Elliptic Integral of the Second Kind

Let the Modulus satisfy . (This may also be written in terms of the Parameter $m\equiv k^2$ or Modular Angle $\alpha \equiv \sin^{-1} k$ .) The incomplete elliptic integral of the second kind is then defined as

$\begin{displaymath} E(\phi,k) \equiv \int_0^\phi \sqrt{1-k^2\sin^2\theta}\,d\theta. \end{displaymath}$

(1)

A generalization replacing $\sin\theta$ with $\sinh\theta$ gives

$\begin{displaymath} -iE(i\phi,-k)=\int_0^\phi \sqrt{1-k^2\sinh^2\theta}\,d\theta. \end{displaymath}$

(2)

To place the elliptic integral of the second kind in a slightly different form, let

$\displaystyle t$	$\textstyle \equiv$	$\displaystyle \sin\theta$	(3)
$\displaystyle dt$	$\textstyle =$	$\displaystyle \cos \theta\,d\theta =\sqrt{1-t^2}\,d\theta,$	(4)

so the elliptic integral can also be written as

$\displaystyle E(\phi,k)$	$\textstyle =$	$\displaystyle \int_0^{\sin\phi} \sqrt{1-k^2t^2}\,{dt\over\sqrt{1-t^2}}$
	$\textstyle =$	$\displaystyle \int_0^{\sin\phi} \sqrt{1-k^2 t^2\over 1-t^2}\,dt.$	(5)

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

$\begin{figure}\begin{center}\BoxedEPSF{EllipticEReIm.epsf scaled 750}\end{center}\end{figure}$

The complete elliptic integral of the second kind, illustrated above as a function of the Parameter , is defined by

$\displaystyle E(k)$	$\textstyle \equiv$	$\displaystyle E({\textstyle{1\over 2}}\pi,k)$	(6)
	$\textstyle =$	$\displaystyle {\pi\over 2}\left\{{1 - \sum_{n=1}^\infty \left[{(2n-1)!!\over (2n)!!)}\right]^2 {k^{2n}\over 2n-1}}\right\}$	(7)
	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\pi\, {}_2F_1(-{\textstyle{1\over 2}}, {\textstyle{1\over 2}}, 1; k^2)$	(8)
	$\textstyle =$	$\displaystyle \int_0^K \mathop{\rm dn}\nolimits ^2 u\,du,$	(9)

where ${}_2F_1(a,b;c;x)$ is the Hypergeometric Function and $\mathop{\rm dn}\nolimits u$ is a Jacobi Elliptic Function. The complete elliptic integral of the second kind satisfies the Legendre Relation

$\begin{displaymath} E(k)K'(k)+E'(k)K(k)-K(k)K'(k)={\textstyle{1\over 2}}\pi, \end{displaymath}$

(10)

where

and

are complete Elliptic Integrals of the First and second kinds, and

and

are the complementary integrals. The Derivative is

$\begin{displaymath} {dE\over dk} = {E(k)-K(k)\over k} \end{displaymath}$

(11)

(Whittaker and Watson 1990, p. 521). If

is a singular value (i.e.,

$\begin{displaymath} k_r=\lambda^*(r), \end{displaymath}$

(12)

where $\lambda^*$ is the Elliptic Lambda Function), and

and the Elliptic Alpha Function $\alpha(r)$ are also known, then

$\begin{displaymath} E(k)={K(k)\over\sqrt{r}}\left[{{\pi\over 3[K(k)]^2}-\alpha(r)}\right]+K(k). \end{displaymath}$

(13)

References

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

Spanier, J. and Oldham, K. B. ``The Complete Elliptic Integrals and '' and ``The Incomplete Elliptic Integrals $F(p;\phi)$ and $E(p;\phi)$ .'' Chs. 61 and 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609-633, 1987.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.