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:
Function category: elliptic functions