Category:Elliptic Integrals of the Third Kind

This category contains results about Elliptic Integrals of the Third Kind.
Definitions specific to this category can be found in Definitions/Elliptic Integrals of the Third Kind.

Incomplete Elliptic Integral of the Third Kind

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

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

Complete Elliptic Integral of the Third Kind

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

is the complete elliptic integral of the third kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$

$n \in \Z$