FLINT3.7.0-dev
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acb_elliptic_k

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

Sets k to the complete elliptic integral of the first kind K(m)=∫0π/2(1−msin⁡2t)−1/2 dtK(m) = \int_0^{\pi/2} (1 - m \sin^2 t)^{-1/2} \, dt, computed as K(m)=π/(2M(1−m))K(m) = \pi / (2 M(\sqrt{1-m})) where M(z)=agm⁡(1,z)M(z) = \operatorname{agm}(1, z) is computed by acb_agm1. The branch cut is m∈[1,∞)m \in [1, \infty).

K(m)=∫0π/2dt1−msin⁡2t=∫01dt1−t21−mt2.K(m) = \int_0^{\pi/2} \frac{dt}{\sqrt{1-m \sin^2 t}} = \int_0^1 \frac{dt}{\sqrt{1-t^2}\sqrt{1-mt^2}}.

The AGM converges quadratically, so the cost is O(log⁡p)O(\log p) square roots at precision pp; see Elliptic integrals, elliptic functions and modular forms.

Cite this page

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