# Definition:Elliptic Integral of the Third Kind/Incomplete

## Special Function

### Definition 1

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

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 $0 \le \phi \le \pi / 2$.

### Definition 2

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

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 $0 \le \phi \le \pi / 2$.

Note that:

$\Pi \left({k, n, \dfrac \pi 2}\right)$

## Also known as

Some sources omit the incomplete from the definition, calling this merely the elliptic integral of the third kind.