Lyness, J. N. Integrating some infinite oscillating tails. (English) Zbl 0574.65013 J. Comput. Appl. Math. 12/13, 109-117 (1985). In numerical integration of a slowly decaying function f(x) with alternative sign, the tail of f(x) is often expressed as \(f(x)=g(x)j(x)\), where j(x) is a Bessel function \(J_ 0(x)\) or \(J_ 1(x)\) and g(x) is a ultimately positive function. For making it easier to calculate, this proposal approximates the consecutive zeros of the Bessel function by \(J_ 1(x)\approx \sqrt{2/\pi x}(\cos (x-(3/4)\pi)).\) Reviewer: Y.Kobayashi Cited in 2 ReviewsCited in 7 Documents MSC: 65D32 Numerical quadrature and cubature formulas 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series Keywords:Euler transformation; oscillatory integrals; trigonometric approximation; Bessel function PDFBibTeX XMLCite \textit{J. N. Lyness}, J. Comput. Appl. Math. 12/13, 109--117 (1985; Zbl 0574.65013) Full Text: DOI Digital Library of Mathematical Functions: §3.5(vii) Oscillatory Integrals ‣ §3.5 Quadrature ‣ Areas ‣ Chapter 3 Numerical Methods References: [1] Davis, P. J.; Rabinowitz, P., Methods of Numerical Integration (1975), Academic Press: Academic Press London · Zbl 0154.17802 [2] Longman, I. M., Note on a method for computing infinite integrals of oscillatory functions, Proc. Cambridge Philos. Soc., 52, 764-768 (1956) · Zbl 0072.33803 [3] Longman, I. M., Tables for the rapid and accurate numerical evaluation of certain infinite integrals involving Bessel functions, MTAC, 11, 166-180 (1957) · Zbl 0081.35003 [4] Longman, I. M., A method for the numerical evaluation of finite integrals of oscillatory functions, Math. Comp., 14, 53-59 (1960) · Zbl 0097.12105 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.