,z,y))=3z{R}_F(0,y,z)+z(y-z)R_D(0,y,z).$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind Referenced by: §19.21(i), §19.22(i), §19.34 Permalink: http://dlmf.nist.gov/19.21.E3 Encodings: TeX, pMML, png See also: Annotations for §19.21(i), §19.21 and Ch.19

The complete cases of $R_F$ and $R_G$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). Upper signs apply if , and lower signs if :

19.21.4		$$R_F(0,z-1,z)=R_F(0,1-z,1)\mp \mathrm{i}{R}_F(0,z,1),$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind and $\mathrm{i}$: imaginary unit Permalink: http://dlmf.nist.gov/19.21.E4 Encodings: TeX, pMML, png See also: Annotations for §19.21(i), §19.21 and Ch.19

19.21.5		$$2R_G(0,z-1,z)=\begin{array}{l}2R_G(0,1-z,1)\pm \mathrm{i}2R_G(0,z,1)+(z-1)R_F(0,1-z,1)\\ \mp \mathrm{i}z{R}_F(0,z,1).\end{array}$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\mathrm{i}$: imaginary unit Permalink: http://dlmf.nist.gov/19.21.E5 Encodings: TeX, pMML, png See also: Annotations for §19.21(i), §19.21 and Ch.19

Let $y$, $z$, and $p$ be positive and distinct, and permute $y$ and $z$ to ensure that $y$ does not lie between $z$ and $p$. The complete case of $R_J$ can be expressed in terms of $R_F$ and $R_D$:

19.21.6		$$(\sqrt{rp}/z)R_J(0,y,z,p)=(r-1)R_F(0,y,z)R_D(p,rz,z)+R_D(0,y,z)R_F(p,rz,z),$$
$r=(y-p)/(y-z)>0$.
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind Referenced by: §19.21(i), §19.21(i) Permalink: http://dlmf.nist.gov/19.21.E6 Encodings: TeX, pMML, png See also: Annotations for §19.21(i), §19.21 and Ch.19

If and $y=z+1$, then as $p\to 0$ (19.21.6) reduces to Legendre’s relation (19.21.1).

$R_D(x,y,z)$ is symmetric only in $x$ and $y$, but either (nonzero) $x$ or (nonzero) $y$ can be moved to the third position by using

19.21.7		$$(x-y)R_D(y,z,x)+(z-y)R_D(x,y,z)=3R_F(x,y,z)-3\sqrt{y/(xz)},$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind Referenced by: §19.21(ii), §19.25(i) Permalink: http://dlmf.nist.gov/19.21.E7 Encodings: TeX, pMML, png See also: Annotations for §19.21(ii), §19.21 and Ch.19

or the corresponding equation with $x$ and $y$ interchanged.

19.21.8		$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)$	$=3{(xyz)}^{-1/2},$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables Referenced by: §19.21(ii), §19.33(iii) Permalink: http://dlmf.nist.gov/19.21.E8 Encodings: TeX, pMML, png See also: Annotations for §19.21(ii), §19.21 and Ch.19
19.21.9		$x{R}_D(y,z,x)+y{R}_D(z,x,y)+z{R}_D(x,y,z)$	$=3R_F(x,y,z).$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind Referenced by: §19.21(i), §19.21(ii) Permalink: http://dlmf.nist.gov/19.21.E9 Encodings: TeX, pMML, png See also: Annotations for §19.21(ii), §19.21 and Ch.19

19.21.10		$$2R_G(x,y,z)=z{R}_F(x,y,z)-\frac{1}{3}(x-z)(y-z)R_D(x,y,z)+\sqrt{xy/z},$$
$z\ne 0$.
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind Referenced by: §19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), §19.36(i), §19.36(ii) Permalink: http://dlmf.nist.gov/19.21.E10 Encodings: TeX, pMML, png See also: Annotations for §19.21(ii), §19.21 and Ch.19

Because $R_G$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\le 0$ if the variables are real, thereby avoiding cancellations when $R_G$ is calculated from $R_F$ and $R_D$ (see §19.36(i)).

19.21.11		$$6R_G(x,y,z)=3(x+y+z)R_F(x,y,z)-\sum x^2{R}_D(y,z,x)=\sum x(y+z)R_D(y,z,x),$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind Referenced by: §19.21(i), §19.21(ii) Permalink: http://dlmf.nist.gov/19.21.E11 Encodings: TeX, pMML, png See also: Annotations for §19.21(ii), §19.21 and Ch.19

where both summations extend over the three cyclic permutations of $x,y,z$.

Connection formulas for $R_{-a}(\mathbf{b};\mathbf{z})$ are given in Carlson (1977b, pp. 99, 101, and 123–124).

Let $x,y,z$ be real and nonnegative, with at most one of them 0. Change-of-parameter relations can be used to shift the parameter $p$ of $R_J$ from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). The latter case allows evaluation of Cauchy principal values (see (19.20.14)).

19.21.12		$$(p-x)R_J(x,y,z,p)+(q-x)R_J(x,y,z,q)=3R_F(x,y,z)-3R_C(\xi ,\eta ),$$
ⓘ 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, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\xi$ and $\eta$ Referenced by: §19.15, §19.20(iii), §19.21(iii), §19.25(i), §19.26(i) Permalink: http://dlmf.nist.gov/19.21.E12 Encodings: TeX, pMML, png See also: Annotations for §19.21(iii), §19.21 and Ch.19

where

19.21.13		$(p-x)(q-x)$	$=(y-x)(z-x),$
		$\xi$	$=yz/x,$
		$\eta$	$=pq/x,$
ⓘ Symbols: $\xi$ and $\eta$ Permalink: http://dlmf.nist.gov/19.21.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.21(iii), §19.21 and Ch.19

and $x,y,z$ may be permuted. Also,

19.21.14		$$\eta -\xi =p+q-y-z=\frac{(p-y)(p-z)}{p-x}=\frac{(q-y)(q-z)}{q-x}=\frac{(p-y)(q-y)}{x-y}=\frac{(p-z)(q-z)}{x-z}.$$
ⓘ Symbols: $\xi$ and $\eta$ Permalink: http://dlmf.nist.gov/19.21.E14 Encodings: TeX, pMML, png See also: Annotations for §19.21(iii), §19.21 and Ch.19

For each value of $p$, permutation of $x,y,z$ produces three values of $q$, one of which lies in the same region as $p$ and two lie in the other region of the same type. In (19.21.12), if $x$ is the largest (smallest) of $x,y$, and $z$, then $p$ and $q$ lie in the same region if it is circular (hyperbolic); otherwise $p$ and $q$ lie in different regions, both circular or both hyperbolic. If $x=0$, then $\xi =\eta =\mathrm{\infty }$ and $R_C(\xi ,\eta )=0$; hence

19.21.15		$$p{R}_J(0,y,z,p)+q{R}_J(0,y,z,q)=3R_F(0,y,z),$$
$pq=yz$.
ⓘ 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 Permalink: http://dlmf.nist.gov/19.21.E15 Encodings: TeX, pMML, png See also: Annotations for §19.21(iii), §19.21 and Ch.19