The lemniscate functions $\text{sl}$ and $\text{cl}$ are the solutions to the differential equation $$ (y')^2+y^4=1$$ with $\ y(0)=0, \ y'(0)=1$ $\ $ or $\ $ $y(0)=1, \ y'(0)=0.$
Using the integral definitions for the functions $$z = \int\limits_0^{\text{sl} z} \frac{\text{dt}}{\sqrt{1-t^4}}=\int\limits_{\text{cl} z}^{1} \frac{\text{dt}}{\sqrt{1-t^4}},$$ how does one analytically continue the functions $\text{sl}$ and $\text{cl}$ to show they are periodic in $ϖ(1\pm i)$?
$ϖ$ denotes the lemniscate constant.