Definition:Elliptic Integral of the Third Kind/Incomplete

Special Function
The integral:


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

is known as 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 $0 \le \phi \le \pi / 2$.

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

Note that:
 * $\Pi \left({k, n, \dfrac \pi 2}\right)$

is the complete elliptic integral of the third kind.

Also known as
Some sources omit the incomplete from the definition, calling this merely the elliptic integral of the third kind.

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:Complete Elliptic Integral of the Third Kind