# Definition:Elliptic Integral of the Third Kind

## Special Function

### Incomplete Elliptic Integral of the Third Kind

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

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

$k$, defined on the interval $0 < k < 1$
$n \in \Z$
$\phi$, defined on the interval $0 \le \phi \le \pi / 2$.

### Complete Elliptic Integral of the Third Kind

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

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

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