Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

On nonoscillating integrals for computing inhomogeneous Airy functions

Author(s): Amparo Gil; Javier Segura; Nico M. Temme.
Journal: Math. Comp. 70 (2001), 1183-1194.
MSC (2000): Primary 33C10, 41A60, 30E10, 65D20
Posted: April 13, 2000
MathSciNet review: 1826580
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Abstract | References | Similar articles | Additional information

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 nonoscillating integrals for complex values of . 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
Email: amparo@titan.ific.uv.es

Javier Segura
Affiliation: Instituto de Bioingeniería, Universidad Miguel Hernández, Edificio La Galia. 03202-Elche (Alicante), Spain
Email: segura@flamenco.ific.uv.es

Nico M. Temme
Affiliation: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
Email: nicot@cwi.nl

DOI: 10.1090/S0025-5718-00-01268-0
PII: S 0025-5718(00)01268-0
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
Posted: April 13, 2000
Copyright of article: Copyright 2000, American Mathematical Society

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