Related papers: Notes and Remarks on certain logarithmic integrals
Henstock-type integrals are considered, for functions defined in a Radon measure space and taking values in a Banach lattice X considering the norm and the order structure of the space. A number of results are obtained, highlighting the…
In this paper, a new calculus on sequences is defined. Also, the $\lambda$-derivative and the $\lambda$-integration are investigated. The fundamental theorem of $\lambda$-calculus is included. A suitable function basis for the…
We discuss a problem initially thought for the Mathematical Olympiad but which has several interpretations. The recurrence sequences involved in this problem may be generalized to recurrence sequences related to a much larger set of…
In this paper, we give explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers. In addition, some formulas for generalized (alternating) harmonic numbers will also be derived.
In this pedagogical note I will discuss one-loop integrals, where (i) different regions of the integration region lead to divergences and (ii) where these divergences cancel in the sum over all regions. These integrals cannot be calculated…
We generalize techniques of Addison to a vastly larger context. We obtain integral representations in terms of the first periodic Bernoulli polynomial for a number of important special functions including the Lerch zeta, polylogarithm,…
In the paper we prove several inequalities involving two isotonic linear functionals. We consider inequalities for functions with variable bounds, for Lipschitz and H\" older type functions etc. These results give us an elegant method for…
Integer partitions express the different ways that a positive integer may be written as a sum of positive integers. Here we explore the analytic properties of a new polynomial $f_\lambda(x)$ that we call the partition polynomial for the…
Estimates of some integrals related to variations of smooth functions are presented.
This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact…
In an attempt to look for the root of nonstandard Lagrangians in the theories of the inverse variational problem we introduce a logarithmic Lagrangian (LL) in addition to the so-called reciprocal Lagrangian (RL) that exists in the…
Real and imaginary part of the limit 2N->infinity of the integral int_{x=1..2N} exp(i*pi*x)*x^(1/x) dx are evaluated to 20 digits with brute force methods after multiple partial integration, or combining a standard Simpson integration over…
We study some "density function" related to the value-distribution of $L$-functions. The first example of such a density function was given by Bohr and Jessen in 1930s for the Riemann zeta-function. In this paper, we construct the density…
The search for new integrable (3+1)-dimensional partial differential systems is among the most important challenges in the modern integrability theory. It turns out that such a system can be associated to any pair of rational functions of…
For a large class of systems of o.d.e.'s which have first integrals, the method of arrays yields the following results: i) The first integrals $I$ can be found by solving systems of linear equations. ii) How the first integral $I$ responds…
Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is…
A two-term functional equation for an infinite series involving the digamma function and a logarithmic factor is derived. A modular relation on page 220 of Ramanujan's Lost Notebook as well as a corresponding recent result for the…
For positive integers d, r, and M, we consider the class of rational functions on real d-dimensional space whose denominators are products of at most r functions of the form 1+Q(x) where each Q is a quadratic form with eigenvalues bounded…
Integrals occurring in Thomas-Fermi theory which contains the logarithm of the Airy function Ai'(x) have been obtained in terms of analytical expressions.
This note presents selected values of definite integrals whose integrand contains a power of the Dedekind function having imaginary argument.