A simple approach to asymptotic expansions for Fourier integrals of singular functions

In this work, we are concerned with the derivation of full asymptotic expansions for Fourier integrals ∫abf(x)e±isxdx as s → ∞, where s is real positive, [a, b] is a finite interval, and the functions f(x) may have different types of algebraic and logarithmic singularities at x = a and x = b. This problem has been treated in the literature by techniques involving neutralizers and Mellin transforms. Here, we derive the relevant asymptotic expansions by a method that employs simpler and less sophisticated tools.