Related papers: Integrals Associated with the Digamma Integral Rep…
A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…
A general framework of Numerical Singular Integrals (NSI) method based on the Integration By Parts (IBP) has been developed for integrals involving singular and nearly singular integrands, or NSI-IBP. Through a general integration by parts…
This thesis is devoted to asymptotic norm estimates for oscillatory integral operators acting on the L^2 space of functions of one real variable. The operators in question have compact support and an oscillatory kernel of the form exp(i…
The pursuit of closed forms for infinite series has long been a focal point of research. In this paper, we contribute to this endeavor by presenting closed forms for the class of digamma series: \[\sum_{k=1}^\infty…
This paper presents an analytical closed-form solution to improper integral $\mu(r)=\int_0^{\infty} x^r dx$, where $r \geq 0$. The solution technique is based on splitting the improper integral into an infinite sum of definite integrals…
We give different integral representations of the Lommel function $s_{\mu,\nu}(z)$ involving trigonometric and hypergeometric $_2F_1$ functions. By using classical results of Polya, we give the distribution of the zeros of $s_{\mu,\nu}(z)$…
We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily-computable exact formulas for quantities that involve expectations and higher moments of the logarithm…
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral…
We obtain asymptotic results for well known summatory arithmetic functions, such as $\psi(x),$ and establish connections to new summatory functions. A new Volterra integral equation is offered, which is solved by summatory arithmetic…
This paper studies the integral of the Riemann xi-function. More generally, it studies a one-parameter family of functions given by Fourier integrals and satisfying a functional equation. Members of this family are shown to have only…
We continue our systematic construction of Baxter Q-operators for spin chains, which is based on certain degenerate solutions of the Yang-Baxter equation. Here we generalize our approach from the fundamental representation of gl(n) to…
By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such…
We discuss a formal derivation of an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicates that their positivity criterion is obeyed, whereby entailing the criticality of the…
We consider the Lommel functions $s_{\mu,\nu}(z)$ for different values of the parameters $(\mu,\nu)$. We show that if $(\mu,\nu)$ are half integers, then it is possible to describe these functions with an explicit combination of polynomials…
This paper is concerned mainly with the deceptively simple integral equation \[ u(x) - \frac{1}{\pi}\int_{-1}^{1} \frac{\alpha\, u(y)}{\alpha^2+(x-y)^2} \, \rd y = 1, \quad -1 \leq x \leq 1, \] where $\alpha$ is a real non-zero parameter…
In this paper we establish some (presumably new) interesting expressions for the composition of some well known fractional integral operators $ I^{\mu}_{a+}, D^{\mu}_{a+} $,$ I^{\gamma , \mu}_{a+}$ and also derive an integral operator…
In this article, we consider the Diophantine equation $\sigma_{2}(n)-n^2=An+B$ with $A=P^2\pm2$. For some $B$, we show that except for finitely many computable solutions in the range $n\leq(|A|+|B|)^{3}$, all the solutions are expressible…
This paper explores closed-form expressions for some polylogarithm integrals with integrands containing five parameters. These closed form expressions are given in terms of the Lerch transcendent function, which reduces, in some cases, to…
We provide two kinds of representations for the Taylor coefficients of the Weierstrass $\sigma$-function $\sigma(\cdot;\Gamma)$ associated to an arbitrary lattice $\Gamma$ in the complex plane $\mathbb{C}=\mathbb{R}^2$ - the first one in…
It is shown that if $\gamma$ is a path of finite $p$ variation ($1\leq p< 2$) in a euclidean vector space and $f,g,h$ are Lipschitz functions on the trace of $\gamma$ then $s\mapsto F(s)=\int_\gamma f^sg dh$ defines an entire holomorphic…