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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 ($0\leq m < 1$)

Comments

(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 $k$.

(2)

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.

Links

Data properties

Entries are of type: real number

Table is complete: no