Numerical Calculation of Integrals with Strongly Oscillating Integrand

In this paper a method is presented for evaluating \[ ∫ _0^N {f(x){eⁱʷˣ}} dx{ where }ω N = p · 2π ,{ }p{ integer}{.}\] 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 ω. 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.