Computation of integrals with oscillatory and singular integrands using Chebyshev expansions

This paper is concerned with evaluation of integrals whose integrands are oscillatory and contain singularities at the endpoints of the interval of integration. A typical form is G(9) ■ f * w(x)e*xf(x) dx, where a and b can be finite or infinite, 9 is a parameter which is usually large, fix) is analytic in the range of integration, and the singularities are encompassed in the weight function w(x). We suppose Ihut fix) can be expanded in series of polynomials which are orthogonal over the interval of integration with respect to w(x). There are two such expansions for fix). One is an infinite series which follows from the usual orthogonality property. The other is a polynomial approximation plus a remainder. The relations between the coefficients in these representations are detailed and methods for the evaluation of these are analyzed. Error analyses are provided. A numerical example is given to illustrate the effectiveness of the schemes developed.