Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders

Abstract

The paper addresses the exact evaluation of the generalized Stieltjes transform Sn[f ] = ∞ 0 f (x)(ω + x) −n dx of integral order n = 1, 2, 3, . . . about ω = 0 from which the asymptotic behavior of Sn[f ] for small parameters ω is directly extracted. An attempt to evaluate the integral by expanding the integrand (ω + x) −n about ω = 0 and then naively integrating the resulting infinite series term by term lead to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard's finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in [E.A. Galapon, Proc. Roy. Soc. A 473, 20160567 (2017)]. It is shown that, when f (x) does not vanish or has zero of order m at the origin such that (n − m) ≥ 1, the dominant terms of Sn[f ] as ω → 0 come from contributions arising from the poles and branch points of the complex valued function f (z)(ω + z) −n . These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

“…When finite part integration of the incomplete generalized Stieltjes tranform (1) lifts the integration from the positive real line into the complex plane as a complex contour integral, the point z = −ω is no longer a pole of order n which was the case with integer-ordered Stieltjes transform [2], but a branch point. Consequently, this will prompt us to handle divergent integrals of the form…”

“…The need for a separate treatment stems from the fact that the nature of the singularity of the complex valued function (ω + z) −λ depends on λ: when λ is an integer n, the integrand in (1) has a pole at z = −ω of order n; on the other hand, when λ is non-integer, the integrand has a branch point at z = −ω instead. Hence, for the present case, neither Lemma 2.1 nor 2.2 of [2] gives the appropriate complex contour integral formulation of equation (1). Instead, we shall faithfully adhere to the sequence of steps prescribed in [2] to properly implement finite part integration and obtain the contour integral representation of the integral in equation (1).…”

“…The divergence occurs at the origin when f (0) = 0. Their corresponding finite parts have already been previously considered in [1,2]. In particular, their contour integral representation is given by [2, Theorem 2.2],…”

Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration

“…Recently, in revisiting the problem of missing terms arising from term by term integration leading to an infinite series of divergent integrals [3,4,5], it was determined that the finite part of divergent integrals can be rigorously used as a means of evaluating convergent integrals, a method we have referred to as finite-part integration [6]. Finite-part integration has been applied in the exact and asymptotic evaluation of the Stieltjes transform of integer [7] and non-integer orders [8]. Applying the method to known Stieltjes integral representations of some special functions has led to new representations of them.…”

“…where the integral \ \ a 0 f (x)x −k−ρ dx is the finite part [12,13,14] of the divergent integral a 0 f (x)x −k−ρ dx and the term ∆(ω) is a contribution coming from the singularity of the kernel of transformation at −ω. However, in [6,7,8] it was assumed that f (x) possesses an entire complex extension f (z). This condition severely restricts the domain of applicability of the method.…”