acb_elliptic_f
Synopsis
#include <flint/acb_elliptic.h>Sets res to the incomplete elliptic integral of the first kind F(\phi, m) = \int_0^\phi (1 - m \sin^2 t)^{-1/2} \, dt, extended by F(\phi + d\pi, m) = 2 d K(m) + F(\phi, m). With times_pi nonzero, \phi is replaced by \pi \phi. In particular F(\pi/2, m) = K(m).
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 = 0 returns \phi, and the case of an exact half-integer \phi with times_pi returns 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 2026-10-20).
@misc{flint-doc-acb-elliptic-acb-elliptic-f,
author = {{The FLINT team}},
title = {{acb\_elliptic\_f (acb\_elliptic.h)}},
howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
year = {2026},
url = {https://flintlib.org/doc/acb_elliptic/acb_elliptic_f.html},
note = {Generated 2026-10-20}
}