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On nonoscillating integrals for computing inhomogeneous Airy functions

Authors: Amparo Gil, Javier Segura and Nico M. Temme
Journal: Math. Comp. 70 (2001), 1183-1194
MSC (2000): Primary 33C10, 41A60, 30E10, 65D20
Published electronically: April 13, 2000
MathSciNet review: 1826580
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Abstract | References | Similar Articles | Additional Information


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 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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Additional Information

Amparo Gil
Affiliation: Instituto de Bioingeniería, Universidad Miguel Hernández, Edificio La Galia. 03202-Elche (Alicante), Spain

Javier Segura
Affiliation: Instituto de Bioingeniería, Universidad Miguel Hernández, Edificio La Galia. 03202-Elche (Alicante), Spain

Nico M. Temme
Affiliation: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

Keywords: Inhomogeneous Airy functions, Scorer functions, method of steepest descent, saddle point method, numerical computation of special functions
Received by editor(s): September 11, 1998
Received by editor(s) in revised form: April 27, 1999, and August 25, 1999
Published electronically: April 13, 2000
Article copyright: © Copyright 2000 American Mathematical Society