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On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions

Amparo Gil·Javier Segura·Nico M. Temme
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Abstract

Integral representations are considered of solutions of the inhomogeneous Airy differential equation $w''-z w=\pm1/\pi$. 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 non-oscillating 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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Amparo Gil, Javier Segura, Nico M. Temme. 2001-09-24. On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions. https://arxiv.org/abs/math/0109187

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