Related papers: Regularized Limit, analytic continuation and finit…
The paper addresses the exact evaluation of the generalized Stieltjes transform $S_{n}[f]=\int_0^{\infty} f(x) (\omega+x)^{-n}\mathrm{d}x$ of integral order $n=1,2, 3,\dots$ about $\omega =0$ from which the asymptotic behavior of $S_{n}[f]$…
The present book gives a systematic overview of function theory and the theory of Stieltjes integral. In particular, we give a detailed account of the theory of functions of bounded variation and of the theory of regulated functions (=…
It is known in the case of the Stieltjes transform that evaluating the integral by expanding the kernel of transformation followed by term by term integration leads to an infinite series of divergent integrals. Moreover, it is known that…
The paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform $S_{\lambda}[f]=\int_0^{\infty} f(x) (\omega+x)^{-\lambda}\mathrm{d}x$ about $\omega = 0$ where now $\lambda$ is a…
Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity…
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…
Stieltjes integral theorem is more commonly known by the phrase 'integration by parts' and enables rearrangement of an otherwise intractable integral to a more amenable form; often permitting completion of an integral in closed form.…
We study various Stieltjes integrals as Poisson-Stieltjes, conjugate Poisson-Stieltjes, Schwartz-Stieltjes and Cauchy-Stieltjes and prove theorems on the existence of their finite angular limits a.e. in terms of the singular…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
In this work, we investigate the regularized solutions and their finite element solutions to the inverse source problems governed by partial differential equations, and establish the stochastic convergence and optimal finite element…
The integral $\int_0^a f(t) t^{-s} \mathrm{d}t$ diverges for $\text{Re}(s) \geq \lambda + 1$, where $\lambda$ is the order of the first non-vanishing derivative of $f(t)$ at the origin. With the assumption that $f(t)$ is analytic at the…
Regularization and interior point approaches offer valuable perspectives to address constrained nonlinear optimization problems in view of control applications. This paper discusses the interactions between these techniques and proposes an…
The Central Limit Theorem (CLT) establishes that sufficiently large sequences of independent and identically distributed random variables converge in probability to a normal distribution. This makes the CLT a fundamental building block of…
We study the existence of Riemann-Stieltjes integrals of bounded functions against a given integrator. We are also concerned with the possibility of computing the resulting integrals by means of related Riemann integrals. In particular, we…
We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between the integrals of functions defined on subsets $\mathbb{I}_1 \subseteq…
Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…
The paper surveys the basic properties of generalized Stieltjes functions including some new ones. We introduce the notion of the exact Stieltjes order and give a criterion of exactness, simple sufficient conditions and some prototypical…
In this paper we prove pointwise and distributional Fourier transform inversion theorems for functions on the real line that are locally of bounded variation, while in a neighbourhood of infinity are Lebesgue integrable or have polynomial…
Motivated by the limitations of the traditional definitions of the Riemann-Stieltjes and Darboux-Stieltjes integrals, we introduce a generalized Darboux-Stieltjes integral that is equivalent to an earlier generalization by Ross \cite{Ross}.…
Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for…