# Definition:Elliptic Integral of the Second Kind/Complete

## Special Function

### Definition 1

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

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

### Definition 2

$\displaystyle E \left({k}\right) = \int \limits_0^1 \dfrac {\sqrt{1 - k^2 v^2} } {\sqrt{1 - v^2}} \, \mathrm d v$

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