FLINT3.7.0-dev
/

acb_hypgeom_erf_1f1a, acb_hypgeom_erf_1f1b, acb_hypgeom_erf_asymp

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_erf_1f1a(acb_t res, const acb_t z, slong prec)source
void acb_hypgeom_erf_1f1b(acb_t res, const acb_t z, slong prec)source
void acb_hypgeom_erf_asymp(acb_t res, const acb_t z, int complementary, slong prec, slong prec2)source

Sets res to erf⁡(z)\operatorname{erf}(z) computed respectively from

erf⁡(z)=2zπ1F1(12,32,−z2),erf⁡(z)=2ze−z2π1F1(1,32,z2),\operatorname{erf}(z) = \frac{2z}{\sqrt{\pi}} {}_1F_1(\tfrac{1}{2}, \tfrac{3}{2}, -z^2), \qquad \operatorname{erf}(z) = \frac{2z e^{-z^2}}{\sqrt{\pi}} {}_1F_1(1, \tfrac{3}{2}, z^2),
erf⁡(z)=zz2(1−e−z2πU(12,12,z2))=zz2−e−z2zπU∗(12,12,z2).\operatorname{erf}(z) = \frac{z}{\sqrt{z^2}} \left(1 - \frac{e^{-z^2}}{\sqrt{\pi}} U(\tfrac{1}{2}, \tfrac{1}{2}, z^2)\right) = \frac{z}{\sqrt{z^2}} - \frac{e^{-z^2}}{z \sqrt{\pi}} U^{*}(\tfrac{1}{2}, \tfrac{1}{2}, z^2).

The first series has terms of alternating sign when zz is near the real axis, the second has positive terms there and is the one to use for real zz; near the imaginary axis the roles are exchanged. The asymptotic version computes the UU term at the second precision prec2, which can be much lower than prec when the term is small, and computes erfc⁡(z)\operatorname{erfc}(z) instead if complementary is nonzero.

Cite this page

The FLINT team, “acb_hypgeom_erf_1f1a, acb_hypgeom_erf_1f1b, acb_hypgeom_erf_asymp (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_erf_1f1a.html (generated 2026-11-02).