Explicit Error Terms for Asymptotic Expansions of Stieltjes Transforms

Distributional Derivation of an Asymptotic Expansion

“…As already pointed out in §2, the Fourier transform (Amtp)'(t) given in (3) exists uniformly for all sufficiently large values of t E R". Hence, by a generalization of the Riemann-Lebesgue lemma, we have Sm(t) = o(\t\-2m) as M ^oo.…”

“…Introduction. Recently, a new method based on the use of distributions has been introduced into the asymptotic evaluation of integrals; see [1], [3], [4], [11]. An advantage of this approach is that it leads to a particularly illuminating construction of error terms associated with the asymptotic expansions.…”

“…An advantage of this approach is that it leads to a particularly illuminating construction of error terms associated with the asymptotic expansions. However, this method has been applied, so far, only to integrals whose kernels are algebraic functions, for instance, the Stieltjes transform [3], the Riemann-Liouville fractional integral [4], and a Mellin convolution of two algebraically dominated functions [11, §11]. Although a different method has been devised recently for the same purpose for integrals with oscillatory kernels, such as the one-sided Fourier integral [5], the Hankel transform [10], and the multidimensional Fourier transform [9], it is nevertheless of interest to know whether the distributional approach can be applied with equal advantage to the second-mentioned class of integrals.…”

Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales

“…Let e s (x) = e icx x −s−α . From [27], the functions f (x), e s (x) and f n (x) generate distributions on S defined as follows:…”

Quasiasymptotic expansion of distributions from S+′ and the asymptotic expansion of the distributional Stieltjes transform

“…McClure and Wong [3,14] studied the asymptotic expansion of the generalized Stieltjes transform of some classes of locally integrable functions characterized by their asymptotic expansions at and 0 + Our approach to the asymptotic expansion of the distributional Stieltjes transform which we study in this paper is quite different from the approach given in [3,14].…”