Complete Elliptic Integral of the Second Kind

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 the Wolfram Language 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)

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.