ヤコビの楕円関数
ヤコビの楕円関数を微分すると、
d d z s n ( z ) = c n ( z ) d n ( z ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {sn} \,(z)=\mathrm {cn} \,(z)\,\mathrm {dn} \,(z)}
d d z c n ( z ) = − s n ( z ) d n ( z ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {cn} \,(z)=-\mathrm {sn} \,(z)\,\mathrm {dn} \,(z)}
d d z d n ( z ) = − k 2 s n ( z ) c n ( z ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {dn} \,(z)=-k^{2}\mathrm {sn} \,(z)\,\mathrm {cn} \,(z)}
となる。