FLINT3.7.0-dev
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acb_hypgeom_0f1_asymp, acb_hypgeom_0f1_direct, acb_hypgeom_0f1

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_0f1_asymp(acb_t res, const acb_t a, const acb_t z, int regularized, slong prec)source
void acb_hypgeom_0f1_direct(acb_t res, const acb_t a, const acb_t z, int regularized, slong prec)source
void acb_hypgeom_0f1(acb_t res, const acb_t a, const acb_t z, int regularized, slong prec)source

Sets res to the confluent hypergeometric limit function {}_0F_1(a, z), or {}_0F_1(a,z)/\Gamma(a) if regularized is nonzero, using asymptotic expansions, direct summation, or an automatic choice.

The asymptotic version uses the connection formulas with Bessel functions

\frac{{}_0F_1(a,z)}{\Gamma(a)} = (-z)^{(1-a)/2} J_{a-1}(2 \sqrt{-z}) = z^{(1-a)/2} I_{a-1}(2 \sqrt{z}),

using the J form in the left half-plane and the I form in the right half-plane, so that the square root is never evaluated on its branch cut, and the asymptotic expansions of the Bessel functions (through acb_hypgeom_u_asymp). The automatic version uses the asymptotic expansion when |z| is large enough for the target precision.

Cite this page

The FLINT team, “acb_hypgeom_0f1_asymp, acb_hypgeom_0f1_direct, acb_hypgeom_0f1 (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_0f1_asymp.html (generated 2026-10-20).