\[\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}%
^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}%
\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.\]