# Definition:Elliptic Integral of the Second Kind

## Special Function

### Incomplete Elliptic Integral of the Second Kind

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

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

$\displaystyle E \left({k}\right) = \int \limits_0^{\pi / 2} \sqrt{1 - k^2 \sin^2 \phi} \, \mathrm d \phi$

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