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

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Special Function

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

is the 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