FLINT3.7.0-dev
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acb_elliptic_e_inc

Synopsis#include <flint/acb_elliptic.h>
void acb_elliptic_e_inc(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 second kind E(ϕ,m)=∫0ϕ(1−msin⁡2t)1/2 dtE(\phi, m) = \int_0^\phi (1 - m \sin^2 t)^{1/2} \, dt, extended by E(ϕ+dπ,m)=2dE(m)+E(ϕ,m)E(\phi + d\pi, m) = 2 d E(m) + E(\phi, m). With times_pi nonzero, ϕ\phi is replaced by πϕ\pi \phi. In particular E(π/2,m)=E(m)E(\pi/2, m) = E(m).

E(ϕ,m)=∫0ϕ1−msin⁡2t dt=∫0sin⁡ϕ1−mt21−t2 dt.E(\phi,m) = \int_0^{\phi} \sqrt{1-m \sin^2 t} \, dt = \int_0^{\sin \phi} \frac{\sqrt{1-mt^2}}{\sqrt{1-t^2}} \, dt.
Cite this page

The FLINT team, “acb_elliptic_e_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_e_inc.html (generated 2026-11-02).