Elliptic Integrals of the Third Kind
Let $0\leqslant \phi\leqslant\frac{\pi}{2}$, $-1 \leqslant k\leqslant 1$, and n any real number with the conditions $k^2\sin^2\phi < 1$ and $n\sin^2\phi < 1$, this calculator evaluates the incomplete elliptic integral of the third kind $\Pi(n,\phi,k)$: \[ \Pi(n,\phi,k)=\int_0^\phi \frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}} \] and if $-1 < k < 1$ and $n < 1$, it also evaluates the complete elliptic integral of the third kind $\Pi(n,k)$: \[\Pi(n,k)=\Pi\left(n,\frac{\pi}{2},k\right)=\int_0^\frac{\pi}{2} \frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}\]