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

Figure 19.17.8 (See in context.)

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}

p\to 0+

, and to

(\frac{3}{2}/p)\ln\left(16/y\right)

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.

ⓘ

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
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