FLINT3.7.0-dev
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acb_calc_integrate_taylor

Synopsis#include <flint/acb_calc.h>
int acb_calc_integrate_taylor(acb_t res, acb_calc_func_t func, void * param, const acb_t a, const acb_t b, const arf_t inner_radius, const arf_t outer_radius, slong accuracy_goal, slong prec)source

Computes ∫abf(t) dt\int_a^b f(t)\, dt along the segment from aa to bb (both finite) by Taylor polynomials of ff centred at uniformly spaced points, with truncation errors bounded by the Cauchy integral formula. The singularities of ff must lie at distance greater than outer_radius from the path; the steps do not exceed inner_radius, which must be smaller (half of outer_radius gives about one bit per term). The truncation error targets 2−accuracy_goal2^{-\mathit{accuracy\_goal}} in absolute terms; arithmetic is done at prec bits. This function uses derivatives of ff (order greater than 1) and is superseded by acb_calc_integrate.

If f(m+x)=∑nanxnf(m + x) = \sum_n a_n x^n, the integral of the tail satisfies

∣∑n=N∞anxn+1n+1∣≤C(m,R) R ∣x∣(R−∣x∣)(N+1)(∣x∣R)N\left|\sum_{n=N}^{\infty} a_n \frac{x^{n+1}}{n+1}\right| \le \frac{C(m, R)\, R\, |x|}{(R - |x|)(N + 1)} \left(\frac{|x|}{R}\right)^N

for ∣x∣<R|x| < R, with C(m,R)C(m, R) from acb_calc_cauchy_bound.

Cite this page

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