$$\begin{array}{rcl}F\left(-\phi ,k\right)& =& -F\left(\phi ,k\right)\\ F\left(\phi +m\pi ,k\right)& =& F\left(\phi ,k\right)+2mK\left(k\right)\\ F\left(\phi ,k\right)& =& \sin \phi R_F\left(\cos {}^2\phi ,1-k^2\sin {}^2\phi ,1\right)\end{array}$$