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

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

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

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

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

19.21.5 | $$2{R}_{G}(0,z-1,z)=\begin{array}{l}2{R}_{G}(0,1-z,1)\pm \mathrm{i}2{R}_{G}(0,z,1)+(z-1){R}_{F}(0,1-z,1)\\ \phantom{\rule{2em}{0ex}}\mp \mathrm{i}z{R}_{F}(0,z,1).\end{array}$$ | ||

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

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

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

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

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

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

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

$z\ne 0$. | |||

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 | $$6{R}_{G}(x,y,z)=3(x+y+z){R}_{F}(x,y,z)-\sum {x}^{2}{R}_{D}(y,z,x)=\sum x(y+z){R}_{D}(y,z,x),$$ | ||

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

where

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

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

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

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

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

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

$pq=yz$. | |||