Uniform asymptotic expansions for the zeros of parabolic cylinder functions

TL;DR

This paper develops uniform asymptotic expansions for the zeros of parabolic cylinder functions, valid for large parameters and across complex domains, with accuracy verified through numerical comparisons.

Contribution

It introduces new uniform asymptotic formulas for zeros of $U(a,z)$ involving Airy function zeros, applicable for large positive or negative $a$ and unbounded $z$.

Findings

Asymptotic expansions accurately approximate zeros for large parameters.

Formulas are valid uniformly for real and complex zeros.

Numerical tests confirm the high precision of the asymptotic approximations.

Abstract

The real and complex zeros of the parabolic cylinder function $U(a,z)$ are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and these are valid for $a$ positive or negative and large in absolute value, uniformly for unbounded $z$ (real or complex). The accuracy of the approximations of the complex zeros is then demonstrated with some comparative tests using a highly precise numerical algorithm for finding the complex zeros of the function.