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Complete elliptic integral of the third kind $\Pi(n,m)$
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Numbers
$n$
$m$
$\Pi(n,m)$
INPUT{numbers.yaml} (not shown in preview)
Definition
The list contains the evaluations of the complete elliptic integral of the third kind $\Pi(n,m) = \int_0^{\pi/2} \frac{1}{(1-n\sin^2 \theta)\sqrt{1-m\sin^2 \theta}} d\theta$ for some parameters $n$ and $m=k^2$, where $k$ is the ellpitic modulus.
Parameters
$n$
—   real number ($0\leq n < 1$)
$m$
—   real number ($0\leq m < 1$)
$\Pi(n,m)$ is also denoted as $\Pi(n,k)$ despite the relation $m=k^2$ between the parameter $m$ and the elliptic modulus $k$.
The parameter $n$ can be larger than $1$, although in that case one has to specify how to integrate through the singulatiry of the integrand, e.g. by taking the Cauchy principal value.