# Definition:Elliptic Integral of the Third Kind/Incomplete

## Special Function

### Definition 1

$\ds \map \Pi {k, n, \phi} = \int \limits_0^\phi \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt{1 - k^2 \sin^2 \phi} }$

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

$\ds \map \Pi {k, n, \phi} = \int \limits_0^x \frac {\d v} {\paren {1 + n v^2} \sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$

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

### Completion

$\map \Pi {k, n, \dfrac \pi 2} = \map \Pi {k, n}$

where $\map \Pi {k, n}$ denotes the complete elliptic integral of the third kind.

## Also known as

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

## Also see

• Results about the incomplete elliptic integral of the third kind can be found here.