elliptic_pi( n , z , m )

The incomplete elliptic integral of the third kind of z, characteristic n and parameter m in SageMath. Defined by

$\Pi(n;z|m) = \int_0^z \frac{ du } { ( 1 - n \sin^2 u ) \sqrt{ 1 - m \sin^2 u } }$

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: