Asymptotic solutions of inhomogeneous differential equations having a turning point

TL;DR

This paper develops uniform asymptotic solutions for inhomogeneous differential equations with turning points, involving Scorer functions, and provides error bounds for these approximations, applicable to complex arguments.

Contribution

It introduces new uniform asymptotic approximations for inhomogeneous equations with turning points, including error bounds and applications to Airy equations with polynomial and exponential forcing.

Findings

Asymptotic solutions involve Scorer functions and are valid for complex arguments.

Error bounds are established for all approximations, including new bounds for Scorer functions.

Applications to inhomogeneous Airy equations demonstrate the method's effectiveness.

Abstract

Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The asymptotic approximations are uniformly valid for unbounded complex values of the argument, and are applied to inhomogeneous Airy equations having polynomial and exponential forcing terms. Error bounds are available for all approximations, including new simple ones for the well-known asymptotic expansions of Scorer functions of large complex argument.