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

Abstract

A new automatic quadrature scheme is proposed for evaluating Cauchy principal value integrals of oscillatory functions: ⨍ - 1 1 f ( x ) exp ( i ω x ) ( x - τ ) - 1 dx ( - 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)∣ < ∞.