FLINT3.7.0-dev
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acb_elliptic_pi_inc

Synopsis#include <flint/acb_elliptic.h>
void acb_elliptic_pi_inc(acb_t res, const acb_t n, const acb_t phi, const acb_t m, int times_pi, slong prec)source

Sets res to the incomplete elliptic integral of the third kind Π(n,ϕ,m)=∫0ϕ(1−nsin⁡2t)−1(1−msin⁡2t)−1/2 dt\Pi(n, \phi, m) = \int_0^\phi (1 - n \sin^2 t)^{-1} (1 - m \sin^2 t)^{-1/2} \, dt, extended by Π(n,ϕ+dπ,m)=2dΠ(n,m)+Π(n,ϕ,m)\Pi(n, \phi + d\pi, m) = 2 d \Pi(n, m) + \Pi(n, \phi, m). With times_pi nonzero, ϕ\phi is replaced by πϕ\pi \phi. In particular Π(n,π/2,m)=Π(n,m)\Pi(n, \pi/2, m) = \Pi(n, m) and Π(0,ϕ,m)=F(ϕ,m)\Pi(0, \phi, m) = F(\phi, m).

Π(n,ϕ,m)=∫0ϕdt(1−nsin⁡2t)1−msin⁡2t=∫0sin⁡ϕdt(1−nt2)1−t21−mt2.\Pi(n, \phi, m) = \int_0^{\phi} \frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} = \int_0^{\sin \phi} \frac{dt}{(1-nt^2) \sqrt{1-t^2} \sqrt{1-mt^2}}.

With times_pi and a non-real ϕ\phi, the function multiplies ϕ\phi by π\pi and proceeds as without the flag, because the exact argument reduction can otherwise land exactly on a branch cut of RJR_J.

Cite this page

The FLINT team, “acb_elliptic_pi_inc (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_inc.html (generated 2026-11-02).