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