Numerical calculation of integrals with strongly oscillating integrand

Abstract

In this paper a method is presented for evaluating \[ ∫ 0 N f ( x ) e i w x d x  where  ω N = p ⋅ 2 π ,   p  integer . \int _0^N {f(x){e^{iwx}}} dx{\text { where }}\omega N = p \cdot 2\pi ,{\text { }}p{\text { integer}}{\text {.}} \] The idea is to approximate f ( x ) f(x) instead of the whole integrand by aid of polynomials. The Romberg-Stiefel algorithm has been extended to this case. The new method is complementary to the usual Romberg-Stiefel algorithm in the sense that it is more advantageous for larger values of ω \omega . An expression for the remainder term is also included. Results for the real part are exact if f ( x ) f(x) is of at most 7th degree and for the imaginary part if f ( x ) f(x) is of at most 8th degree.