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

Special Function

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

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

Also see

 * Equivalence of Definitions of Elliptic Integral of the Third Kind


 * Definition:Incomplete Elliptic Integral of the First Kind
 * Definition:Complete Elliptic Integral of the First Kind


 * Definition:Incomplete Elliptic Integral of the Second Kind
 * Definition:Complete Elliptic Integral of the Second Kind


 * Definition:Complete Elliptic Integral of the Third Kind