Jacobi amplitude

Summary

The Jacobi amplitude $\operatorname{am}(u, k)$ is the inverse function of the incomplete elliptic integral of the first kind $u = F(\varphi, k)$. It is the angle $\varphi$ whose sine and cosine are the Jacobi elliptic functions $\operatorname{sn} u$ and $\operatorname{cn} u$. For modulus $k = 0$ it is the identity, and for $k = 1$ it is the Gudermannian function.