Related papers: A solution to Ahmed's Integral(II)
In 2001-2002, I happened to have proposed a new definite integral in the American Mthematical Monthly (AMM),which later came to be known in my name (Ahmed). In the meantime, this integral has been mentioned in mathematical encyclopedias and…
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
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…
In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging…
The paper suggests a slightly more rigorous justification to Wang et al.'s work from 2007, and introduces the Slanted Line Integral.
We consider some applications of the non-homogeneous second order integral equation of Fox. Some new solutions to Fox's integral equation are discussed in relation to number theory.
The method introduced in (Yehia H M 2006 J. Phys. A: Math. Gen. 39 5807-5824) and (Yehia H M 2012 J. Phys. A: Math. Gen. 45 395209) is extended to construct new families of several-parameter integrable systems, which admit a complementary…
In the first part, by the first author's work of 1972, an integral representation for an ultraspherical polynomial of higher index in terms of one of lower index and an infinite series was obtained. While this representation works well from…
A modified Gauss's algorithm for solving a system of linear equations in an integral ring is proposed, as well as an appropriate algorithm for calculating the elements of the adjoint matrix.
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…
The two parameter generalization of the complete elliptic integral of the second kind discussed recently by Barsan is expressed in terms of ordinary complete elliptic integrals.
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…
Several new formulas are developed that enable the evaluation of a family of definite integrals containing the product of two Whittaker W-functions. The integration is performed with respect to the second index, and the first index is…
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…
Part I. Some Facts From p-Adic Analysis. Part II. Tables of Integrals.
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and…
The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…
We propose a rather elementary method to compute a certain family of integrals on the half line, depending on the integer parameters $n\geq q\geq 1$.