Relationships Between Squares of Jacobi Elliptic Functions

In this post, we derive a number of relationships between squares of Jacobi elliptic functions. The derivations are trivial, but the results will be required for subsequent posts involving derivatives and integrals of Jacobi elliptic functions. The numbering scheme will also be referenced in future posts.

We drop the arguments for convenience, thus \(\mathrm{cn}(u,k)=\mathrm{cn}\).

The 12 Jacobi Elliptic Functions

Beginning with \(\mathrm{cn}, \mathrm{dn}, \mathrm{sn}\) we define 9 other functions using Glaisher’s notation.
\begin{equation}
\mathrm{nc} = \frac{1}{\mathrm{cn}} \qquad \mathrm{nd} = \frac{1}{\mathrm{dn}} \qquad \mathrm{ns} = \frac{1}{\mathrm{sn}}
\end{equation}

\begin{equation}
\mathrm{cd} = \frac{\mathrm{cn}}{\mathrm{dn}} \qquad \mathrm{cs} = \frac{\mathrm{cn}}{\mathrm{sn}} \qquad \mathrm{ds}=\frac{\mathrm{dn}}{\mathrm{sn}}
\end{equation}

\begin{equation}
\mathrm{dc} = \frac{\mathrm{dn}}{\mathrm{cn}} \qquad \mathrm{sc} = \frac{\mathrm{sn}}{\mathrm{cn}} \qquad \mathrm{sd}=\frac{\mathrm{sn}}{\mathrm{dn}}
\end{equation}

Relationships Between Squares of Jacobi Elliptic Functions

We begin with 3 results from Theta Functions and Jacobi Elliptic Functions.

\begin{equation}
\mathrm{cn}^{2} + \mathrm{sn}^{2} = 1
\label{eq:pythag-1}
\tag{1}
\end{equation}

\begin{equation}
\mathrm{dn}^{2} + k^{2}\mathrm{sn}^{2} = 1
\label{eq:pythag-2}
\tag{2}
\end{equation}

\begin{equation}
k^{2} + {k^{\prime}}^{2} = 1
\label{eq:pythag-3}
\tag{3}
\end{equation}

Divide equation \eqref{eq:pythag-1} by \(\mathrm{cn}^{2}\) to obtain:
\begin{equation}
1 + \mathrm{sc}^{2} = \mathrm{nc}^{2}
\label{eq:pythag-4}
\tag{4}
\end{equation}

Divide equation \eqref{eq:pythag-1} by \(\mathrm{sn}^{2}\) to obtain:
\begin{equation}
\mathrm{cs}^{2} + 1 = \mathrm{ns}^{2}
\label{eq:pythag-5}
\tag{5}
\end{equation}

Divide equation \eqref{eq:pythag-2} by \(\mathrm{dn}^{2}\) to obtain:
\begin{equation}
\mathrm{cd}^{2} + \mathrm{sd}^{2} = \mathrm{nd}^{2}
\label{eq:pythag-6}
\tag{6}
\end{equation}

Divide equation \eqref{eq:pythag-2} by \(\mathrm{dn}^{2}\) to obtain:
\begin{equation}
1 + k^{2}\mathrm{sd}^{2} = \mathrm{nd}^{2}
\label{eq:pythag-7}
\tag{7}
\end{equation}

Divide equation \eqref{eq:pythag-2} by \(\mathrm{cn}^{2}\) to obtain:
\begin{equation}
\mathrm{dc}^{2} + k^{2}\mathrm{sc}^{2} = \mathrm{nc}^{2}
\label{eq:pythag-8}
\tag{8}
\end{equation}

Divide equation \eqref{eq:pythag-2} by \(\mathrm{sn}^{2}\) to obtain:
\begin{equation}
\mathrm{ds}^{2} + k^{2} = \mathrm{ns}^{2}
\label{eq:pythag-9}
\tag{9}
\end{equation}

Eliminate \(\mathrm{nd}^{2}\) from equations \eqref{eq:pythag-6} and \eqref{eq:pythag-7} and use equation \eqref{eq:pythag-3} to obtain:
\begin{equation}
\mathrm{cd}^{2} + {k^{\prime}}^{2}\mathrm{sd}^{2} = 1
\label{eq:pythag-10}
\tag{10}
\end{equation}

Eliminate \(\mathrm{sd}^{2}\) from equations \eqref{eq:pythag-6} and \eqref{eq:pythag-7} and use equation \eqref{eq:pythag-3} to obtain:
\begin{equation}
1 = k^{2}\mathrm{cd}^{2} + {k^{\prime}}^{2}\mathrm{nd}^{2}
\label{eq:pythag-11}
\tag{11}
\end{equation}

Eliminate \(\mathrm{nc}^{2}\) from equations \eqref{eq:pythag-4} and \eqref{eq:pythag-8} and use equation \eqref{eq:pythag-3} to obtain:
\begin{equation}
1 + {k^{\prime}}^{2}\mathrm{sc}^{2} = \mathrm{dc}^{2}
\label{eq:pythag-12}
\tag{12}
\end{equation}

Eliminate \(\mathrm{sc}^{2}\) from equations \eqref{eq:pythag-4} and \eqref{eq:pythag-8} and use equation \eqref{eq:pythag-3} to obtain:
\begin{equation}
\mathrm{dc}^{2} = k^{2} + {k^{\prime}}^{2}\mathrm{nc}^{2}
\label{eq:pythag-13}
\tag{13}
\end{equation}

Eliminate \(\mathrm{ns}^{2}\) from equations \eqref{eq:pythag-5} and \eqref{eq:pythag-9} and use equation \eqref{eq:pythag-3} to obtain:
\begin{equation}
\mathrm{cs}^{2} + {k^{\prime}}^{2} = \mathrm{ds}^{2}
\label{eq:pythag-14}
\tag{14}
\end{equation}

Multiply equations \eqref{eq:pythag-11} by \(\mathrm{ns}^{2}\) to obtain:
\begin{equation}
\mathrm{ds}^{2} = k^{2}\mathrm{cs}^{2} + {k^{\prime}}^{2}\mathrm{ns}^{2}
\label{eq:pythag-15}
\tag{15}
\end{equation}

Eliminate \(\mathrm{sn}^{2}\) from equations \eqref{eq:pythag-1} and \eqref{eq:pythag-2} and use equation \eqref{eq:pythag-3} to obtain:
\begin{equation}
\mathrm{dn}^{2} = k^{2}\mathrm{cn}^{2} + {k^{\prime}}^{2}
\label{eq:pythag-16}
\tag{16}
\end{equation}

Eliminate 1 from equations \eqref{eq:pythag-1} and \eqref{eq:pythag-2} and use equation \eqref{eq:pythag-3} to obtain:
\begin{equation}
\mathrm{dn}^{2} = \mathrm{cn}^{2} + {k^{\prime}}^{2}\mathrm{sn}^{2}
\label{eq:pythag-17}
\tag{17}
\end{equation}

References

1. Handbook of Elliptic Integrals for Engineers and Scientists – Byrd and Friedman

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