Numerical and Asymptotic Aspects of Parabolic Cylinder Functions

TL;DR

This work delivers a comprehensive suite of uniform asymptotic expansions for the parabolic cylinder functions $U(a,z)$ and $V(a,z)$ in regimes with large parameters, incorporating both elementary-function and Airy-type representations. Building on Olver's foundational expansions, the authors introduce modified forms with a double asymptotic property, derive precise recursion relations for coefficient functions, and connect the results to integral representations via Laplace-type methods and contour integrals. They also provide detailed numerical verifications, including Wronskian-based error assessments and tabulated accuracy across parameter ranges, validating the practical reliability of the proposed expansions. The findings enable robust and scalable computation of parabolic cylinder functions in real-parameter regimes, with explicit guidance for choosing expansions and for implementing stable numerical algorithms. Overall, the paper advances both the theory and practice of uniform asymptotics for Weber parabolic cylinder functions, relevant to applications in physics and harmonic analysis.

Abstract

Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.