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Asymptotic solutions of inhomogeneous differential equations having a turning point. (English) Zbl 1457.34126

Summary: 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.

MSC:

34M04 Nonlinear ordinary differential equations and systems in the complex domain
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
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