On nonoscillating integrals for computing inhomogeneous Airy functions

Gil, Amparo; Segura, Javier; Temme, Nico

doi:10.1090/S0025-5718-00-01268-0

On nonoscillating integrals for computing inhomogeneous Airy functions
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by Amparo Gil, Javier Segura and Nico M. Temme PDF

Math. Comp. 70 (2001), 1183-1194 Request permission

Abstract:

Integral representations are considered of solutions of the inhomogeneous Airy differential equation $w”-z w=\pm 1/\pi$. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain nonoscillating integrals for complex values of $z$. In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.

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