Abstract
In this paper, we study efficient numerical methods for the computation of highly oscillatory Bessel transform ∫a+∞f(x)Jν(ωg(x))dx, g(x)≠0, g′(x)≠0 for all x∈[a,+∞). We first derive an asymptotic expansion formula for this integral by using integration by parts. Then we present an improved numerical steepest descent method, by applying complex integration theory to the remainder term of asymptotic expansion, which is also an oscillatory integral with Bessel kernel. In addition, the asymptotic orders about ω of the presented methods are given. The efficiency and accuracy of the proposed methods are investigated through theoretical results and numerical experiments.
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