# Definition:Elliptic Integral of the Third Kind/Incomplete/Definition 2

## Special Function

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

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

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