Lame function is the solution of the following equation, $$\frac{d^2w}{dz^2}+\left(A+B\wp(z)\right)w=0,$$ where $A$ and $B$ are constants and $\wp(z)$ is the Weierstrass elliptic function. Wiki says that the most important case corresponds to $B\wp(z)=-\kappa^2\mathrm{sn}^2(\kappa,z)$, where $\mathrm{sn}(\kappa,z)$ is the elliptic sine function. Moreover, the Lame equation with elliptic sine function can be quite simply transformed into the Mathieu equation by setting $\kappa\rightarrow 0^{+}$, which gives $$\kappa^2\mathrm{sn}^2(\kappa,z)=\kappa^2\sin z+\mathcal{O}(\kappa^3).$$
I would like to understand the following:
- Is it possible to obtain the elliptic sine function from the Weierstrass elliptic function?
- Is it possible to obtain the sine/cosine function from the Weierstrass elliptic function?
I have a feeling that it is tightly related to the Inozemtsev limit, which corresponds to setting one of the periods in Weierstrass function to infinity with some additional tricks.