On quadrature of highly oscillatory integrals with logarithmic singularities

In this paper a quadrature rule is discussed for highly oscillatory integrals with logarithmic singularities. At the same time, its error depends on the frequency ω and the computation of its moments are given. The new rule is implemented by interpolating f at Chebyshev nodes and singular point where the interpolation polynomial satisfies some conditions. Numerical experiments conform the efficiency for obtaining the approximations.