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Parabolic Cylinder Functions: Examples of Error Bounds for Asymptotic Expansions

Nico M. Temme

CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

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and

Raimundas Vidunas

Department of Mathematics and Computer Science, University of Antwerp, Universiteitsplein 1, B-2610 Wilrijk, Belgium

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https://doi.org/10.1142/S0219530503000168Cited by:1 (Source: Crossref)

Abstract

Several asymptotic expansions of parabolic cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular, Poincaré-type expansions for large values of the argument z and uniform expansions for large values of the parameter are considered. It is shown how expansions can be derived by using the differential equation, and, for a special case, how an integral representation can be used. The expansions are based on those given in Olver (1959) and on modifications of these expansions given in Temme (2000). Computer algebra techniques are used for obtaining representations of the bounds and for numerical computations.

Keywords:

AMSC: 41A60, 33C15, 33F05, 65D20

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