An Analytic Exact Form of the Unit Step Function

Home , Heaviside step function, Operational calculus, Step function

Mathematics and Statistics 2(7): 235-237, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020702

J. Venetis

Section of Mechanics, Faculty of Applied Mathematics and Physical Sciences, National Technical University of Athens *Corresponding Author: [email protected]

Abstract In this paper, the author obtains an analytic Meanwhile, there are many smooth analytic exact form of the unit step function, which is also known as approximations to the unit step function as it can be seen in Heaviside function and constitutes a fundamental concept of the literature [4,5,6]. Besides, Sullivan et al [7] obtained a the Operational Calculus. Particularly, this function is linear algebraic approximation to this function by means of a equivalently expressed in a closed form as the summation of linear combination of exponential functions. two inverse trigonometric functions. The novelty of this However, the majority of all these approaches lead to work is that the exact representation which is proposed here closed – form representations consisting of non - elementary is not performed in terms of non – elementary special special functions, e.g. Logistic function, Hyperfunction, or functions, e.g. Dirac delta function or Error function and Error function and also most of its algebraic exact forms are also is neither the limit of a function, nor the limit of a expressed in terms generalized integrals or infinitesimal sequence of functions with point wise or uniform terms, something that complicates the related computational convergence. Therefore it may be much more appropriate in procedures. the computational procedures which are inserted into 2. Towards an Exact Form of the Unit Step Function Operational Calculus techniques. Let us introduce the following inverse trigonometric single – π π Keywords Unit Step Function, Algebraic Representation, valued functions f1 : R → (− , ) and Inverse Trigonometric Function 2 2 π π f : R* → (− , ) such that f (x) = arctan(x−1) 2 2 2 1  2 − x  1. Introduction and f2 (x) = arctan   x  The Heaviside step function, or unit step function, which Thus, one can define a single – valued function is usually notated by the symbols H or u, is a discontinuous * f : R → [0,1] single – valued function, whose value is zero for negative such that argument and one for positive argument [1]. This function 1  3π  2 − x was introduced by Oliver Heaviside, who was an important f (x) =  + arctan(x−1) + arctan  (1) pioneer in the study of electronics and also made a π  4  x  remarkable contribution to the field of Operational Calculus [2]. A very important characteristic of this function is that is 3. Claim able to be represented either as a piecewise constant function or as a generalized function [1,3]. The function f coincides with the unit step function The unit step function is mainly used in the calculation over its domain of definition. processes of Control Theory and signal processing in order to represent a signal which switches on at a specified time and stays switched on indefinitely. 4. Proof It is also applied along with its derivative, i.e. Dirac delta function, in structural engineering to describe various types We will prove that the value of the function f is zero for of structural loads, e.g. off – axis four point bending of negative arguments and one for strictly positive arguments. simply supported or fully constrained beams. Hence, it is To this end, let as make the following substitution: very useful for the necessary calculations dealing with x = t +1. Apparently, the variable t cannot be equal to conceptual and embodiment design procedures from the −1. engineering viewpoint. Hence it implies 236 An Analytic Exact Form of the Unit Step Function

1  3π 1− t  1 1 f (x) ≡ φ(t) =  + arctant + arctan  (2) − < k < ⇒ k = 0 (8) π  4 1+ t  2 4 where a single – valued function φ : R\{−1}→ [0,1] Thus it follows has been defined, by means of the above formula. 1− t π arctant + arctan = ⇔ Thus in order the above claim to hold, it is enough to be 1+ t 4 proved that φ(t) = 0∀t < −1 and φ(t) = 1∀t > −1. In continuing, let us distinguish the following four cases 1  3π 1− t  concerning the auxiliary variable t . φ(t) =  + arctant + arctan  = 1 (9) π  4 1+ t  i) The variable t belongs to the interval (−1,0) ii) The variable t belongs to the interval (0,1) = By the substitutions arctant a and By the same substitutions, i.e. arctant = a and 1− t = 1− t arctan b the following two inequalities are evident arctan = b the following two inequalities emerge 1+ t 1+ t π π − < a < 0 (3) 0 < a < (10) 4 4 π π 0 < b < (4) 0 < b < (11) 2 4 Besides, it is known from Trigonometry that the following Hence, by the use of eqn. (5) and following the same identity holds procedure as in the first case, we obtain again tan a + tanb π tan(a + b) ≡ (5) a + b = kπ + ,k ∈ Z 1− tan a ⋅ tanb 4 Thus we can write out According to (10) and (11) we deduce 1− t 3π t + 0 < a + b < ⇒ k = 0 + 4 tan(a + b) = 1 t ⇔ 1− t 1− t ⋅ Thus we infer that + 1 t 1− t π arctant + arctan = ⇔ t 2 +1 1+ t 4 + = ⇔ tan(a b) 2 t +1 1  3π 1− t  φ(t) =  + arctant + arctan  = 1 tan(a + b) =1⇒ π  4 1+ t  π iii) The variable t belongs to (1,+∞) tan(a + b) = tan (6) 4 By the same substitutions as previous we obtain now the following inequalities Hence the following relationship arises π π π < a < (12) a + b = kπ + ,k ∈ Z (7) 4 2 4 π The latter expression in accordance with inequalities (3) − < b < 0 (13) and (4) yields 2 π π Hence, via the same process as before we deduce − < a + b < ⇔ π π π 4 2 − < kπ + < (14) π π π 4 4 2 − < kπ + < ⇔ 4 4 2 and therefore Mathematics and Statistics 2(7): 235-237, 2014 237

1 1 function was equivalently represented in an explicit − < k < ⇒ k = 0 algebraic form as the summation of two inverse 2 4 trigonometric single – valued functions. Here, we should Thus we obtain elucidate that a shortcoming of our formula is that the inverse trigonometric functions do not have unique definitions. 1  3π 1− t  φ(t) =  + arctant + arctan  =1 On the other hand, to define H(0) as 1/2 something that is π  4 1+ t  taken for granted in the majority of the approximations of On the other hand, for t = 0 or t =1 one can easily this function, is a fact that cannot be in accordance with our verify that φ(t) =1 and therefore we conclude that for suggested formula according to which the argument of the t > −1the value of the function φ is 1.  2 − x term arctan  equals infinity at this point and iv) The variable t belongs to (−∞,−1)  x  therefore must be a priori excluded, since it may correspond By the same substitutions as before we derive here the π following inequalities to any arc in the form kπ ± ,k ∈ Z and therefore in such 2 π π − < a < − (15) a case, Eqn. (1) would actually introduce a binary relation 2 4 between the variables x and f (x) instead of a single – π valued function. − < b < 0 (16) Nevertheless, since this formula constitutes a purely 2 algebraic representation and does not contain generalized Thus, by means of the same reasoning as in the other cases integrals or any infinitesimal quantities, it may have good we infer prospects towards the computational procedures that concern the applications of the unit step function in Operational π π −π < kπ + < − (17) Calculus, as well as in engineering practices. 4 4 and therefore 5 1 − < k < − ⇒ k = −1 (18) REFERENCES 4 2 [1] Abramowitz, M. and Stegun, I. A. (1972). Handbook of Thus we deduce Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications INC 1− t π 3π a + b = arctant + arctan = −π + = − ⇒ [2] Hunt, B. J. (2012). Oliver Heaviside. Phys. Today, 65(11), 48. 1+ t 4 4 [3] Bracewell, R. (2000). Heaviside's Unit Step Function,H(x). 1  3π 1− t  The Fourier Transform and its Applications, McGraw – Hill φ(t) =  + arctant + arctan  = 0 (19) π  4 1+ t  [4] Berg L. (1967). Introduction to the Operational Calculus, North – Holland Publishing Company Hence, we conclude that for t < −1 the value of the function φ vanishes. [5] Kanwal, R. P. (1998). Generalized Functions Theory and Consequently, since x = t +1, we have proved that the Technique: Theory and Technique. 2nd ed. Boston, MA: Birkhäuser value of the function f is zero for negative arguments and one for strictly positive arguments and therefore this function [6] Spanier, J. and Oldham, K. B. (1987). The Unit-Step u(x-a) coincides with the unit step function over its domain of and Related Functions Ch. 8 from: An Atlas of Functions. definition, i.e. the set R* . Washington, DC: Hemisphere, 63-69. [7] Sullivan J. Crone L. Jalickee J. (1980) Approximation of the Unit Step Function by a Linear Combination of Exponential 5. Discussion Functions, Journal of. Approximation Theory, 28, 299 – 308 The objective of this work was to propose an analytic exact form of the unit step function. Specifically, this special