An Analytic Exact form of Heaviside Function

Abstract

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula 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. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

An Explicit Form of Heaviside Step Function

“…(1) does not describe only one function but in fact describes a family of functions, that all have the same property (to be synonymous with Heaviside function) and this is an advantage over the mathematical formula proposed in Ref. [8]. Finally, on the basis of eqn.…”

“…In Ref. [8] an analytic exact form of the Unit Step Function was proposed as a summation of two inverse tangent functions. Nonetheless, according to this simplified approach the singularity structure was left ambiguous.…”

“…In the present study, in the sense of Ref. [8] an analytic exact form of the Unit Step Function has been proposed as a summation of four inverse tangent functions. Indeed, this formula constitutes a purely algebraic representation since it does not contain special functions, generalized integrals or any other infinitesimal quantities.…”

Analytic Exact Forms of Heaviside and Dirac Delta Function

“…In Ref. [8] an analytic exact form of the Unit Step Function was proposed as a sum of two inverse tangent functions. Nonetheless, according to this approach the singularity structure was left ambiguous.…”

Asymptotic Graph Neural Networks Improve Adversarial Robustness in Machine Vision

“…I focus on the match detector instead of the mismatch detector from Chapter 1 to simplify many of the equations in this section. The analytic form of the unit step function shifted to the right by 1 − δ is defined as follows (Venetis, 2014).…”