\[\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sd}\left(z,k\right)=(k%
^{2}-{k^{\prime}}^{2})\operatorname{sd}\left(z,k\right)-2k^{2}{k^{\prime}}^{2}%
{\operatorname{sd}}^{3}\left(z,k\right),\]