jacobi_am( z , m )

The Jacobi amplitude of z and parameter m in SageMath. Defined as the inverse of the incomplete elliptic integral of the first kind:

$u = \int_0^\phi \frac{ d \theta }{ \sqrt{ 1 - m \sin^2 \theta } } \qquad \rightarrow \qquad \operatorname{am}( u | m ) = \phi$

Note that all Jacobi elliptic functions in SageMath use the parameter rather than the elliptic modulus k, which is related to the parameter by $$m = k^2$$.

Plot on the real axis: