Notes on the Heaviside Operational Calculus 



By JOHN R. CARSON 



This paper briefly discusses the following topics: (1) the asymptotic solu- 

 tion of operational equations; (2) Bromwich's formulation of the Heaviside 

 problem, and its relation to the classical Fourier integral; and (3) the 

 existence of solutions of the operational equation. The paper closes with 

 some general remarks on the interpretation of the operator and the opera- 

 tional equation, emphasizing the purely symbolic character of the latter. 



THE large amount of work done in the past thirteen years, start- 

 ing with important papers by Bromwich ^ and K. W. Wagner,^ 

 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 prob- 

 lem 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 asymp- 

 totic 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:' 

 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 / = 0, when it is suddenly acted upon 

 by a 'unit' force (zero before, unity after time / = 0) ; the subsequent 

 behavior of the system is required. In the solution of this problem, 

 Heaviside's first step was the purely formal and symbolic one of 

 replacing the differential operator d/dt by the symbol p, thereby 



1" Normal Coordinates in Dynamical Systems," Proc. Lond. Math. Soc. (2), 

 15, 1916. 



2"Uber eine Formel von Heaviside zur Berechnung von Einschaltvorgange," 

 Archiv. Elektrotechnik, Vol. 4, 1916. 



^ Since this paper is addressed largely to physicists and engineers, we shall employ 

 to some extent the language of circuit theory rather than pure mathematics; no loss 

 of essential generality is involved. 



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