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 \int_a^b f(t)\, dt along the segment from a to b (both finite) by Taylor polynomials of f centred at uniformly spaced points, with truncation errors bounded by the Cauchy integral formula. The singularities of f 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^{-\mathit{accuracy\_goal}} in absolute terms; arithmetic is done at prec bits. This function uses derivatives of f (order greater than 1) and is superseded by acb_calc_integrate.

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

\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, with 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 2026-10-20).