Definition:Elliptic Integral of the Third Kind/Incomplete

Special Function

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

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

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

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

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

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

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