Proofs of two conjectures on the real zeros of the cylinder and Airy functions

TL;DR

This paper proves two conjectures regarding the asymptotic behavior of large real zeros of cylinder and Airy functions by constructing analytic functions and analyzing their contour integrals, confirming the enveloping property of their divergent expansions.

Contribution

It introduces a novel method of constructing analytic functions to accurately describe the large zeros of these special functions, confirming conjectures about their asymptotic properties.

Findings

Confirmed the enveloping property of divergent asymptotic expansions.

Derived asymptotic expansions with efficient error estimates.

Validated conjectures for the zeros of cylinder and Airy functions.

Abstract

We prove the enveloping property of the known divergent asymptotic expansions of the large real zeros of the cylinder and Airy functions, and thereby answering in the affirmative two conjectures posed by Elbert and Laforgia and by Fabijonas and Olver, respectively. The essence of the proof is the construction of analytic functions that return the zeros when evaluated along certain discrete sets of real numbers. By manipulating contour integrals of these functions, we derive the asymptotic expansions of the large zeros truncated after a finite number of terms plus remainders that can be estimated efficiently. The conjectures are then deduced as corollaries of these estimates. An analogous result for the associated phase function is also discussed.