Elementary Special Functions Research Articles

An Explicit Expression of the Unit Step Function

In this paper, an analytical form of the Unit Step Function (or Heaviside Step Function) is presented. In particular, this piecewise – defined function, which constitutes a fundamental concept of Operational Calculus and is also involved in many other areas of applied and engineering mathematics, is explicitly performed in a very simple manner by the aid of purely algebraic representations. The novelty of this work, when compared with other analytical approximations to this discontinuous function, is that the proposed exact formula is not performed in terms of non – elementary special functions, e.g. Gamma function or Error function. In addition, this function does not contain any infinitesimal quantities and also is not the limit of a sequence of functions with a pointwise or uniform convergence. Hence, it may be much more appropriate and useful to the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

Constructive Formulations of Resonant Maxwell's Equations

In this article we propose new constructive weak formulations for resonant time-harmonic wave equations with singular solutions. Our approach follows the limiting absorption principle and combines standard weak formulations of PDEs with properties of elementary special functions adapted to the singularity of the solutions, called manufactured solutions. We show the well-posedness of several formulations obtained by these means for the limit problem in dimension one, and propose a generalization in dimension two.

Some applications of a hypergeometric identity

In this paper, we use a general identity for generalized hypergeometric series to obtain some new applications. The first application is a hypergeometric-type decomposition formula for elementary special functions and the second one is a generalization of the well-known Euler identity \(e^{i\,x} = \cos x + i\,\sin x\) and an extension of hyperbolic functions in the sequel. Applying the mentioned identity on classical hypergeometric orthogonal polynomials and deriving summation formulae for some classical summation theorems are two further applications of this identity.

An Analytic Exact Form of the Unit Step Function

In this paper, the author obtains an analytic exact form of the unit step function, which is also known as Heaviside function and constitutes a fundamental concept of the Operational Calculus. Particularly, this function is equivalently expressed in a closed form as the summation of two inverse trigonometric functions. The novelty of this work is that the exact representation which is proposed here is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Therefore it may be much more appropriate in the computational procedures which are inserted into Operational Calculus techniques.

Representation Theory over Tropical Semifield and Langlands Duality

Recently we propose a class of infinite-dimensional integral representations of classical \({\mathfrak{gl}_{\ell+1}}\)-Whittaker functions and local Archimedean local L-factors using two-dimensional topological field theory framework. The local Archimedean Langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories. In this note we introduce elementary analogs of the Whittaker functions and the Archimedean L-factors given by Ul+1-equivariant symplectic volumes of appropriate Kahler Ul+1-manifolds. We demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids \({GL_{\ell+1}(\mathcal{R}),\; \mathcal{R}}\) being the tropical semifield. We also show that the elementary Whittaker functions can be obtained from the non-Archimedean Whittaker functions over \({\mathbb{Q}_p}\) by taking the formal limit p→ 1. Hence the elementary special functions constructed in this way might be considered as functions over the mysterious field \({\mathbb{Q}_1}\). The existence of two representations for the elementary Whittaker functions, one as an equivariant volume and the other as a matrix element, should be considered as a manifestation of a hypothetical elementary analog of the local Langlands duality for number fields. We would like to note that the elementary local L-factors coincide with L-factors introduced previously by Kurokawa.

The effects of neglecting displacement currents when studying transient wave propagation in the Earth

Diffusion approximations are often used to study the interaction of transient electromagnetic waves with the Earth. If a frequency-independent conductivity and permittivity are assumed, then neglecting the displacement currents in the Earth reduces the wave equation to the diffusion equation, which sometimes can be solved analytically using elementary special functions. We study the validity of the diffusion approximation for a transient electromagnetic plane wave obliquely incident on the Earth which is modeled as a conductive half-space. The limitations of the diffusion approximation are investigated by comparing the approximate diffusion results with exact solutions, which are computed using closed-form expressions for the transient fields. We find that the approximate diffusion fields violate causality and only yield accurate results at late times.

Some inequalities in elementary special functions with applications to nonparametric statistical inference

Some inequalities are established in P1(r, s) and P1(r+1, s), where P1(r, s) is the confidence coefficient of Wilks’ (1962) outer confidence interval (X(r) X(s)) for the quantile interval (ξp1, ξp2). An inequality concerning incomplete beta functions is also presented and it is shown to be an improved version of one of Koti's (1989) inequalities.

Multiple integrals involving theH-function of two variables

In this paper, we establish two multiple integrals involving the products of generalised hypergeometric function,H-function of one and two variables. The integrals evaluated are very general in nature and generalise the well-known integrals due to Mittal and Gupta, Goyal, Olkha, Dahiya and others. Some interesting integrals involving the products of elementary special functions and orthogonal polynomials have also been obtained as particular cases of the main results.