Computation of integrals with oscillatory and singular integrands using Chebyshev expansions

We present a general method for computing oscillatory integrals of the form ∫−11f(x)G(x)eiωxdx, where ff is sufficiently smooth on [−1,1][−1,1], ωω is a positive parameter and GG is a product of singular factors of algebraic or logarithmic type. Based on a Chebyshev expansion of ff and the properties of Chebyshev polynomials, the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor eiωxeiωx. Furthermore, due to numerically stable recurrence relations for the modified moments, the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration. Numerical examples are provided to confirm our analysis.