An Efficient Quadrature Rule for the Oscillatory Infinite Generalized Bessel Transform with a General Oscillator and Its Error Analysis

Our recent work (Kang and Wang in J Sci Comput 82:1–33, 2020) performed a complete asymptotic analysis and proposed a modified Filon-type method for a class of oscillatory infinite Bessel transform with a general oscillator. In this paper, we present and analyze a different method by converting the integration path to the complex plane for this class of oscillatory infinite Bessel transform. In particular, we establish a series of new quadrature rules for this transform and carry out rigorous analysis, including the cases that the oscillator g(x) has either zeros or stationary points. The error analysis shows the advantages that this approach exhibits high asymptotic order, and the accuracy improves significantly as either the frequency ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} or the number of nodes n increases. Furthermore, the constructed method shows higher accuracy and error order by comparing with the existing modified Filon-type method in our recent work (Kang and Wang 2020) at the same computational cost. Some numerical experiments are provided to verify the theoretical results and demonstrate the efficiency of the proposed method.