FLINT3.7.0-dev
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arb_hypgeom_airy_zero

Synopsis#include <flint/arb_hypgeom.h>
void arb_hypgeom_airy_zero(arb_t a, arb_t a_prime, arb_t b, arb_t b_prime, const fmpz_t n, slong prec)source

Sets a, a_prime, b, b_prime to the nn-th real zero ana_n, an′a'_n, bnb_n, bn′b'_n of Ai⁡\operatorname{Ai}, Ai⁡′\operatorname{Ai}', Bi⁡\operatorname{Bi}, Bi⁡′\operatorname{Bi}' respectively; any of them may be NULL. The zeros are negative and indexed by increasing magnitude starting with n=1n = 1, as in the literature. An index n≤0n \le 0 raises an exception.

A starting enclosure comes from a table for n≤10n \le 10 and otherwise from the asymptotic expansions an∼−T(38π(4n−1))a_n \sim -T(\tfrac38 \pi (4n-1)), bn∼−T(38π(4n−3))b_n \sim -T(\tfrac38\pi(4n-3)), an′∼−U(38π(4n−3))a'_n \sim -U(\tfrac38\pi(4n-3)), bn′∼−U(38π(4n−1))b'_n \sim -U(\tfrac38\pi(4n-1)) [NIST2012, §9.9], whose truncations give proven bounds for ana_n and bnb_n [PS1991]. The enclosure is refined by interval Newton iteration with doubling precision; for the zeros of the derivatives, for which the asymptotic bounds are not proved, an interval Newton step verifies that the enclosure contains the zero.

References#

  • [NIST2012] National Institute of Standards and Technology, Digital Library of Mathematical Functions (2012), https://dlmf.nist.gov/
  • [PS1991] G. Pittaluga and L. Sacripante, "Inequalities for the zeros of the Airy functions", SIAM J. Math. Anal. 22:1 (1991), 260-267.
Cite this page

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