Analytic Exact Forms of Heaviside and Dirac Delta Function

Venetis, J.

doi:10.37622/adsa/15.1.2020.115-121

Cited by 3 publications

(2 citation statements)

References 4 publications

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“…On the other hand, in Ref. [18] a single -valued function was introduced, which was proved to be synonymous with the Unit Step Function. This formula consists of purely algebraic representations and does not contain either generalized integrals or any other infinitesimal quantities.…”

Section: Introductionmentioning

confidence: 99%

A Closed-Form Expression for the Unit Step Function

Venetis¹

2022

Preprint

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In this paper, the author obtains an analytic exact form of the Unit Step Function (or Heaviside step function) which evidently constitutes a fundamental concept of Operational Calculus. This important function is also involved in many other fields of applied and engineering mathematics. Heaviside step function is performed here in a very simple manner, by the use of a finite number of standard operations. In particular it is expressed as the summation of six inverse tangent functions. The novelty of this work when compared with other analytic representations, is that the proposed exact formula contains two arbitrary single - valued continuous functions which satisfy only one restriction. In addition, the proposed explicit representation is not exhibited in terms of miscellaneous special functions, e.g. Bessel functions, Error function, Beta function etc and also 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.

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Section: Introductionmentioning

confidence: 99%

A Closed-Form Expression for the Unit Step Function

Venetis¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Finally in Ref. [14] a singlevalued function was introduced, which was proved to be synonymous with the Unit Step Function. This formula consists of purely algebraic representations and does not contain either generalized integrals or any other infinitesimal quantities.…”

Section: Introductionmentioning

confidence: 99%

An Explicit Form of Heaviside Step Function

Venetis¹

2021

Preprint

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In this paper, the author derives an explicit form of Heaviside Step Function, which evidently constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics.In particular, this special function is exhibited in a very simple manner as a summation of four inverse tangent functions. The novelty of this work is that the proposed exact formulae are not performed in terms of miscellaneous special functions, e.g. Bessel functions, Error function, Beta function etc and also are 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 appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

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An Analytical Simulation of Boundary Roughness for Incompressible Viscous Flows

Venetis

2021

Wseas Transactions on Applied and Theoretical Mechanics

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The intention of this paper is to investigate the boundary roughness of a mounted obstacle which is inserted into an incompressible, external and viscous flow field of a Newtonian fluid. In particular, the present study focuses on the cross – sectional area of the obstacle, which is assumed to be a non deformable body (rigid object) with a predefined shape of random roughness. For facility reasons and without violating the generality, one may select the cross – section of the body which contains its center of gravity and is perpendicular to the main flow direction. The boundary of this cross – sectional area is mathematically simulated as the polygonal path of the length of a single – valued continuous function. Evidently, this function should be of bounded variation. The novelty of this work is that the formulation of the random roughness of the boundary has been carried out in a deterministic manner.

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Analytic Exact Forms of Heaviside and Dirac Delta Function

Cited by 3 publications

References 4 publications

A Closed-Form Expression for the Unit Step Function

A Closed-Form Expression for the Unit Step Function

An Explicit Form of Heaviside Step Function

An Analytical Simulation of Boundary Roughness for Incompressible Viscous Flows

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