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19 Elliptic IntegralsSymmetric Integrals

§19.17 Graphics

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Keywords:
graphics, symmetric elliptic integrals
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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
See accompanying text
Figure 19.17.2: R_{G}\left(x,y,1\right) for 0\leq x\leq 1, y=0,\,0.1,\,0.5,\,1. y=1 corresponds to \frac{1}{2}(R_{C}\left(x,1\right)+\sqrt{x}). 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.

See accompanying text
Figure 19.17.3: R_{D}\left(x,y,1\right) for 0\leq x\leq 2, y=0,\,0.1,\,1,\,5,\,25. y=1 corresponds to \frac{3}{2}(R_{C}\left(x,1\right)-\sqrt{x})/(1-x), x\neq 1. Magnify
See accompanying text
Figure 19.17.4: R_{J}\left(x,y,1,2\right) for 0\leq x\leq 1, y=0,\,0.1,\,0.5,\,1. y=1 corresponds to 3(R_{C}\left(x,1\right)-R_{C}\left(x,2\right)). Magnify
See accompanying text
Figure 19.17.5: R_{J}\left(x,y,1,0.5\right) for 0\leq x\leq 1, y=0,\,0.1,\,0.5,\,1. y=1 corresponds to 6(R_{C}\left(x,0.5\right)-R_{C}\left(x,1\right)). Magnify
See accompanying text
Figure 19.17.6: Cauchy principal value of R_{J}\left(x,y,1,-0.5\right) for 0\leq x\leq 1, y=0,\,0.1,\,0.5,\,1. y=1 corresponds to 2(R_{C}\left(x,-0.5\right)-R_{C}\left(x,1\right)). Magnify
See accompanying text
Figure 19.17.7: Cauchy principal value of R_{J}\left(0.5,y,1,p\right) for y=0,\,0.01,\,0.05,\,0.2,\,1, -1\leq p<0. y=1 corresponds to 3(R_{C}\left(0.5,p\right)-(\pi/\sqrt{8}))/(1-p). As p\to 0 the curve for y=0 has the finite limit -8.10386\dots; see (19.20.10). Magnify 3D Help
See accompanying text
Figure 19.17.8: R_{J}\left(0,y,1,p\right), 0\leq y\leq 1, -1\leq p\leq 2. Cauchy principal values are shown when p<0. The function is asymptotic to \frac{3}{2}\pi/\sqrt{yp} as p\to 0+, and to (\frac{3}{2}/p)\ln\left(16/y\right) as y\to 0+. As p\to 0- it has the limit (-6/y)R_{G}\left(0,y,1\right). When p=1, it reduces to R_{D}\left(0,y,1\right). If y=1, then it has the value \frac{3}{2}\pi/(p+\sqrt{p}) when p>0, and \frac{3}{2}\pi/(p-1) when p<0. See (19.20.10), (19.20.11), and (19.20.8) for the cases p\to 0\pm, y\to 0+, and y=1, respectively. Magnify 3D Help
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