acb_elliptic_pi
Synopsis
#include <flint/acb_elliptic.h>Sets r to the complete elliptic integral of the third kind \Pi(n, m) = \int_0^{\pi/2} (1 - n \sin^2 t)^{-1} (1 - m \sin^2 t)^{-1/2} \, dt. For n = 0 this is K(m) and for m = 0 it is \frac{\pi}{2}(1-n)^{-1/2}; otherwise it is computed as R_F(0, 1-m, 1) + \frac{n}{3} R_J(0, 1-m, 1, 1-n), the same reduction as for the incomplete integral, which is slower than the AGM methods for K and E.
\Pi(n, m) = \int_0^{\pi/2}
\frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} =
\int_0^1
\frac{dt}{(1-nt^2) \sqrt{1-t^2} \sqrt{1-mt^2}}.
Cite this page
The FLINT team, “acb_elliptic_pi (acb_elliptic.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_elliptic/acb_elliptic_pi.html (generated 2026-10-20).
@misc{flint-doc-acb-elliptic-acb-elliptic-pi,
author = {{The FLINT team}},
title = {{acb\_elliptic\_pi (acb\_elliptic.h)}},
howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
year = {2026},
url = {https://flintlib.org/doc/acb_elliptic/acb_elliptic_pi.html},
note = {Generated 2026-10-20}
}