FLINT3.7.0-dev
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arb_hypgeom_1f1_integration

Synopsis#include <flint/arb_hypgeom.h>
void arb_hypgeom_1f1_integration(arb_t res, const arb_t a, const arb_t b, const arb_t z, int regularized, slong prec)source

Computes 1F1(a,b,z){}_1F_1(a,b,z) (regularized if regularized is nonzero) by rigorous numerical integration of

1F1(a,b,z)=Γ(b)Γ(a)Γ(b−a)∫01eztta−1(1−t)b−a−1 dt.{}_1F_1(a,b,z) = \frac{\Gamma(b)}{\Gamma(a) \Gamma(b-a)} \int_0^1 e^{zt} t^{a-1} (1-t)^{b-a-1} \, dt.

The result is finite only if a≥1a \ge 1 and b−a≥1b - a \ge 1.

The integral is computed with acb_calc_integrate, which bounds the error of Gauss–Legendre quadrature rigorously [Joh2018a]; the bounds for the integrand on the edges of the integration region are computed with intervals of doubles (interval arithmetic with doubles). The method is useful when the parameters are large, where the series expansions need many terms and lose accuracy to cancellation. acb_hypgeom_m falls back to it for real arguments when its other methods give less than half the requested accuracy.

References#

Cite this page

The FLINT team, “arb_hypgeom_1f1_integration (arb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/arb_hypgeom/arb_hypgeom_1f1_integration.html (generated 2026-11-02).