Addition formulas for Jacobi amplitude function Are there any known summation formulas for the Jacobi amplitude function? I need a formula like $\mathrm{am}(t+x)=\mathrm{am}(t) + f(x)$. I have plotted some graphs and it seems that $f(x)$ is periodic but I wasn't able to figure out what this function is.
Maybe someone can give me references?
 A: One can use addition theorems for Jacobi elliptic functions. In particular, since
\begin{align}
\sin(\mathrm{am}\;u)&=\mathrm{sn}\,u,\\
\cos(\mathrm{am}\;u)&=\mathrm{cn}\,u,\\
\sqrt{1-k^2\sin^2(\mathrm{am}\;u)}&=\mathrm{dn}\,u,
\end{align}
and, say, 
\begin{align}
\mathrm{sn}(u+v)=\frac{\mathrm{sn}\,u\;\mathrm{cn}\,v\;\mathrm{dn}\,v+
\mathrm{sn}\,v\;\mathrm{cn}\,u\;\mathrm{dn}\,u}{1-k^2\mathrm{sn}^2u\;\mathrm{sn}^2v},
\end{align}
one finds
\begin{align}
&\qquad \mathrm{am}(u+v)=\\
&=\arcsin\left(\frac{\sin(\mathrm{am}\;u)\cos(\mathrm{am}\;v)
\sqrt{1-k^2\sin^2(\mathrm{am}\;v)}+
\sin(\mathrm{am}\;v)\cos(\mathrm{am}\;u)
\sqrt{1-k^2\sin^2(\mathrm{am}\;u)}}{1-k^2\sin^2(\mathrm{am}\;u)\sin^2(\mathrm{am}\;v)}\right).
\end{align}
A: Actually there is the following formula:
$$ \operatorname{am}(u+v)=x + y , $$
where
$$\tan x = \frac{\operatorname{sn} u\,\operatorname{dn} v}{\operatorname{cn} u},$$
$$\tan y = \frac{\operatorname{sn} v\,\operatorname{dn} u}{\operatorname{cn} v}.$$
One has to choose carefully, however, which determinations of the arctan to use.
