Efficient evaluation of oscillatory Bessel Hilbert transforms

In this paper, we study the asymptotics and evaluation of the oscillatory Bessel Hilbert transform ⨍0∞f(x)x−τJν(ωx)dx with 0<τ<∞0<τ<∞. The singularity of the Hilbert transform is transferred to an individual oscillatory integral independent of f(x)f(x). 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.