Jacobi elliptic functions

Summary

The Jacobi elliptic functions $\operatorname{sn}$, $\operatorname{cn}$ and $\operatorname{dn}$ are elliptic functions that depend on a modulus $k$ and generalise the sine and cosine. They come from inverting the incomplete elliptic integral of the first kind, through the Jacobi amplitude. They satisfy $\operatorname{sn}^2 + \operatorname{cn}^2 = 1$ and simple differential equations, and they solve the pendulum equation exactly. Nine further functions such as $\operatorname{sd} = \operatorname{sn}/\operatorname{dn}$ are formed as quotients.