# §19.21 Connection Formulas

## §19.21(i) Complete Integrals

Legendre’s relation (19.7.1) can be written

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

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 $3R_{F}\left(0,y,z\right)=zR_{D}\left(0,y,z\right)+yR_{D}\left(0,z,y\right).$ ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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 and Ch.19
 19.21.3 $6R_{G}\left(0,y,z\right)=yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right))=3% zR_{F}\left(0,y,z\right)+z(y-z)R_{D}\left(0,y,z\right).$

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 $0<\operatorname{ph}z<\pi$, and lower signs if $-\pi<\operatorname{ph}z<0$:

 19.21.4 $R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)\mp\mathrm{i}R_{F}\left(0,z% ,1\right),$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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 $2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1\right)\pm\mathrm{i}2R_{G}\left(% 0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)\mp\mathrm{i}zR_{F}\left(0,z,1% \right).$

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 $R_{J}$ can be expressed in terms of $R_{F}$ and $R_{D}$:

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

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

$R_{D}\left(x,y,z\right)$ 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 $(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z% \right)-3y^{1/2}x^{-1/2}z^{-1/2},$ ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Referenced by: §19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19 Permalink: http://dlmf.nist.gov/19.21.E7 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 §19.21(ii), §19.21 and Ch.19

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

 19.21.8 $\displaystyle R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)$ $\displaystyle=3x^{-1/2}y^{-1/2}z^{-1/2},$ ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables Referenced by: §19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19 Permalink: http://dlmf.nist.gov/19.21.E8 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 §19.21(ii), §19.21 and Ch.19 19.21.9 $\displaystyle xR_{D}\left(y,z,x\right)+yR_{D}\left(z,x,y\right)+zR_{D}\left(x,% y,z\right)$ $\displaystyle=3R_{F}\left(x,y,z\right).$ ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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 and Ch.19
 19.21.10 $2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)R_{D}% \left(x,y,z\right)+x^{1/2}y^{1/2}z^{-1/2},$ $z\neq 0$. ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10) Permalink: http://dlmf.nist.gov/19.21.E10 Encodings: TeX, pMML, png Correction (effective with 1.1.7): An incorrect factor of 3 was removed in the last term on the right-hand side. Suggested 2022-06-27 by Abdulhafeez Abdulsalam 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 §19.21(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 $(x-z)(y-z)\leq 0$ 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 $6R_{G}\left(x,y,z\right)=3(x+y+z)R_{F}\left(x,y,z\right)-\sum x^{2}R_{D}\left(% y,z,x\right)=\sum x(y+z)R_{D}\left(y,z,x\right),$

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

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

## §19.21(iii) Change of Parameter of $R_{J}$

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

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), §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 $\xi=\eta=\infty$ and $R_{C}\left(\xi,\eta\right)=0$; hence

 19.21.15 $pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right),$ $pq=yz$. ⓘ 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 Permalink: http://dlmf.nist.gov/19.21.E15 Encodings: TeX, pMML, png See also: Annotations for §19.21(iii), §19.21 and Ch.19