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.

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.

We devise a prescription to utilize a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral and its generalizations [Galapon E.A Proc.R. Soc A 473, 20160567(2017)] to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments which we treated as Hadamard's finite part integrals. The result allowed us to compute the groundstate energy of the quartic, sextic anharmonic oscillators as well as the PT symmetric cubic oscillator, and the funnel potential across all perturbation regimes from a single expansion that is built from the divergent weak-coupling perturbation series and incorporates the known leading-order strong-coupling behavior of the spectra.

We devise a prescription to use a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral and its generalizations (Galapon EA. 2017 Proc. R. Soc A 473 , 20160567) to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments, which we treated as Hadamard’s finite part integrals. The result allowed us to compute the ground state energy of the quartic, sextic anharmonic oscillators as well as the P T symmetric cubic oscillator, and the funnel potential across all perturbation regimes from a single expansion that is built from the divergent weak-coupling perturbation series and incorporates the known leading-order strong-coupling behaviour of the spectra.