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Integrating some infinite oscillating tails. (English) Zbl 0574.65013

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

MSC:

65D32 Numerical quadrature and cubature formulas
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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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
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