The Elliptic Functions in a First-Order System

Abstract

We investigate the first-order system `$s\,' = c^3, \, c\,' = - s^3; \, s(0) = 0, \, c(0) = 1$'. Its solutions have the property that $s \, c$, $s^2$ and $c^2$ extend to simply-poled elliptic functions, which we explicitly identify in terms of the lemniscatic Weierstrass $\wp$ function.