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

Abstract

The paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform S λ [f ] = ∞ 0 f (x)(ω + x) −λ dx about ω = 0 where now λ is a non-integer positive real number. Divergent integrals with singularities at the origin are induced by writing (ω + x) −λ as a binomial expansion about ω = 0 and interchanging the order of operations of integration and summation. The prescription of finite part integration is then implemented by interpreting these divergent integrals as finite part integrals which are rigorously represented as complex contour integrals. The same contour is then used to express S λ [f ] itself as a complex contour integral. This led to the recovery of the terms missed by naive term-wise integration which themselves are finite parts of divergent integrals whose singularity is at the finite upper limit of integration. When the function f (x) has a zero at the origin of order m = 0, 1, . . . such that m − λ < 0 the correction terms missed out by naive term by term integration give the dominant contribution to S λ [f ] as ω → 0. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naive term by term integration. We apply these results by obtaining exact and asymptotic representations of the Kummer and Gauss hypergeometric functions by evaluating their known Stieltjes integral representations. We then apply the method of finite part integration to obtain the asymptotic behavior of a generalization of the Stieltjes integral which is relevant in the calculation of the effective index of refraction of a shallow potential well.

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

“…where ∆ S is a contribution arising from the pole or branch point singularity of the kernel (ω + z) −ρ at z = −ω, for which reason we have refered to ∆ S as the singular contribution [7,8]. In general, when z = −ω is a branch point, the singular contribution is a progenic finite-part integral.…”

“…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.…”

“…The Stieltejs integral representations (1.5) and (1.7) involve the transform of functions that have singularities competing with the singularity of the kernel of transformation, a circumstance which is outside the scope of [6,7,8]. Here we will perform finite-part integration in the presence of such singularities.…”

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

“…On the other hand, for integral orders λ = n, ν = 0 and entire f (z), the Stieltjes transform will assume the representation given by equation (20); this is implemented in Section-5. For non-integral order λ = n, a contour integral representation other than equations (19) and (20) will have to be devised in [22].…”