The fundamental importance of the Heaviside operational calculus

Abstract

It is an essential part of Heaviside's operational calculus that the symbol p is an operator equivalent to d/dt and that p and p−1 the inverse or integrating operator, are commutative. This is ensured if the operation of integration is not confused with the operation of selection, i.e. if the lower limit of integration is taken as minus infinity, and is only raised to a finite value when we are sure that this change has no effect. We have first set forth in as explicit a manner as possible what we believe to be the basis of Heaviside's own work, with particular attention to the properties of the unit function H(t) and the way in which differentiation and integration can be extended to include functions containing H(t) or its derivatives as factors. A continuous function approximating to H(t) is considered in an Appendix. The kinds of function (of time) that can occur in nature are carefully considered; the case of both passive and active networks is discussed. Heaviside's contemporaries were not prepared to accept his premises and methods even if they were forced to accept his results. The relation between Heaviside's calculus and Fourier analysis, symbolic calculus and Laplace transforms is therefore considered; the advocates of symbolic calculus and particularly of Laplace transforms have introduced difficulties and even errors which need not have occurred if they had followed Heaviside more faithfully.The full power and universality of Heaviside's approach (in which the mathematics was always subordinate to the physics) is made clear in Section 4, where the relation between input and output is considered for any system, not necessarily electrical; this relation is expressed by a single operational equation, but there are several possible ways of handling that equation and it is important not to choose too early which of these ways should be used.‡