Lemniscate elliptic functions

Summary

The lemniscate sine $\operatorname{sl}$ and lemniscate cosine $\operatorname{cl}$ are elliptic functions. They play for the lemniscate of Bernoulli the role that sine and cosine play for the circle. The lemniscate sine is the inverse of the integral $\int_0^x dt/\sqrt{1 - t^4}$, which measures arc length on the lemniscate. Their period lattice is the square lattice of Gaussian integers multiplied by $(1 + i)\varpi$, with $\varpi$ the lemniscate constant. So they have complex multiplication by $i$. They were studied by Fagnano, Euler and Gauss, and Abel's work on dividing the lemniscate grew out of them.