Uniform approximations to Cauchy principal value integrals of oscillatory functions

This paper presents an interpolatory type integration rule for the numerical evaluation of Cauchy principal value integrals of oscillatory integrands ⨍-11eiωxf(x)x-τdx, where -1<τ<1-1<τ<1, for a given smooth function f(x)f(x). The proposed method is constructed by interpolating f(x)f(x) at practical Chebyshev points and subtracting out the singularity. A numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast Fourier transform. The validity of the method has been demonstrated by several numerical experiments.