High-precision evaluation of the Bessel functions via Hadamard series

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:224

Article

Paris, R. B.

关键词: Hadamard series;    Hyperasymptotics;    Bessel functions;    Confluent hypergeometric functions;   

DOI  :  10.1016/j.cam.2008.04.025

来源: Elsevier

【 摘 要 】

We present a method of high-precision calculation of the Bessel functions using Hadamard series. Such series are absolutely convergent expansions involving the normalised incomplete gamma function P(a, z) = gamma (a, z)/Gamma(a) and possess early terms that behave like those in an asymptotic expansion. In the case of real variables the function P(a. z) acts as a smoothing factor oil the terms of the series. We show how these series representing the Bessel functions of complex argument can be chosen so as to produce rapidly convergent series that possess terms decaying at the geometric rate nu(k), where 0 < nu < 1 and k is the ordinal number of the series. We give numerical examples with nu = 1/2, 1/3 and 1/4. The theory is extended to cover the confluent hypergeometric functions (1)F(1)(a: b: z) and U (a, b, z), thereby dealing with many of the special functions arising in mathematical physics. (C) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

Free