Definition:Elliptic Integral of the Second Kind

Special Function

Incomplete Elliptic Integral of the Second Kind

$\ds \map E {k, \phi} = \int \limits_0^\phi \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$

is the incomplete elliptic integral of the second kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$

$\phi$, defined on the interval $0 \le \phi \le \pi / 2$.

Complete Elliptic Integral of the Second Kind

$\ds \map E k = \int \limits_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$

is the complete elliptic integral of the second kind, and is a function of $k$, defined on the interval $0 < k < 1$.

Also see

• Definition:Elliptic Integral of the First Kind
• Definition:Elliptic Integral of the Third Kind

• Definition:Elliptic Integral

• Results about elliptic integrals of the second kind can be found here.