On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions

Details On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions

2001-09-24

arxiv.org/abs/math/0109187v1

Math. Comput. 70, 1183-1194 (2000)

Mathematics

Numerical Analysis

12 pages, 5 figures

Scientific paper

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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