- §19.21(i) Complete Integrals
- §19.21(ii) Incomplete Integrals
- §19.21(iii) Change of Parameter of ${R}_{J}$

Legendre’s relation (19.7.1) can be written

19.21.1 | $${R}_{F}\left(0,z+1,z\right){R}_{D}\left(0,z+1,1\right)+{R}_{D}\left(0,z+1,z\right){R}_{F}\left(0,z+1,1\right)=3\pi /\left(2z\right),$$ | ||

$z\in \mathrm{\u2102}\backslash \left(-\mathrm{\infty},0\right]$. | |||

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 | $$3{R}_{F}\left(0,y,z\right)=z{R}_{D}\left(0,y,z\right)+y{R}_{D}\left(0,z,y\right).$$ | ||

19.21.3 | $$6{R}_{G}\left(0,y,z\right)=yz\left({R}_{D}\left(0,y,z\right)+{R}_{D}\left(0,z,y\right)\right)=3z{R}_{F}\left(0,y,z\right)+z\left(y-z\right){R}_{D}\left(0,y,z\right).$$ | ||

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 $$, and lower signs if $$:

19.21.4 | $${R}_{F}\left(0,z-1,z\right)={R}_{F}\left(0,1-z,1\right)\mp \mathrm{i}{R}_{F}\left(0,z,1\right),$$ | ||

19.21.5 | $$2{R}_{G}\left(0,z-1,z\right)=2{R}_{G}\left(0,1-z,1\right)\pm \mathrm{i}2{R}_{G}\left(0,z,1\right)+\left(z-1\right){R}_{F}\left(0,1-z,1\right)\mp \mathrm{i}z{R}_{F}\left(0,z,1\right).$$ | ||

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 | $$\left(\sqrt{rp}/z\right){R}_{J}\left(0,y,z,p\right)=\left(r-1\right){R}_{F}\left(0,y,z\right){R}_{D}\left(p,rz,z\right)+{R}_{D}\left(0,y,z\right){R}_{F}\left(p,rz,z\right),$$ | ||

$r=\left(y-p\right)/\left(y-z\right)>0$. | |||

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

${R}_{D}\left(x,y,z\right)$ 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 | $$\left(x-y\right){R}_{D}\left(y,z,x\right)+\left(z-y\right){R}_{D}\left(x,y,z\right)=3{R}_{F}\left(x,y,z\right)-3\sqrt{y/\left(xz\right)},$$ | ||

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

19.21.8 | ${R}_{D}\left(y,z,x\right)+{R}_{D}\left(z,x,y\right)+{R}_{D}\left(x,y,z\right)$ | $=3{\left(xyz\right)}^{-1/2},$ | ||

19.21.9 | $x{R}_{D}\left(y,z,x\right)+y{R}_{D}\left(z,x,y\right)+z{R}_{D}\left(x,y,z\right)$ | $=3{R}_{F}\left(x,y,z\right).$ | ||

19.21.10 | $$2{R}_{G}\left(x,y,z\right)=z{R}_{F}\left(x,y,z\right)-\frac{1}{3}\left(x-z\right)\left(y-z\right){R}_{D}\left(x,y,z\right)+\sqrt{xy/z},$$ | ||

$z\ne 0$. | |||

Because ${R}_{G}$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $\left(x-z\right)\left(y-z\right)\le 0$ 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 | $$6{R}_{G}\left(x,y,z\right)=3\left(x+y+z\right){R}_{F}\left(x,y,z\right)-\sum {x}^{2}{R}_{D}\left(y,z,x\right)=\sum x\left(y+z\right){R}_{D}\left(y,z,x\right),$$ | ||

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).

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

where

19.21.13 | $\left(p-x\right)\left(q-x\right)$ | $=\left(y-x\right)\left(z-x\right),$ | ||

$\xi $ | $=yz/x,$ | |||

$\eta $ | $=pq/x,$ | |||

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

19.21.14 | $$\eta -\xi =p+q-y-z=\frac{\left(p-y\right)\left(p-z\right)}{p-x}=\frac{\left(q-y\right)\left(q-z\right)}{q-x}=\frac{\left(p-y\right)\left(q-y\right)}{x-y}=\frac{\left(p-z\right)\left(q-z\right)}{x-z}.$$ | ||

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 ${R}_{C}\left(\xi ,\eta \right)=0$; hence

19.21.15 | $$p{R}_{J}\left(0,y,z,p\right)+q{R}_{J}\left(0,y,z,q\right)=3{R}_{F}\left(0,y,z\right),$$ | ||

$pq=yz$. | |||