Related papers: Ahmed's Integral: the maiden solution
In the year 2000, Ahmed proposed a family of integrals in the American Mathematical Monthly which invoked a considerable response then. Here I would like to present another solution to this family of integrals. I propose to call this as…
The so-called Ahmed integral $$ \int_{0}^{1}\frac{\arctan\left(\sqrt{2+x^{2}}\right)}{(1+x^{2})\sqrt{2+x^{2}}}\,\mathrm{d} x=\frac{5\pi^{2}}{96}, $$ has attracted considerable interest since its appearance in the "American Mathematical…
We show that Ahmed's Integral is closely related to the fourth power of the Probability Integral
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…
We present a historiographical review of algorithms and computer codes developed for solving integration-by-parts relations for Feynman integrals. This procedure is one of the key steps in the evaluation of Feynman integrals, since it…
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
This work deals with Adem relations in the Dyer-Lashof algebra from a modular invariant point of view. The main result is to provide an algorithm which has two effects: Firstly, to calculate the hom-dual of an element in the Dyer-Lashof…
Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the \emph{Journal of the Indian Mathematical Society}, notebooks, Quarterly Reports to the University of…
We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both…
This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
In this article, we explore a series of elementary yet insightful results involving integrals related to Gaussian sums. Using techniques rooted in classical calculus, we derive several identities and evaluate nontrivial definite integrals…
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.
Some divergent trigonometric integrals have appeared in standard tables for many years, listed as converging. We give a simple proof that these integrals diverge and trace their history. The original error was made when a (startlingly)…
The article shows how two choices are possible whenever computing the geometric mean, and the repetition of this process can in general yield 2-to-the power N different values when the choices are compounded in the first N steps of…
In this short note, we provide an elementary complex analytic method for converting known real integrals into numerous strange and interesting looking real integrals.
This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that…
This paper is an enhanced version of a more than decade-older paper with a similar title. Many formulae involving both finite and infinite sums of digamma and polygamma functions up to quadratic order, few of which appear in standard…
Differential lambda-calculus was first introduced by Thomas Ehrhard and Laurent Regnier in 2003. Despite more than 15 years of history, little work has been done on a differential calculus with integration. In this paper, we shall propose a…
We define a more general type of integral on time scales. The new diamond integral is a refined version of the diamond-alpha integral introduced in 2006 by Sheng et al. A mean value theorem for the diamond integral is proved, as well as…