Definition:Elliptic Integral of the Third Kind/Complete

Special Function
The integral:


 * $\displaystyle \int_0^{\pi / 2} \frac {\mathrm d \phi} {\left({1 + n \sin^2 \phi}\right) \sqrt{1 - k^2 \sin^2 \phi} }$

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

It is denoted $\Pi \left({k, n, \pi / 2}\right)$.

Also see

 * 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:Incomplete Elliptic Integral of the Third Kind