Digital Library of Mathematical Functions
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19 Elliptic IntegralsSymmetric Integrals19.17 Graphics

Figure 19.17.8 (See in context.)

See accompanying text
Figure 19.17.8: \mathop{R_{J}\/}\nolimits\!\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)\mathop{\ln\/}\nolimits\!\left(16/y\right) as y\to 0+. As p\to 0- it has the limit (-6/y)\mathop{R_{G}\/}\nolimits\!\left(0,y,1\right). When p=1, it reduces to \mathop{R_{D}\/}\nolimits\!\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.
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