My understanding
The Jacobi amplitude is defined as the inverse of the incomplete elliptic integral of the first kind, \begin{equation} \mathrm{am}(\theta\,|\,k) = F^{-1}(\theta\,|\,k), \end{equation} where the elliptic integral is the integral function \begin{equation} F(\theta\,|\,k) = \int_0^\theta\frac{\mathrm{d}\phi}{\sqrt{1-k^2\sin(\phi)^2}}. \end{equation} By definition, the derivative of the Jacobi amplitude obeys the identity \begin{equation} \mathrm{am}'(\theta\,|\,k) = \sqrt{1-k^2\sin\big(\mathrm{am}(\theta\,|\,k)\big)^2}. \end{equation} One usually assumes $0\le k\le1$, but I am interested in the case $k>1$. In this case the square root is properly defined only for $\phi\in(-\phi_k,\phi_k)+n\pi$, where $\phi_k = \arcsin(\frac{1}{k})$ and $n\in\mathbb{Z}$, but one can see the integral defining $F(\theta\,|\,k)$ converges at $\theta=\pm\phi_k$. Moreover, \begin{equation} \mathrm{am}'\big(\pm F(\phi_k\,|\,k)\,|\,k\big) = \sqrt{1-k^2\sin(\phi_k)^2} = 0. \end{equation}
The question
It follows that when $k>1$, the Jacobi amplitude ought to be defined only for $\theta\in[-F(\phi_k\,|\,k),\,F(\phi_k\,|\,k)\,]$. However most softwares have it defined for all $\theta\in\mathbb{R}$, either by extending it periodically and continuously, or with these weird jumps (as in the case of Mathematica, see the figure). How can this extension be justified, besides the fact that the Jacobi amplitude approaches the enpoints of its "normal" domain with horizontal tangent?
$\mbox{A graph of the functions for }k=1.1$" />