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Acta Applicandae Mathematicae

, Volume 150, Issue 1, pp 141–177 | Cite as

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions

  • Gergő NemesEmail author
Article

Abstract

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.

Keywords

Asymptotic expansions Error bounds Remainder terms Bessel functions Hankel functions 

Mathematics Subject Classification

41A60 30E15 33C10 

Notes

Acknowledgements

The author’s research was supported by a research grant (GRANT11863412/ 70NANB15H221) from the National Institute of Standards and Technology. The author greatly appreciates the help of Dorottya Sziráki in improving the presentation of the paper. The author thanks the anonymous referees for their helpful comments and suggestions on the manuscript.

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of MathematicsThe University of EdinburghEdinburghUK

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