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