FLINT3.7.0-dev
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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 \operatorname{erf}(z) computed respectively from

\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),
\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 z is near the real axis, the second has positive terms there and is the one to use for real z; near the imaginary axis the roles are exchanged. The asymptotic version computes the U term at the second precision prec2, which can be much lower than prec when the term is small, and computes \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 2026-10-20).