Numerical calculation of integrals with strongly oscillating integrand

Abstract:

In this paper a method is presented for evaluating \[ \int _0^N {f(x){e^{iwx}}} dx{\text { where }}\omega N = p \cdot 2\pi ,{\text { }}p{\text { integer}}{\text {.}}\] The idea is to approximate $f(x)$ 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 $\omega$. An expression for the remainder term is also included. Results for the real part are exact if $f(x)$ is of at most 7th degree and for the imaginary part if $f(x)$ is of at most 8th degree.