The evaluation of Bessel functions via exp-arc integrals

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Abstract

A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J.M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an "exp-arc" integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions. © 2007 Elsevier Inc. All rights reserved.

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Borwein, D., Borwein, J. M., & Chan, O. Y. (2008). The evaluation of Bessel functions via exp-arc integrals. Journal of Mathematical Analysis and Applications, 341(1), 478–500. https://doi.org/10.1016/j.jmaa.2007.10.003

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