19 Elliptic Integrals Symmetric Integrals 19.16 Definitions 19.18 Derivatives and Differential Equations

§19.17 Graphics

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Keywords:: graphics, symmetric elliptic integrals
Notes:: These graphics were produced at NIST.
Permalink:: http://dlmf.nist.gov/19.17
See also:: Annotations for Ch.19

See Figures 19.17.1–19.17.8 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 $2R_{G}\left(0,y,1\right)$ for complex $y$ , put $y=1-k^{2}$ , use (19.25.1), and see Figures 19.3.7–19.3.12.

See accompanying text — Figure 19.17.1: $R_{F}\left(x,y,1\right)$ for $0\leq x\leq 1$ , $y=0,\,0.1,\,0.5,\,1$ . $y=1$ corresponds to $R_{C}\left(x,1\right)$ . Magnify

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

19.16 Definitions 19.18 Derivatives and Differential Equations

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