# §19.17 Graphics

See Figures 19.17.119.17.8 for symmetric elliptic integrals with real arguments.

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 $\mathop{R_{F}\/}\nolimits$, $\mathop{R_{G}\/}\nolimits$, and $\mathop{R_{J}\/}\nolimits$, 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 $0\leq x\leq 1$ and $0\leq y\leq 1$. The cases $x=0$ or $y=0$ correspond to the complete integrals. The case $y=1$ corresponds to elementary functions.

To view $\mathop{R_{F}\/}\nolimits\!\left(0,y,1\right)$ and $2\!\mathop{R_{G}\/}\nolimits\!\left(0,y,1\right)$ for complex $y$, put $y=1-k^{2}$, use (19.25.1), and see Figures 19.3.719.3.12.

To view $\mathop{R_{F}\/}\nolimits\!\left(0,y,1\right)$ and $2\!\mathop{R_{G}\/}\nolimits\!\left(0,y,1\right)$ for complex $y$, put $y=1-k^{2}$, use (19.25.1), and see Figures 19.3.719.3.12.