文摘
In this paper, we study the asymptotics and evaluation of the oscillatory Bessel Hilbert transform with . The singularity of the Hilbert transform is transferred to an individual oscillatory integral independent of . For this singular integral, we present two methods. One is the combination of a Filon-type method and a complex integration method, the other is the combination of a Filon-type method and an adaptive Clenshaw-Curtis quadrature. The remaining integral which is nonsingular can be well calculated with a combination of a Filon-type method and a Gauss-Laguerre quadrature. The efficiency and accuracy of the proposed methods are illustrated by numerical examples.