\[\operatorname{sn}\left(z,k\right)=z-\left(1+k^{2}\right)\frac{z^{3}}{3!}+\left%
(1+14k^{2}+k^{4}\right)\frac{z^{5}}{5!}-\left(1+135k^{2}+135k^{4}+k^{6}\right)%
\frac{z^{7}}{7!}+O\left(z^{9}\right),\]