FLINT3.7.0-dev
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acb_elliptic_f

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

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

F(ϕ,m)=∫0ϕdt1−msin⁡2t=∫0sin⁡ϕdt1−t21−mt2.F(\phi,m) = \int_0^{\phi} \frac{dt}{\sqrt{1-m \sin^2 t}} = \int_0^{\sin \phi} \frac{dt}{\sqrt{1-t^2}\sqrt{1-mt^2}}.

The case m=0m = 0 returns ϕ\phi, and the case of an exact half-integer ϕ\phi with times_pi returns 2ϕK(m)2\phi K(m).

Cite this page

The FLINT team, “acb_elliptic_f (acb_elliptic.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_elliptic/acb_elliptic_f.html (generated 2026-11-02).