Absence of Zeros and Asymptotic Error Estimates for Airy and Parabolic Cylinder Functions

TL;DR

This paper develops WKB approximations with error bounds for Airy and parabolic cylinder functions, proves zero-free regions for these functions, and explores their asymptotic limits, advancing understanding of their complex behavior.

Contribution

It introduces new WKB approximation techniques with error estimates and establishes zero-free regions for Airy and parabolic cylinder functions.

Findings

All zeros of the Airy function lie on a specific ray in the complex plane.

Parabolic cylinder functions have no zeros in the complex plane.

The paper analyzes the asymptotic limits of these functions in the Airy and Airy-WKB regimes.

Abstract

We derive WKB approximations for a class of Airy and parabolic cylinder functions in the complex plane, including quantitative error bounds. We prove that all zeros of the Airy function lie on a ray in the complex plane, and that the parabolic cylinder functions have no zeros. We also analyze the Airy and Airy-WKB limit of the parabolic cylinder functions.