FLINT3.7.0-dev
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acb_theta_sum_jet_radius

Synopsis#include <flint/acb_theta.h>
void acb_theta_sum_jet_radius(arf_t R2, arf_t eps, const arb_mat_t cho, arb_srcptr v, slong ord, slong prec)source

Sets R2 and eps as acb_theta_sum_radius does, but for the derivatives up to order ord, where v=−CY−1yv = -C Y^{-1} y.

The differentiated series is

∂∣k∣θa,b∂z0k0⋯∂zg−1kg−1(z,τ)=(2πi)∣k∣eπyTY−1y∑n∈Zg+a2n0k0⋯ng−1kg−1ξne−π(n+Y−1y)TY(n+Y−1y)\frac{\partial^{|k|}\theta_{a,b}}{\partial z_0^{k_0}\cdots \partial z_{g-1}^{k_{g-1}}}(z,\tau) = (2\pi i)^{|k|} e^{\pi y^T Y^{-1} y} \sum_{n\in \mathbb{Z}^g + \tfrac a2} n_0^{k_0} \cdots n_{g-1}^{k_{g-1}} \xi_n e^{-\pi (n + Y^{-1}y)^T Y (n + Y^{-1}y)}

with ∣ξn∣=1|\xi_n| = 1, and with m=Cn+vm = Cn + v, ∣n0k0⋯ng−1kg−1∣≤(∥C−1∥∞∥m∥2+∥Y−1y∥∞)∣k∣|n_0^{k_0}\cdots n_{g-1}^{k_{g-1}}| \le (\lVert C^{-1}\rVert_\infty \lVert m\rVert_2 + \lVert Y^{-1}y\rVert_\infty)^{|k|}. The function first calls acb_theta_sum_radius with ord = 0. If R≤∥Y−1y∥∞/∥C−1∥∞R \le \lVert Y^{-1}y\rVert_\infty/\lVert C^{-1}\rVert_\infty, it multiplies eps by max⁡(1,2∥Y−1y∥∞)ord\max(1, 2 \lVert Y^{-1}y \rVert_\infty)^{\mathit{ord}}. Otherwise it calls acb_theta_sum_radius with the given ord, takes the maximum of R2 and (∥Y−1y∥∞/∥C−1∥∞)2(\lVert Y^{-1}y \rVert_\infty /\lVert C^{-1} \rVert_\infty)^2, and multiplies eps by max⁡(1,2∥C−1∥∞)ord\max(1, 2\lVert C^{-1}\rVert_\infty)^{\mathit{ord}}.

Cite this page

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