# §19.21 Connection Formulas

## §19.21(i) Complete Integrals

Legendre’s relation (19.7.1) can be written

 19.21.1 $\mathop{R_{F}\/}\nolimits\!\left(0,z+1,z\right)\mathop{R_{D}\/}\nolimits\!% \left(0,z+1,1\right)+\mathop{R_{D}\/}\nolimits\!\left(0,z+1,z\right)\mathop{R_% {F}\/}\nolimits\!\left(0,z+1,1\right)=3\pi/(2z),$ $z\in\mathbb{C}\setminus(-\infty,0]$.

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 $3\!\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)=z\mathop{R_{D}\/}\nolimits\!% \left(0,y,z\right)+y\mathop{R_{D}\/}\nolimits\!\left(0,z,y\right).$ Symbols: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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)
 19.21.3 $6\!\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)=yz(\mathop{R_{D}\/}\nolimits% \!\left(0,y,z\right)+\mathop{R_{D}\/}\nolimits\!\left(0,z,y\right))=3z\mathop{% R_{F}\/}\nolimits\!\left(0,y,z\right)+z(y-z)\mathop{R_{D}\/}\nolimits\!\left(0% ,y,z\right).$

The complete cases of $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{G}\/}\nolimits$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). Upper signs apply if $0<\mathop{\mathrm{ph}\/}\nolimits z<\pi$, and lower signs if $-\pi<\mathop{\mathrm{ph}\/}\nolimits z<0$:

 19.21.4 $\mathop{R_{F}\/}\nolimits\!\left(0,z-1,z\right)=\mathop{R_{F}\/}\nolimits\!% \left(0,1-z,1\right)\mp\mathrm{i}\!\mathop{R_{F}\/}\nolimits\!\left(0,z,1% \right),$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Permalink: http://dlmf.nist.gov/19.21.E4 Encodings: TeX, pMML, png See also: Annotations for 19.21(i)
 19.21.5 $2\!\mathop{R_{G}\/}\nolimits\!\left(0,z-1,z\right)=2\!\mathop{R_{G}\/}% \nolimits\!\left(0,1-z,1\right)\pm\mathrm{i}2\!\mathop{R_{G}\/}\nolimits\!% \left(0,z,1\right)+(z-1)\mathop{R_{F}\/}\nolimits\!\left(0,1-z,1\right)\mp% \mathrm{i}z\!\mathop{R_{F}\/}\nolimits\!\left(0,z,1\right).$

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 $\mathop{R_{J}\/}\nolimits$ can be expressed in terms of $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{D}\/}\nolimits$:

 19.21.6 $(\sqrt{rp}/z)\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)={(r-1)}\mathop{R_% {F}\/}\nolimits\!\left(0,y,z\right)\mathop{R_{D}\/}\nolimits\!\left(p,rz,z% \right)+\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)\mathop{R_{F}\/}\nolimits% \!\left(p,rz,z\right),$ $r=(y-p)/(y-z)>0$.

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

## §19.21(ii) Incomplete Integrals

$\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ 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 $(x-y)\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right)+(z-y)\mathop{R_{D}\/}% \nolimits\!\left(x,y,z\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)% -3\sqrt{y/(xz)},$ Symbols: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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)

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

 19.21.8 $\displaystyle\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right)+\mathop{R_{D}\/}% \nolimits\!\left(z,x,y\right)+\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=3(xyz)^{-1/2},$ Symbols: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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) 19.21.9 $\displaystyle x\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right)+y\mathop{R_{D}\/}% \nolimits\!\left(z,x,y\right)+z\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ $\displaystyle=3\!\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right).$ Symbols: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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)
 19.21.10 $2\!\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=z\mathop{R_{F}\/}\nolimits\!% \left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)\mathop{R_{D}\/}\nolimits\!\left(x,y,% z\right)+\sqrt{xy/z},$ $z\neq 0$.

Because $\mathop{R_{G}\/}\nolimits$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\leq 0$ if the variables are real, thereby avoiding cancellations when $\mathop{R_{G}\/}\nolimits$ is calculated from $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{D}\/}\nolimits$ (see §19.36(i)).

 19.21.11 $6\!\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=3(x+y+z)\mathop{R_{F}\/}% \nolimits\!\left(x,y,z\right)-\sum x^{2}\mathop{R_{D}\/}\nolimits\!\left(y,z,x% \right)=\sum x(y+z)\mathop{R_{D}\/}\nolimits\!\left(y,z,x\right),$

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

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

## §19.21(iii) Change of Parameter of $\mathop{R_{J}\/}\nolimits$

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 $\mathop{R_{J}\/}\nolimits$ 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 $(p-x)\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)+(q-x)\mathop{R_{J}\/}% \nolimits\!\left(x,y,z,q\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(x,y,z% \right)-3\!\mathop{R_{C}\/}\nolimits\!\left(\xi,\eta\right),$

where

 19.21.13 $\displaystyle(p-x)(q-x)$ $\displaystyle=(y-x)(z-x),$ $\displaystyle\xi$ $\displaystyle=yz/x,$ $\displaystyle\eta$ $\displaystyle=pq/x,$ 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)

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)

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 $\xi=\eta=\infty$ and $\mathop{R_{C}\/}\nolimits\!\left(\xi,\eta\right)=0$; hence

 19.21.15 $p\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)+q\mathop{R_{J}\/}\nolimits\!% \left(0,y,z,q\right)=3\!\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right),$ $pq=yz$.