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

			(1)
			(2)
			(3)
			(4)

where is an incomplete elliptic integral of the second kind, is the hypergeometric function, and is a Jacobi elliptic function.

It is implemented in Mathematica as EllipticE[m], where is the parameter.

can be computed in closed form in terms of and the elliptic alpha function for special values of , where is a called an elliptic integral singular value. Other special values include

			(5)
			(6)

The complete elliptic integral of the second kind satisfies the Legendre relation

(7)

where and are complete elliptic integrals of the first and second kinds, respectively, and and are the complementary integrals. The derivative is

(8)

(Whittaker and Watson 1990, p. 521).

The solution to the differential equation

(9)

(Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is given by

(10)

If is a singular value (i.e.,

(11)

where is the elliptic lambda function), and and the elliptic alpha function are also known, then

(12)

RELATED WOLFRAM SITES: http://functions.wolfram.com/EllipticIntegrals/EllipticE/

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.

CITE THIS AS:

Weisstein, Eric W. "Complete Elliptic Integral of the Second Kind." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html