On computation of oscillatory integrals with exponential kernel and Jacobi-type singularities

Abstract In this communication two rules are paid close attentions for computing the oscillatory integrals with exponential kernel and Jacobi-type singularities. An asymptotic rule is presented with the aid of integration by parts. An asymptotic-Filon-type quadrature rule is constructed based on two-point Taylor interpolation polynomial. The error analysis on the proposed rules in inverse powers of the large frequency ω is given. Numerical examples are tested in order to illustrate that two rules are very efficient in obtaining very high precision approximations if ω is sufficiently large.