I am reading a physics book that uses a Jacobian elliptic function for a special purpose. For that i have to evaluate $\text{sn}(\zeta ; \gamma)$. However, both parameters are purely imaginary and the absolute of $\gamma$ is larger one. While i found an appropriate transformations to obtain $|\gamma|<1$, i do not know how to appropriately handle an imaginary $\gamma$.
Probably, the problem lies within the definition of the elliptic function, which looks different from what you commonly find. The book defines
\begin{equation}\zeta = \int_0^{\xi}\frac{\text{d}\xi}{\sqrt{(1-\xi^2)(1-\gamma^2\xi^2)}}\end{equation}
with
\begin{equation}\xi = \text{sn}(\zeta ; \gamma).\end{equation}
And it is not a typo from my side that integration variable and upper limit are both denoted $\xi$.
Does anybody have an idea what i have to do to compute $\text{sn}(\zeta ; \gamma)$?