\[\operatorname{sd}\left(2z,k\right)=\frac{2\operatorname{sd}\left(z,k\right)%
\operatorname{cd}\left(z,k\right)\operatorname{nd}\left(z,k\right)}{1+k^{2}{k^%
{\prime}}^{2}{\operatorname{sd}}^{4}\left(z,k\right)},\]