FLINT3.7.0-dev
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acb_hypgeom_bessel_i_asymp, acb_hypgeom_bessel_i_0f1, acb_hypgeom_bessel_i, acb_hypgeom_bessel_i_scaled

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)source
void acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)source
void acb_hypgeom_bessel_i(acb_t res, const acb_t nu, const acb_t z, slong prec)source
void acb_hypgeom_bessel_i_scaled(acb_t res, const acb_t nu, const acb_t z, slong prec)source

Sets res to the modified Bessel function of the first kind Iν(z)=zν(iz)−νJν(iz)I_\nu(z) = z^\nu (iz)^{-\nu} J_\nu(iz), computed respectively with the asymptotic expansion (see acb_hypgeom_bessel_j_asymp), the convergent series

Iν(z)=1Γ(ν+1)(z2)ν0F1(ν+1,z24),I_{\nu}(z) = \frac{1}{\Gamma(\nu+1)} \left(\frac{z}{2}\right)^{\nu} {}_0F_1\left(\nu+1, \frac{z^2}{4}\right),

or an automatic choice. acb_hypgeom_bessel_i_scaled computes e−zIν(z)e^{-z} I_\nu(z); the asymp and 0f1 versions compute the scaled function if scaled is nonzero. For real arguments with an inaccurate result, the automatic versions also try arb_hypgeom_bessel_i_integration.

Cite this page

The FLINT team, “acb_hypgeom_bessel_i_asymp, acb_hypgeom_bessel_i_0f1, acb_hypgeom_bessel_i, acb_hypgeom_bessel_i_scaled (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_i_asymp.html (generated 2026-11-02).