An Explicit Expression of the Unit Step Function

Abstract

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.