# Definition:Elliptic Integral of the Third Kind/Complete

## Special Function

### Definition 1

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

is the complete elliptic integral of the third kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$
$n \in \Z$

### Definition 2

$\displaystyle \Pi \left({k, n}\right) = \int \limits_0^1 \frac {\mathrm d v} {\left({1 + n v^2}\right) \sqrt{\left({1 - v^2}\right) \left({1 - k^2 v^2}\right)} }$

is the complete elliptic integral of the third kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$
$n \in \Z$