Sharp error bounds for turning point expansions

TL;DR

This paper derives precise, computable error bounds for asymptotic expansions involving Airy functions in differential equations with a turning point, validated through numerical examples including Bessel functions.

Contribution

It introduces sharp, computable error bounds for turning point asymptotic expansions involving Airy functions, enhancing accuracy and reliability.

Findings

Error bounds are shown to be sharp through numerical tests.

Application to Bessel functions demonstrates practical effectiveness.

Abstract

Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the bounds is illustrated numerically with an application to Bessel functions of large order.