19 Elliptic IntegralsSymmetric Integrals19.16 Definitions19.18 Derivatives and Differential Equations

Because the $R$-function is homogeneous, there is no loss of generality in giving one variable the value $1$ or $-1$ (as in Figure 19.3.2). For ${R}_{F}$, ${R}_{G}$, and ${R}_{J}$, which are symmetric in $x,y,z$, we may further assume that $z$ is the largest of $x,y,z$ if the variables are real, then choose $z=1$, and consider only $0\le x\le 1$ and $0\le y\le 1$. The cases $x=0$ or $y=0$ correspond to the complete integrals. The case $y=1$ corresponds to elementary functions.

To view ${R}_{F}\left(0,y,1\right)$ and $2{R}_{G}\left(0,y,1\right)$ for complex $y$, put $y=1-{k}^{2}$, use (19.25.1), and see Figures 19.3.7–19.3.12.