Elliptic Integral of the Third Kind

Let . The incomplete elliptic integral of the third kind is then defined as

$\displaystyle \Pi(n;\phi,k)$	$\textstyle =$	$\displaystyle \int_0^\phi {d\theta \over (1-n\sin^2\theta)\sqrt{1-k^2 \sin^2\theta}}$	(1)
	$\textstyle =$	$\displaystyle \int_0^{\sin \phi} {dt \over (1-nt^2)\sqrt{(1-t^2)(1-k^2t^2)}},$	(2)

where

is a constant known as the Characteristic.

$\begin{figure}\begin{center}\BoxedEPSF{EllipticPi.epsf}\end{center}\end{figure}$

The complete elliptic integral of the second kind

$\begin{displaymath} \Pi(n\vert m)=\Pi(n; {\textstyle{1\over 2}}\pi \vert m) \end{displaymath}$

(3)

is illustrated above.

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Elliptic Integrals'' and ``Elliptic Integrals of the Third Kind.'' Ch. 17 and §17.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.