FLINT3.7.0-dev
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acb_elliptic_zeta

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

Sets res to the Weierstrass zeta function ζ(z,τ)=z−1+∑ω≠0((z−ω)−1+ω−1+zω−2)\zeta(z, \tau) = z^{-1} + \sum_{\omega \ne 0} ((z - \omega)^{-1} + \omega^{-1} + z \omega^{-2}), which satisfies ζ′=−℘\zeta' = -\wp and is quasi-periodic: ζ(z+1)=ζ(z)+2η1\zeta(z + 1) = \zeta(z) + 2\eta_1, ζ(z+τ)=ζ(z)+2η3\zeta(z + \tau) = \zeta(z) + 2\eta_3 with η1=ζ(1/2)\eta_1 = \zeta(1/2) and η3=ζ(τ/2)\eta_3 = \zeta(\tau/2). It is computed as ζ(z)=2η1z+θ1′(z,τ)/θ1(z,τ)\zeta(z) = 2 \eta_1 z + \theta_1'(z, \tau)/\theta_1(z, \tau), where η1=−θ1′′′(0,τ)/(6θ1′(0,τ))\eta_1 = -\theta_1'''(0, \tau) / (6 \theta_1'(0, \tau)).

Cite this page

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