Lemniscate elliptic functions

Remarks

The function $\operatorname{sl}$ is odd, satisfies $\operatorname{sl}(0) = 0$, $\operatorname{sl}(\varpi/2) = 1$, and $(\operatorname{sl}')^2 = 1 - \operatorname{sl}^4$. It is an elliptic function whose periods form the lattice spanned by $(1 + i)\varpi$ and $(1 - i)\varpi$; in particular $2\varpi$ and $2i\varpi$ are periods. It satisfies $\operatorname{sl}(iz) = i\operatorname{sl}(z)$. The two functions are linked by $\operatorname{sl}^2 + \operatorname{cl}^2 + \operatorname{sl}^2\operatorname{cl}^2 = 1$. Example: at $z = \varpi/2$ one has $\operatorname{sl} = 1$ and $\operatorname{cl} = \operatorname{sl}(0) = 0$, and indeed $1 + 0 + 0 = 1$. Non-example: the inverse of $\int_0^x dt/\sqrt{1 - t^2}$ is $\sin$, which is not the lemniscate sine: it has the single period $2\pi$ and satisfies $(\sin')^2 = 1 - \sin^2$ instead of $1 - \operatorname{sl}^4$. In terms of the Jacobi elliptic functions, $\operatorname{sl}(z) = \operatorname{sn}(z; k)$ with $k^2 = -1$, since this $\operatorname{sn}$ solves the same equation $(y')^2 = (1 - y^2)(1 + y^2)$ with $y(0) = 0$, $y'(0) = 1$.