Elliptic Integrals of the Second Kind

Let $0\leqslant \phi\leqslant\frac{\pi}{2}$ and $-1 \leqslant k\leqslant 1$, this calculator evaluates the incomplete elliptic integral of the second kind $E(\phi,k)$: \[ E(\phi,k)=\int_0^\phi \sqrt{1-k^2\sin^2\theta}\; d\theta \] and the complete elliptic integral of the first kind $E(k)$: \[E(k)=E\left(\frac{\pi}{2},k\right)=\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2\theta}\;d\theta\]

Elliptic Integrals of the Second Kind Calculator