FLINT3.7.0-dev
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acb_elliptic_pi

Synopsis#include <flint/acb_elliptic.h>
void acb_elliptic_pi(acb_t r, const acb_t n, const acb_t m, slong prec)source

Sets r to the complete elliptic integral of the third kind Π(n,m)=∫0π/2(1−nsin⁡2t)−1(1−msin⁡2t)−1/2 dt\Pi(n, m) = \int_0^{\pi/2} (1 - n \sin^2 t)^{-1} (1 - m \sin^2 t)^{-1/2} \, dt. For n=0n = 0 this is K(m)K(m) and for m=0m = 0 it is π2(1−n)−1/2\frac{\pi}{2}(1-n)^{-1/2}; otherwise it is computed as RF(0,1−m,1)+n3RJ(0,1−m,1,1−n)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 KK and EE.

Π(n,m)=∫0π/2dt(1−nsin⁡2t)1−msin⁡2t=∫01dt(1−nt2)1−t21−mt2.\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 2026-11-02).