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

In the present note we consider the numerical evaluation of integrals over strongly oscillating functions of the type

$\text{∫}\text{0}\text{∞}\text{f}\text{(}\text{x}\text{)?(}\text{x}\text{)}\text{dx}$

where f(x) is periodic function and where the asymptotic behaviour of ? (x) for large values of x is given by $\text{?(}\text{x}\text{) ～}\text{c}\text{x}{}^{\mathrm{\alpha }}\text{+0(}\text{1}\text{x}{}^1\text{+α}\text{)}$ 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 ? (x) given by $\text{?(}\text{x}\text{) ～}\text{c}\text{x}{}^1\text{+α}\text{+0(}\text{1}\text{x}{}^2\text{+α}\text{)}$, the other one being a proper integral.