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:

Series expansion about the origin:

Special values:

Related functions:   elliptic_f   elliptic_e

Function category: elliptic integrals sagemath-docs