Complete Elliptic Integral of the Second Kind
The complete elliptic integral of the second kind, illustrated above as a function of k, is defined by
![画像:pi/2{1-sum_(n=1)^(infty)[((2n-1)!!)/((2n)!!)]^2(k^(2n))/(2n-1)}](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/CompleteEllipticIntegraloftheSecondKind/Inline7.svg){kind=link}
where E(phi,k) is an incomplete elliptic integral of the second kind, _2F_1(a,b;c;x) is the hypergeometric function, and dn(u,k) is a Jacobi elliptic function.
It is implemented in the Wolfram Language as EllipticE [m], where m=k^2 is the parameter.
E(k) can be computed in closed form in terms of K(k_n) and the elliptic alpha function alpha(n) for special values of k=k_n, where k_n is a called an elliptic integral singular value. Other special values include
The complete elliptic integral of the second kind satisfies the Legendre relation
| E(k)K^'(k)+E^'(k)K(k)-K(k)K^'(k)=1/2pi, |
(7)
|
where K(k) and E(k) are complete elliptic integrals of the first and second kinds, respectively, and K^'(k) and E^'(k) are the complementary integrals. The derivative is
| [画像: (dE)/(dk)=(E(k)-K(k))/k ] |
(8)
|
(Whittaker and Watson 1990, p. 521).
The solution to the differential equation
(Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is given by
| y=C_1E(k)+C_2[E^'(k)-K^'(k)]. |
(10)
|
If k_r is a singular value (i.e.,
| k_r=lambda^*(r), |
(11)
|
where lambda^* is the elliptic lambda function), and K(k_r) and the elliptic alpha function alpha(r) are also known, then
![画像: E(k)=(K(k))/(sqrt(r))[pi/(3[K(k)]^2)-alpha(r)]+K(k).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/CompleteEllipticIntegraloftheSecondKind/NumberedEquation6.svg){kind=link}
See also
Complete Elliptic Integral of the First Kind, Complete Elliptic Integral of the Third Kind, Elliptic Alpha Function, Elliptic Integral of the Second Kind, Elliptic Integral Singular ValueRelated Wolfram sites
https://functions.wolfram.com/EllipticIntegrals/EllipticE/Explore with Wolfram|Alpha
More things to try:
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997.Referenced on Wolfram|Alpha
Complete Elliptic Integral of the Second KindCite this as:
Weisstein, Eric W. "Complete Elliptic Integral of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html