On the evaluation of Cauchy principal value integrals of oscillatory functions

The problem of the numerical evaluation of Cauchy principal value integrals of oscillatory functions ∫−11eiωxf(x)x−τdx, where −1<τ<1−1<τ<1, has been discussed. Based on analytic continuation, if ff is analytic in a sufficiently large complex region GG containing [−1, 1], the integrals can be transformed into the problems of integrating two integrals on [0,+∞)[0,+∞) with the integrand that does not oscillate, and that decays exponentially fast, which can be efficiently computed by using the Gauss–Laguerre quadrature rule. The validity of the method has been demonstrated in the provision of two numerical experiments and their results.