\[F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{\sqrt{1-k^{2}{%
\sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{%
1-k^{2}t^{2}}},\]