$$\begin{array}{rcl}\sqrt{\left(1-n\right)\left(1-\frac{k^2}{n}\right)}\prod \left(n,\phi ,k\right)& =& \sqrt{\left(1-N\right)\left(1-\frac{k^2}{N}\right)}\prod \left(N,\phi ,k\right)\\ & +& \frac{k^2}{p_2}F\left(\phi ,k\right)\\ & +& {\tan }^{-1}\left(\frac{p_2}{2}\frac{\sin 2\phi }{\text{?}\left(\phi \right)}\right)\\ N& =& \frac{\left(k^2-n\right)}{\left(1-n\right)}\\ p_2& =& \sqrt{-\frac{n}{1-n}\left(k^2-n\right)}\end{array}$$