FLINT3.7.0-dev
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acb_hypgeom_bessel_k_0f1

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)source

Sets res to Kν(z)K_\nu(z) (or ezKν(z)e^z K_\nu(z) if scaled is nonzero) computed from

Kν(z)=12[(z2)−νΓ(ν)0F1(1−ν,z24)−(z2)νπνsin⁡(πν)Γ(ν)0F1(ν+1,z24)]K_{\nu}(z) = \frac{1}{2} \left[ \left(\frac{z}{2}\right)^{-\nu} \Gamma(\nu) {}_0F_1\left(1-\nu, \frac{z^2}{4}\right) - \left(\frac{z}{2}\right)^{\nu} \frac{\pi}{\nu \sin(\pi \nu) \Gamma(\nu)} {}_0F_1\left(\nu+1, \frac{z^2}{4}\right) \right]

for ν∉Z\nu \notin \mathbb{Z}, and the limit computed with acb_hypgeom_bessel_k_0f1_series for ν∈Z\nu \in \mathbb{Z}. The output is indeterminate if ν\nu is inexact and contains an integer.

Cite this page

The FLINT team, “acb_hypgeom_bessel_k_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_bessel_k_0f1.html (generated 2026-11-02).