Related papers: A class of logarithmic integrals

By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta…

It is investigated how graded variants of integral and complete integral closures behave under coarsening functors and under formation of group algebras.

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…

Extending the work of Freese and Cook, which develop the basic theory of calculus and power series over real associative algebras, we examine what can be said about the logarithmic functions over an algebra. In particular, we find that for…

This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…

In this paper, we introduce a new type of $ pq $-calculus. The $ pq $-derivative and $ pq $-integration are investigated and various properties of these concepts are given. The fundamental theorem of $ pq $-calculus and formulas of $ pq…

The rational Landen transformations are used to produce a highly efficient numerical method for the integration of rational functions.

The complexes of integral forms on the quantum Euclidean group $E_q(2)$ and the quantum plane are defined and their isomorphisms with the corresponding de Rham complexes are established.

A class of rational functions characterized by some wonderful properties is studied. The properties that identify this class include simple algebra (their inverses can be expressed in radicals), simple topology (the total space of the…

We describe a solving semi-decision method based on examination of the rational structures of the generalized integrating factors of first-order ODEs. We propose a conjecture that for some family of equations of the type…

We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group $G$ is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of $G$. We also provide a…

We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm…

We present some completely monotonic functions involving the$q$-polygamma functions, our result generalizes some known results.

We give a survey of basic facts of $q$-holonomic functions of one or several variables, following Zeilberger and Sabbah. We provide detailed proofs and examples.

In this paper, we establish a q-analog of partial fraction decomposition formula. By using formula, we develop new closed form representations of sums of q-harmonic numbers and reciprocal q-binomial coefficients. Moreover, we give explicit…

The dynamical structure of the rational map $ax+1/x$ on the projective line over the field Q2 of $2$-adic numbers, is fully described.

Let $n$ be a positive integer. In this paper we estimate the size of the set of linear forms $b_1\log a_1 + b_2\log a_2+...+b_n\log a_n$, where $|b_i|\leq B_i$ and $1\leq a_i\leq A_i$ are integers, as $A_i,B_i\to \infty$.

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…

To entirely determine the resulting functions of one-loop integrals it is necessary to find the correct analytic continuation to all relevant kinematical regions. We argue that this continuation procedure may be performed in a general and…

Consider a superelliptic integral $I=\int P/(Q S^{1/k}) dx$ with $\mathbb{K}=\mathbb{Q}(\xi)$, $\xi$ a primitive $k$th root of unity, $P,Q,S\in\mathbb{K}[x]$ and $S$ has simple roots and degree coprime with $k$. Note $d$ the maximum of the…