\[\operatorname{cn}\left(z,k\right)=1-\frac{z^{2}}{2!}+\left(1+4k^{2}\right)%
\frac{z^{4}}{4!}-\left(1+44k^{2}+16k^{4}\right)\frac{z^{6}}{6!}+O\left(z^{8}%
\right),\]