Numerical evaluation and error analysis of many different oscillatory Bessel transforms via confluent hypergeometric function

Abstract

In this paper we study both the fast computation and the related error analysis of many different finite and infinite oscillatory Bessel transforms. By transforming the Bessel function into Confluent hypergeometric function and using its asymptotic expansion, we develop an efficient approach by the contours in the complex plane, corresponding to the paths of steepest descent. Furthermore, we conduct their error analyses in inverse powers of the frequency ω. The related error analyses show that the precision can be improved by either adding more Gauss-Laguerre nodes or increasing the frequency ω. In particular, the method exhibits the high error order. Meanwhile, we apply the proposed method to some other kinds of oscillatory integrals through some simple changes of variables. Numerical experiments can verify our theoretical analysis. It is also shown that the new algorithms are more efficient than the existing Filon-type method and can achieve the same level of accuracy with the method (the XMX method) proposed in Xu, Milovanović and Xiang [46] by some numerical experiments under the given conditions. A combination of the n1-point Gauss-Laguerre quadrature rule and the n2-point generalized Gauss-Laguerre quadrature rule are used in the XMX method. However, we only use the Gauss-Laguerre quadrature rule for one time in the established method, which makes the theoretical and error analysis simpler than the XMX method.