elliptic_kc( z , m )

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

$K(m) = \int_0^{ \frac{\pi}{2} } \frac{ du }{ \sqrt{ 1 - m \sin^2 u } } = F \left( \frac{\pi}{2} \middle| m \right)$

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: