A Closed-Form Expression for the Unit Step Function

Abstract

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.