Elliptic Integral of the Second Kind

Let the elliptic modulus satisfy . (This may also be written in terms of the parameter or modular angle .) The incomplete elliptic integral of the second kind is then defined as

(1)

The elliptic integral of the second kind is implemented in the Wolfram Language as EllipticE[phi, m] (note the use of the parameter instead of the modulus ).

The complete elliptic integral of the second kind is defined by

(2)

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

			(3)
			(4)

so the elliptic integral can also be written as

			(5)
			(6)

A generalization replacing with in (1) gives

(7)

The incomplete elliptic integral of the second kind of the form can be written in terms of complete elliptic integrals of the first and second kinds as

(8)

for .