Derivatives of Jacobi Elliptic Functions – Part 2

Here we derive derivatives of combinations of Jacobi elliptic functions that will be useful in the evaluation of integrals of Jacobi elliptic functions which will be presented in future posts. We make use of results from Derivatives of Jacobi Elliptic Functions and Relationships Between Squares of Jacobi Elliptic Functions.

\begin{equation}
\frac{\partial}{\partial u} \mathrm{cn}^2 = -2\mathrm{cn}\,\mathrm{dn}\,\mathrm{sn}
\label{eq:dpjef-1}
\tag{1}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{dn}^2 = -2k^{2}\mathrm{cn}\,\mathrm{dn}\,\mathrm{sn}
\label{eq:dpjef-2}
\tag{2}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{sn}^2 = 2\mathrm{cn}\,\mathrm{dn}\,\mathrm{sn}
\label{eq:dpjef-3}
\tag{3}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{nc}^2 = \frac{\partial}{\partial u} \frac{1}{\mathrm{cn}^2}
= 2\,\mathrm{nc}\,\mathrm{sc}\,\mathrm{dc}
= 2\frac{\mathrm{dn}\,\mathrm{sn}}{\mathrm{cn}^3}
\label{eq:dpjef-4}
\tag{4}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{nd}^2 = \frac{\partial}{\partial u} \frac{1}{\mathrm{dn}^2}
= 2k^{2}\mathrm{nd}\,\mathrm{sd}\,\mathrm{cd}
= 2k^{2}\frac{\mathrm{cn}\,\mathrm{sn}}{\mathrm{dn}^3}
\label{eq:dpjef-5}
\tag{5}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{ns}^2 = \frac{\partial}{\partial u} \frac{1}{\mathrm{sn}^2}
= -2\,\mathrm{ns}\,\mathrm{cs}\,\mathrm{ds}
= -2\frac{\mathrm{cn}\,\mathrm{dn}}{\mathrm{sn}^3}
\label{eq:dpjef-6}
\tag{6}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{cn}\,\mathrm{dn} = -k^{2}\mathrm{cn}^{2}\mathrm{sn}\,-\mathrm{dn}^{2}\mathrm{sn}
\label{eq:dpjef-7}
\tag{7}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{cn}\,\mathrm{sn} = \mathrm{cn}^{2}\,\mathrm{dn} –
\mathrm{dn}\,\mathrm{sn}^{2}
\label{eq:dpjef-8}
\tag{8}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{dn}\,\mathrm{sn} = 2\mathrm{cn}\,\mathrm{dn}^{2}\,-\mathrm{cn}
\label{eq:dpjef-9}
\tag{9}
\end{equation}

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