Exact Remainders for Asymptotic Expansions of Fractional Integrals

This is a continuation of work begun in an earlier paper inwhich we used the theory of distributions to derive explicitexpressions for the remainder terms associated with the asymptoticexpansions of the Stieltjes transform. In this paper similarresults are obtained for the fractional integral of order definedby 1f(x)=1/f()^x_o(x-)^x-1 f(t)dt, >. Heref(t) is a locally integrable function on [0, ) and satisfies f(t) a_st-5–0(ó >0), s=0 as      

Explicit Error Terms for Asymptotic Expansions of Stieltjes Transforms

A technique is developed which gives explicit expressions forthe error terms associated with the asymptotic expansions ofthe Stieltjes transform of f, S _f(z) = ₀ f(t)/t + z dt. Here z is a complex parameter in the cut plane \arg z\ <, and f(t) is a locally integrable function on [0, ). Near infinitywe assume that f(t) is either algebraically decaying or oscillatory.From the explicit expressions, strict and realistic error boundscan be obtained. Explicit error terms are also given for asymptoticexpansions of Laplace and Fourier transforms of small argument.Our approach is based on the theory of distributions.