Aquileo | DLMF: Figure 19.17.8 ‣ §19.17 Graphics ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals

Figure 19.17.8 (See in context.) 3D Help

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

. 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

, it reduces to $R_{D}\left(0,y,1\right)$ . If

, then it has the value $\frac{3}{2}\pi/(p+\sqrt{p})$ when

, and $\frac{3}{2}\pi/(p-1)$ when

. See (19.20.10), (19.20.11), and (19.20.8) for the cases $p\to 0\pm$ , $y\to 0+$ , and

, respectively. 3D Help

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Annotations:

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\pi$ : the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$ : principal branch of logarithm function
Permalink:: http://dlmf.nist.gov/19.17.F8.mag
Encodings:: Magnified png, Vizualization, pdf
See also:: Annotations for §19.17 and Ch.19