elliptic_eu( z , m )

The incomplete elliptic integral of the second kind of a modified argument z and parameter m in SageMath. Defined by

$E(z|m) = \int_0^{ \sin[ \operatorname{sn}^{-1}z ] } du \sqrt{ 1 - m \sin^2 u }$

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: