Efficient method for the computation of oscillatory Bessel transform and Bessel Hilbert transform

In this paper, we study the numerical methods for the evaluation of two kinds of highly oscillatory Bessel transforms. Firstly, we rewrite both integrals as the sum of two integrals. By rewriting the Bessel function as a linear combination of Whittaker W function, we then transform one of integrals into the Fourier type, which can be transformed into the integrals on [0,+∞)[0,+∞), and can be computed by some proper Gaussian quadrature, which take into account the asymptotic property of Whittaker W function as x→0x→0. The other can be efficiently computed based on the evaluation of special functions. In addition, error analysis for the presented methods is given. Moreover, we also give an explicit formula for the integral ⨍0+∞Jν(ωx)x−τdx in terms of Meijer GG-function, and then apply the method for the oscillatory Bessel transforms to the computation of highly oscillatory Bessel Hilbert transforms. Theoretical results and numerical examples demonstrate the efficiency and accuracy of the proposed methods.