Uniform enclosures for the phase and zeros of Bessel functions and their derivatives

Abstract

We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.

38 pages (including Appendices), 13 figures, 3 tables, v3: minor corrections and edits, some references added. To appear in SIAM J. Math. Anal. The accompanying Mathematica script and its printout are available for download at https://michaellevitin.net/bessels.html