Article
Version 2
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A Closed-Form Expression for the Unit Step Function
Version 1
: Received: 25 June 2022 / Approved: 27 June 2022 / Online: 27 June 2022 (09:34:58 CEST)
Version 2 : Received: 24 February 2023 / Approved: 1 March 2023 / Online: 1 March 2023 (04:20:33 CET)
Version 2 : Received: 24 February 2023 / Approved: 1 March 2023 / Online: 1 March 2023 (04:20:33 CET)
How to cite: Venetis, J. A Closed-Form Expression for the Unit Step Function. Preprints 2022, 2022060357. https://doi.org/10.20944/preprints202206.0357.v2 Venetis, J. A Closed-Form Expression for the Unit Step Function. Preprints 2022, 2022060357. https://doi.org/10.20944/preprints202206.0357.v2
Abstract
In this paper, an analytical form of the Unit Step Function (or Heaviside Step function) is presented. This important function constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics. In particular, Heaviside Step Function is performed in a very simple manner by the use of a finite number of standard operations, as the summation of six inverse tangent functions. The novelty of this work when compared with other analytical expressions, is that the proposed exact formula contains two arbitrary single - valued continuous functions which satisfy only one restriction. In addition, the proposed representation is not exhibited in terms of miscellaneous special functions, e.g. Bessel functions, Beta function, Logistic function. Besides, it is neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence. Hence, this formula may be much more practical, flexible and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.
Keywords
Unit Step Function; closed - form expression; inverse tangent function
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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