Definition:Elliptic Integral of the Third Kind/Incomplete

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Special Function

Definition 1

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

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


Definition 2

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

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

$k$, defined on the interval $0 < k < 1$
$n \in \Z$
$x = \sin \phi$, where $\phi$ is defined on the interval $0 \le \phi \le \pi / 2$.


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