Get your copy here from the publisher. Compare with the original version of the paper, especially Section 3. Tica-Galapon Journal of Mathematical Physics 60, 013502 (2019) (accepted version)
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 »
Integrals are a mainstay in physics. Sooner or later we will have to evaluate an integral and when one arises its either we pick up the venerable Table of Integrals, Series, and Products by Gradhsteyn and Ryzhik (GR) or tap our keyboards and let Mathematica or Maple do the integration for us. We are lucky… Read More »
https://www.sciencedirect.com/science/article/pii/S0003491618302173
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 »
We demonstrate that the quantum corrections to the classical arrival time for a quantum object in a potential free region of space, as computed in Phys. Rev. A 80, 030102(R) (2009), can be eliminated up to a given order of ℏ by choosing an appropriate position-dependent phase for the object’s wave function. This then implies… 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 »