Elliptic Integrals of the First Kind

Let $0\leqslant \phi\leqslant\frac{\pi}{2}$ and $-1 \leqslant k\leqslant 1$ with the condition $k^2\sin^2\phi < 1$, this calculator evaluates the incomplete elliptic integral of the first kind $F(\phi,k)$: \[ F(\phi,k)=\int_0^\phi \frac{d\theta}{\sqrt{1-k^2\sin^2\theta}} \] and if $-1 < k < 1$, it also evaluates the complete elliptic integral of the first kind $K(k)$: \[K(k)=F\left(\frac{\pi}{2},k\right)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}\]

Elliptic Integrals of the First Kind Calculator