A numerical evaluation of very oscillatory infinite integrals by asymptotics
Let f be a real smooth function integrable at infinity and satisfying a linear differential equation \(f(t)=p(t)f'(t)\). Assume that p has an asymptotic expansion p(t)\(\sim \sum^{r}_{-\infty}a_ kt^ k\) with \(r\leq 0\) and \(a_ r\neq 0\) and all derivatives of p have asymptotic expansions obtained by differentiating the expansion of p term by term. If p(t)f(t)\(\to 0\) as \(t\to \infty\) one obtains by integrating by parts n times a representation \(\int^{\infty}_{x}f(t)dt=(A_ 1(x)+...+A_ n(x))f(x)+R(x)\) where the \(A_ n\) are functions of p and its derivatives. The remainder R(x) is claimed to be \(o(x^{-n-r})\) in theorem 1 but the proof yields only \(o(x^{1-n-r})\). With some modifications this can also be applied to the case \(r=1\) and to the case where f satisfies a linear differential equation of the second order. This is applied to the evaluation of integrals of functions of the type \(f(t)=h(t)\exp (b(t)+ia(t))\) where a, b, h have suitable asymptotic expansions.
scientific article; zbMATH DE number 4176330