Definition:Elliptic Integral of the Third Kind/Complete
< Definition:Elliptic Integral of the Third Kind(Redirected from Definition:Complete Elliptic Integral of the Third Kind)
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Special Function
Definition 1
- $\ds \map \Pi {k, n} = \int \limits_0^{\pi / 2} \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt {1 - k^2 \sin^2 \phi} }$
is the complete elliptic integral of the third kind, and is a function of the variables:
- $k$, defined on the interval $0 < k < 1$
- $n \in \Z$
Definition 2
- $\ds \map \Pi {k, n} = \int \limits_0^1 \frac {\d v} {\paren {1 + n v^2} \sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$
is the complete elliptic integral of the third kind, and is a function of the variables:
- $k$, defined on the interval $0 < k < 1$
- $n \in \Z$
Also see
- Results about the complete elliptic integral of the third kind can be found here.