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Elliptic Integral of the Second Kind

Let the Modulus $k$ satisfy $0 < k^2 < 1$. (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

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
-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{EllipticEReIm.epsf scaled 750}\end{center}\end{figure}

The complete elliptic integral of the second kind, illustrated above as a function of the Parameter $m$, 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
E(k)K'(k)+E'(k)K(k)-K(k)K'(k)={\textstyle{1\over 2}}\pi,
\end{displaymath} (10)

where $E$ and $K$ are complete Elliptic Integrals of the First and second kinds, and $E'$ and $K'$ are the complementary integrals. The Derivative is
{dE\over dk} = {E(k)-K(k)\over k}
\end{displaymath} (11)

(Whittaker and Watson 1990, p. 521). If $k_r$ is a singular value (i.e.,
\end{displaymath} (12)

where $\lambda^*$ is the Elliptic Lambda Function), and $K(k_r)$ and the Elliptic Alpha Function $\alpha(r)$ are also known, then
E(k)={K(k)\over\sqrt{r}}\left[{{\pi\over 3[K(k)]^2}-\alpha(r)}\right]+K(k).
\end{displaymath} (13)

See also Elliptic Integral of the First Kind, Elliptic Integral of the Third Kind, Elliptic Integral Singular Value


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 $K(p)$ and $E(p)$'' 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.

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