A practical algorithm for computing Cauchy principal value integrals of oscillatory functions

摘要

A new automatic quadrature scheme is proposed for evaluating Cauchy principal value integrals of oscillatory functions: ⨍⨍-11f(x)exp(iωx)(x-τ)-1dx(-1<τ<1,ω∈R). The desired approximation is obtained by expanding the function f in the series of Chebyshev polynomials of the first kind, and then by constructing the indefinite integral for a properly modified integrand, to overcome the singularity. The method is proved to converge uniformly, with respect to both τ and ω, for any function f satisfying max−1⩽x⩽1∣f′(x)∣ < ∞.