Exact Remainders for Asymptotic Expansions of Fractional Integrals

Abstract

This is a continuation of work begun in an earlier paper in which we used the theory of distributions to derive explicit expressions for the remainder terms associated with the asymptotic expansions of the Stieltjes transform. In this paper similar results are obtained for the fractional integral of order θ defined by 1θf(x)=1/f(θ)σxo(x-σ)x-1 f(t)dt, θ>. Heref(t) is a locally integrable function on [0, θ) and satisfies f(t)∼σ ast-5–0(ó >0), s=0 as θ