This is a major revision of the original preprint. The one-sided and full Hilbert transforms are evaluated exactly by means of the method of finite-part integration [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. In general, the result consists of two terms—the first is an infinite series of finite-part of divergent integrals, and the… Read More »
A prescription is devised to utilize a novel convergent expansion in the strong-asymptotic regime of 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 were treated as Hadamard’s finite… Read More »
Finite-part integration is a recently introduced method of evaluating well-defined integrals using the finite-part of divergent integrals [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. Here finite-part integration is applied in the exact evaluation of the one-sided, , and full, , Hilbert transforms, together with the reductions of the later transform when is a… Read More »
The values of the Riemann zeta function at odd positive integers, , are shown to admit a representation proportional to the finite-part of the divergent integral . Integral representations for are then deduced from the finite-part integral representation. Certain relations between and are likewise deduced, from which integral representations for are obtained.
Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, Proc. R. Soc., A 473, 20160567 (2017)]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the… Read More »
Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform under the assumption that the extension of in the complex… Read More »
In (Galapon 2017) we considered the problem of missing terms arising from evaluating the incomplete Stieltjes transform, (1) following from binomially expanding the kernel about and then integrating the resulting infinite series term by term. The interchange of summation and integration leads to the infinite series whose terms are divergent integrals. Assigning values to… Read More »
The paper addresses the exact evaluation of the generalized Stieltjes transform about from which the asymptotic behavior of for small parameters is directly extracted. An attempt to evaluate the integral by expanding the integrand about and then naively integrating the resulting infinite series term by term lead to an infinite series whose terms are divergent… Read More »
The operations of integration and summation cannot be interchanged arbitrarily. Some uniformity conditions must be satisfied in order for the interchange to be performed. If the conditions are not satisfied and the interchange is nevertheless carried out, some outrageous things happen. For example, the interchange may lead to an infinite series of divergent integrals; that… Read More »
Methods abound in evaluating convergent integrals, for example, by substitution or by differential equations. In this post I show how divergent integrals may be used in evaluating convergent integrals by applying our recent results in the complex contour integral representations of the finite part of divergent integrals found here and here. To be concrete, let… Read More »