§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 $R_{F}$, $R_{G}$, 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 $0\leq x\leq 1$ 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 $R_{F}\left(0,y,1\right)$ and $2\!R_{G}\left(0,y,1\right)$ for complex $y$, put $y=1-k^{2}$, use (19.25.1), and see Figures 19.3.719.3.12. Figure 19.17.1: RF⁡(x,y,1) for 0≤x≤1, y=0, 0.1, 0.5, 1. y=1 corresponds to RC⁡(x,1). Magnify

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 19.3.719.3.12. Figure 19.17.3: RD⁡(x,y,1) for 0≤x≤2, y=0, 0.1, 1, 5, 25. y=1 corresponds to 32⁢(RC⁡(x,1)-x)/(1-x), x≠1. Magnify Figure 19.17.5: RJ⁡(x,y,1,0.5) for 0≤x≤1, y=0, 0.1, 0.5, 1. y=1 corresponds to 6⁢(RC⁡(x,0.5)-RC⁡(x,1)). Magnify Figure 19.17.7: Cauchy principal value of RJ⁡(0.5,y,1,p) for y=0, 0.01, 0.05, 0.2, 1, -1≤p<0. y=1 corresponds to 3⁢(RC⁡(0.5,p)-(π/8))/(1-p). As p→0 the curve for y=0 has the finite limit -8.10386⁢…; see (19.20.10). Magnify 3D Help