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

Special Function

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

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

Also see

 * Equivalence of Definitions of Incomplete Elliptic Integral of the Third Kind