FLINT3.7.0-dev
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acb_elliptic_e

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

Sets res to the complete elliptic integral of the second kind E(m)=∫0π/2(1−msin⁡2t)1/2 dtE(m) = \int_0^{\pi/2} (1 - m \sin^2 t)^{1/2} \, dt, computed from K(m)K(m) and K′(m)K'(m) by E(m)=(1−m)(2mK′(m)+K(m))E(m) = (1-m)(2m K'(m) + K(m)), where K′K' is the derivative (not the complementary integral).

E(m)=∫0π/21−msin⁡2t dt=∫011−mt21−t2 dt.E(m) = \int_0^{\pi/2} \sqrt{1-m \sin^2 t} \, dt = \int_0^1 \frac{\sqrt{1-mt^2}}{\sqrt{1-t^2}} \, dt.

The special values E(0)=π/2E(0) = \pi/2 and E(1)=1E(1) = 1 are handled directly.

Cite this page

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