FLINT3.7.0-dev
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acb_elliptic_sigma

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

Sets res to the Weierstrass sigma function σ(z,τ)=z∏ω≠0(1−z/ω)exp⁡(z/ω+z2/(2ω2))\sigma(z, \tau) = z \prod_{\omega \ne 0} (1 - z/\omega) \exp(z/\omega + z^2/(2\omega^2)), which satisfies σ′/σ=ζ\sigma'/\sigma = \zeta, computed as σ(z)=eη1z2θ1(z,τ)/θ1′(0,τ)\sigma(z) = e^{\eta_1 z^2} \theta_1(z, \tau) / \theta_1'(0, \tau).

It is quasi-periodic with σ(z+1)=−e2η1(z+1/2)σ(z)\sigma(z + 1) = -e^{2 \eta_1 (z + 1/2)} \sigma(z) and σ(z+τ)=−e2η3(z+τ/2)σ(z)\sigma(z + \tau) = -e^{2 \eta_3 (z + \tau/2)} \sigma(z).

Cite this page

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