Related papers: The Ross-Darboux-Stieltjes Integral

In this work we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on…

We solve problem x proposed by O. Oloa, AMM xxx 2012 {\bf 119?} (to appear), p. yyy for certain definite logarithmic integrals. A number of generating functions are developed with certain coefficients $p_n$, and some extensions are…

This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that…

We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise…

We establish a general "boundedness implies convergence" principle for a family of evolving Riemannian metrics. We then apply this principle to collapsing Calabi-Yau metrics and normalized K\"ahler-Ricci flows on torus fibered minimal…

We obtain a variety of series and integral representations of the digamma function $\psi(a)$. These in turn provide representations of the evaluations $\psi(p/q)$ at rational argument and for the polygamma function $\psi^{(j)}$. The…

Convergence is proved for solutions of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated in the context of Markov processes associated with general…

In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval $(0,1)$, and we use it to find closed forms for some numerical series related…

We define $\lambda(r)$-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson-Stiltjes integrals for general kernels…

We extend the Heine-Stieltjes Theorem to concern all (non-degenerate) differential operators preserving the property of having only real zeros. This solves a conjecture of B. Shapiro. The new methods developed are used to describe intricate…

This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and K\"ahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and…

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse…

Assume that $\,I\subseteq\mathbb{R}\,$ is an interval and $\,\beta:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a single fixed point $\,s_0\in I\,$, satisfying $\,(s_0-t)(\beta(t)-t)\leq 0\,$ for all $\,t\in I$,…

A detailed study of a double integral representation of the Catalan's constant allows us to identify a duality identity for the Stieltjes transform on which it is based. This duality identity is then extended to an arbitrary dimensional…

The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In…

In this paper, algorithms are developed for computing the Stirling transform and the inverse Stirling transform; specifically, we investigate a class of sequences satisfying a two-term recurrence. We derive a general identity which…

We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with…

By using structural and asymptotic properties of the Kontorovich-Lebedev transform associated with Minkowski's question mark function, we give an affirmative answer to the question posed by R. Salem (Trans. Amer. Math. Soc., 53 (3), (1943),…

The aim of this paper is to establish some new inequalities similar to the Ostrowski's inequalities which are more generalized than the inequalities of Dragomir and Cerone. The current article obtains bounds for the deviation of a function…

The aim of the paper is twofold. We establish refined Strichartz estimates for the Schr\"odinger equation on tori within the framework of partial regularity. As a result, we reveal that the solution of the free Schr\"odinger equation has…