Simplified error bounds for turning point expansions

TL;DR

This paper provides explicit, elementary-function-based error bounds for simplified asymptotic expansions involving Airy functions in linear differential equations with a turning point, improving the classical bounds' simplicity.

Contribution

It introduces explicit error bounds using elementary functions for simplified turning point expansions, making them more accessible and easier to compute.

Findings

Error bounds involve only elementary functions

Simplifies classical bounds of Olver

Enhances computational accuracy of asymptotic expansions

Abstract

Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of F. W. J. Olver.