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

Special Function

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

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 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