acb_elliptic_k_jet
Synopsis
#include <flint/acb_elliptic.h>Sets w to the Taylor coefficients of K(m + x) at x = 0, truncated to length len. The first two coefficients come from (M, M') evaluated by acb_agm1_cpx, the others from the linear recurrence implied by the hypergeometric differential equation of K.
Since K(m) = \frac{\pi}{2} \, {}_2F_1(\tfrac12, \tfrac12; 1; m), the coefficients c_k of K(m + x) satisfy
4(k+1)(k+2) \, m(m-1) \, c_{k+2} = -(2k+1)^2 c_k - 4(k+1)^2 (2m-1) c_{k+1},
At m = 0 it uses c_k = \frac{\pi}{2} \left(\frac{(1/2)_k}{k!}\right)^2 directly.
Cite this page
The FLINT team, “acb_elliptic_k_jet (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_jet.html (generated 2026-10-20).
@misc{flint-doc-acb-elliptic-acb-elliptic-k-jet,
author = {{The FLINT team}},
title = {{acb\_elliptic\_k\_jet (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_k_jet.html},
note = {Generated 2026-10-20}
}