Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations

TL;DR

This paper derives uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations, including inhomogeneous cases, using special functions with explicit error bounds for large parameters.

Contribution

It provides new uniform asymptotic expansions involving exponential, Airy, and Scorer functions for these differential equations, including inhomogeneous cases.

Findings

Expansions are valid for large parameters and unbounded arguments.

Explicit error bounds are provided for all approximations.

The methods facilitate accurate computation of solutions in complex domains.

Abstract

Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions involve exponential, Airy and Scorer functions and slowly varying analytic coefficient functions involving simple coefficients. The approximations are uniformly valid for large values of the parameter and unbounded real and complex values of the argument. Explicit and readily computable error bounds are either furnished or available for all approximations.