Definition:Complete Elliptic Integral of the Third Kind/Definition 1

Special Function

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

is the complete elliptic integral of the third kind, and is a function of the variables:
 * $k$, defined on the interval $0 < k < 1$
 * $n \in \Z$

Also see

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