Notes on the Heaviside Operational Calculus

HE large amount of work done in the past thirteen years, starting with important papers by Bromwich 1 and K. W. Wagner,2 has served to remove whatever mystery may have surrounded the Heaviside operator, and has placed his operational calculus on a quite secure and logical foundation. However, certain phases of the problem still do not appear to the writer to have as clear or adequate treatment as perhaps might be desired; these it is the object of the present paper to discuss. The topics dealt with are (1) the asymptotic solution of operational equations; (2) Bromwich's very important formula and its relation to the classical Fouiier integral; and (3) the existence of solutions of the operational equation. In the following it will be assumed that the reader has a general acquaintance with the Heaviside operational calculus as well as the Fourier integral, but a brief sketch of the former may not be out of place. It will be recalled that the Heaviside processes were originally developed in connection with the solution of electrical problems:3 more precisely, the determination of the oscillations of a linearly connected system specified by a set of linear differential equations with constant coefficients or a partial differential equation of the type of the wave equation. This system is supposed to be in a state of equilibrium at reference time t = 0, when it is suddenly acted upon by a 'unit' force (zero before, unity after time t = 0); the subsequent behavior of the system is required.