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 »
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 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 »
Term by term integration is one of the frequently used methods in evaluating integrals and in constructing asymptotic expansions of integrals. It typically involves expanding the integrand or a factor of it and then interchanging the order of summation and integration. Sometimes this formal manipulation leads to an infinite series involving divergent integrals. This calls… Read More »
In the publication found here or here, I introduced the concept of analytic principal value (APV) to allow meaningful assignment of values to the class of divergent integrals given by (1) with . The basic assumption in the definition of the APV is that the function has a complex extension , obtained by replacing… Read More »
Equations 30 and 31 should read as follows: (1) (2) In the original paper, the correct index in the third terms of the right-hand sides of equations 30 and 31 was erroneously encoded as . The post has been appropriately updated. Thanks to Avram Guiterrez, Kenneth Jhon Remo and Aster Niel Abellano (PUP… Read More »