A note on the numerical evaluation of integrals over strongly oscillating functions

摘要

In the present note we consider the numerical evaluation of integrals over strongly oscillating functions of the type ∫0∞f(x)ϕ(x)dx where f(x) is periodic function and where the asymptotic behaviour of ϕ (x) for large values of x is given by ϕ(x) ∼ cxα+0(1x1+α) with α ⩾ 1. We show how the integral (1) can be transformed into a sum of two integrals, one of them being of the same type but with an asymptotic expansion for ϕ (x) given by ϕ(x) ∼ cx1+α+0(1x2+α), the other one being a proper integral.