FLINT3.7.0-dev
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acb_elliptic_k_jet

Synopsis#include <flint/acb_elliptic.h>
void acb_elliptic_k_jet(acb_ptr w, const acb_t m, slong len, slong prec)source

Sets w to the Taylor coefficients of K(m+x)K(m + x) at x=0x = 0, truncated to length len. The first two coefficients come from (M,M′)(M, M') evaluated by acb_agm1_cpx, the others from the linear recurrence implied by the hypergeometric differential equation of KK.

Since K(m)=π2 2F1(12,12;1;m)K(m) = \frac{\pi}{2} \, {}_2F_1(\tfrac12, \tfrac12; 1; m), the coefficients ckc_k of K(m+x)K(m + x) satisfy

4(k+1)(k+2) m(m−1) ck+2=−(2k+1)2ck−4(k+1)2(2m−1)ck+1,4(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=0m = 0 it uses ck=π2((1/2)kk!)2c_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 2026-11-02).