# §22.10 Maclaurin Series

## §22.10(i) Maclaurin Series in $z$

Initial terms are given by

 22.10.1 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=z-\left(1+k^{2}\right)\frac{z^% {3}}{3!}+\left(1+14k^{2}+k^{4}\right)\frac{z^{5}}{5!}-\left(1+135k^{2}+135k^{4% }+k^{6}\right)\frac{z^{7}}{7!}+\mathop{O\/}\nolimits\!\left(z^{9}\right),$
 22.10.2 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=1-\frac{z^{2}}{2!}+\left(1+4k^% {2}\right)\frac{z^{4}}{4!}-\left(1+44k^{2}+16k^{4}\right)\frac{z^{6}}{6!}+% \mathop{O\/}\nolimits\!\left(z^{8}\right),$
 22.10.3 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=1-k^{2}\frac{z^{2}}{2!}+k^{2}% \left(4+k^{2}\right)\frac{z^{4}}{4!}-k^{2}\left(16+44k^{2}+k^{4}\right)\frac{z% ^{6}}{6!}+\mathop{O\/}\nolimits\!\left(z^{8}\right).$

Further terms may be derived by substituting in the differential equations (22.13.13), (22.13.14), (22.13.15). The full expansions converge when $|z|<\min\left(\mathop{K\/}\nolimits\!\left(k\right),\mathop{{K^{\prime}}\/}% \nolimits\!\left(k\right)\right)$.

## §22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$

Initial terms are given by

 22.10.4 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=\mathop{\sin\/}\nolimits z-% \frac{k^{2}}{4}(z-\mathop{\sin\/}\nolimits z\mathop{\cos\/}\nolimits z)\mathop% {\cos\/}\nolimits z+\mathop{O\/}\nolimits\!\left(k^{4}\right),$
 22.10.5 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\mathop{\cos\/}\nolimits z+% \frac{k^{2}}{4}(z-\mathop{\sin\/}\nolimits z\mathop{\cos\/}\nolimits z)\mathop% {\sin\/}\nolimits z+\mathop{O\/}\nolimits\!\left(k^{4}\right),$
 22.10.6 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=1-\frac{k^{2}}{2}{\mathop{\sin% \/}\nolimits^{2}}z+\mathop{O\/}\nolimits\!\left(k^{4}\right),$
 22.10.7 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=\mathop{\tanh\/}\nolimits z-% \frac{{k^{\prime}}^{2}}{4}(z-\mathop{\sinh\/}\nolimits z\mathop{\cosh\/}% \nolimits z){\mathop{\mathrm{sech}\/}\nolimits^{2}}z+\mathop{O\/}\nolimits\!% \left({k^{\prime}}^{4}\right),$
 22.10.8 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\mathop{\mathrm{sech}\/}% \nolimits z+\frac{{k^{\prime}}^{2}}{4}(z-\mathop{\sinh\/}\nolimits z\mathop{% \cosh\/}\nolimits z)\mathop{\tanh\/}\nolimits z\mathop{\mathrm{sech}\/}% \nolimits z+\mathop{O\/}\nolimits\!\left({k^{\prime}}^{4}\right),$
 22.10.9 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=\mathop{\mathrm{sech}\/}% \nolimits z+\frac{{k^{\prime}}^{2}}{4}(z+\mathop{\sinh\/}\nolimits z\mathop{% \cosh\/}\nolimits z)\mathop{\tanh\/}\nolimits z\mathop{\mathrm{sech}\/}% \nolimits z+\mathop{O\/}\nolimits\!\left({k^{\prime}}^{4}\right).$

Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). The radius of convergence is the distance to the origin from the nearest pole in the complex $k$-plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime}$-plane in the case of (22.10.7)–(22.10.9); see §22.17.