am( z, m )

The Jacobi amplitude function 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 Math use the parameter rather than the elliptic modulus k, which is related to the parameter by $$m = k^2$$.

Real part on the real axis:

Imaginary part on the real axis is zero.

Real part on the imaginary axis is zero.

Imaginary part on the imaginary axis:

Real part on the complex plane:

Imaginary part on the complex plane:

Absolute value on the complex plane:

Related functions:   sn   cn   dn

Function category: elliptic functions