elliptic_ec( z , m )

The complete elliptic integral of the second kind of parameter m in SageMath. Defined by

\[ E(m) = \int_0^{ \frac{\pi}{2} } du \sqrt{ 1 - m \sin^2 u } = E \left( \frac{\pi}{2} \middle| m \right) \]

Note that all elliptic integrals 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:

Series expansion about the origin:

Special values:

Related functions:   elliptic_e   elliptic_eu

Function category: elliptic integrals sagemath-docs