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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 $\text{sn}(\zeta ; \gamma)$?

user25485
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  • Using two times $\xi$ is horrible, take a look at https://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions#Integral_equation which is the same after the change of variable $s=z^2$ in $\int_0^{\xi}\frac{\text{d}z}{\sqrt{(1-z^2)(1-\gamma^2z^2)}}$. From this definition of $\wp,\zeta$ we obtain the differential equation directly, and the problem is to show the double periodicity and the equivalence with the series of poles definition. – reuns Mar 19 '19 at 11:00
  • I recommend DLMF Chapter 22: Jacobian Elliptic Functions for the information you need. – Somos Mar 19 '19 at 11:21
  • Ok, thanks for the hints. I have to give that a try. I will get back, if I run into further trouble. – user25485 Mar 19 '19 at 11:53
  • Personally, I would leave the integral's upper limit blank, and write $\int_0 f'(\xi),d\xi=f(\xi)-f(0)$. – mr_e_man May 07 '19 at 21:10

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