Editing Jacobi elliptic functions (section)

=== Addition theorems ===
The functions satisfy the two square relations (dependence on ''m'' suppressed)
<math display="block">\operatorname{cn}^2(u) + \operatorname{sn}^2(u) = 1,\,</math>

<math display="block">\operatorname{dn}^2(u) + m \operatorname{sn}^2(u) = 1.\,</math>

From this we see that (cn, sn, dn) parametrizes an [[elliptic curve]] which is the intersection of the two [[quadric]]s defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions<ref name="DLMF22" />

<math display="block">
\begin{align}
\operatorname{cn}(x+y) & =
{\operatorname{cn}(x) \operatorname{cn}(y)
- \operatorname{sn}(x) \operatorname{sn}(y) \operatorname{dn}(x) \operatorname{dn}(y)
\over {1 - m \operatorname{sn}^2 (x) \operatorname{sn}^2 (y)}}, \\[8pt]
\operatorname{sn}(x+y) & =
{\operatorname{sn}(x) \operatorname{cn}(y) \operatorname{dn}(y) +
\operatorname{sn}(y) \operatorname{cn}(x) \operatorname{dn}(x)
\over {1 - m \operatorname{sn}^2 (x) \operatorname{sn}^2 (y)}}, \\[8pt]
\operatorname{dn}(x+y) & =
{\operatorname{dn}(x) \operatorname{dn}(y) - m \operatorname{sn}(x) \operatorname{sn}(y) \operatorname{cn}(x) \operatorname{cn}(y)
\over {1 - m \operatorname{sn}^2 (x) \operatorname{sn}^2 (y)}}.
\end{align}
</math>

The Jacobi epsilon and zn functions satisfy a quasi-addition theorem:
<math display="block">\begin{align}\mathcal{E}(x+y,m)&=\mathcal{E}(x,m)+\mathcal{E}(y,m)-m\operatorname{sn}(x,m)\operatorname{sn}(y,m)\operatorname{sn}(x+y,m),\\
\operatorname{zn}(x+y,m)&=\operatorname{zn}(x,m)+\operatorname{zn}(y,m)-m\operatorname{sn}(x,m)\operatorname{sn}(y,m)\operatorname{sn}(x+y,m).\end{align}</math>

Double angle formulae can be easily derived from the above equations by setting ''x''&nbsp;=&nbsp;''y''.<ref name="DLMF22" /> Half angle formulae<ref name="WolframJE" /><ref name="DLMF22" /> are all of the form:

<math display="block">\operatorname{pq}(\tfrac{1}{2}u,m)^2 = f_{\mathrm p}/f_{\mathrm q}</math>

where:
<math display="block">f_{\mathrm c} = \operatorname{cn}(u,m)+\operatorname{dn}(u,m)</math>
<math display="block">f_{\mathrm s} = 1-\operatorname{cn}(u,m)</math>
<math display="block">f_{\mathrm n} = 1+\operatorname{dn}(u,m)</math>
<math display="block">f_{\mathrm d} = (1+\operatorname{dn}(u,m))-m(1-\operatorname{cn}(u,m))</math>