The Heaviside Operational Cdlculus 



By JOHN R. CARSON 



Synopsis: The art of electrical communication owes a great and incrcas- 

 inj;!y recognized debt to Oliver Heaviside for his work in developing and 

 emphasizing a correct theory of electrical transmission along wires and in 

 particular for his insistance on the importance of inrluctance. His oper- 

 ational methods of solving the differential etjuations which are fundamental 

 of the theory of electric circuits, although not widely known, are important. 

 These methods are peculiarly applicable to many important problems of 

 electrical transmission. The present paper, while theoretical in char- 

 acter, therefore deals with a subject of practical importance to the com- 

 munication engineer. 



Without attempting to give any adequate idea of the striking originality 

 and ingenuity- of Hcaviside's methods, his operational calculus may be 

 very briefly explained as follows. Problems in electric circuit theory are 

 described by a set of differential equations involving the differential oper- 

 ator -p. These differential equations may be reduced formally to alge- 



dt 

 braic equations by replacing the differential operator by the symbol p and 

 by this expedient a purely symbolic solution is obtained. This syrnbolic 

 solution is called the operational formula of the problem. 



In order to interpret the purely symbolic operational formula, Heaviside 

 proceeded as fellows: By direct comparison of the operational fornmla of 

 specific problems with their known explicit solutions he was led to assign 

 a definite significance to the operator p. Thereupon, he obtained by in- 

 duction generalized specific criteria or rules for solving the operational 

 formula. 



The present paper, by attacking the problem from a different standpoint, 

 shows that the Heaviside operational formula is a shorthand equivalent of 

 an integral equation from which the methods and rules of his operational 

 calculus are deducible. — Editor. 



AVERY interesting and by no means the least valuable part of 

 Heaviside's researches relates to operational methods of solving 

 the differential equations of a class of physical problems of which 

 electric circuit theory problems are typical; in fact Volume II of his 

 Electromagnetic Theory is almost entirely devoted to this subject. 

 The methods of solution which he originated and employed are of 

 extraordinary directness and simplicity in a very large class of prob- 

 lems in applied mathematics. In fact it would be difhcult to exagger- 

 ate the value of his work along this line, and nowhere is it more im- 

 mediately and usefully applicable than in the theoretical problems 

 of electro-technics. 



Heaviside is, however, by no means easy reading and, in spite of 

 the considerable number of published studies relating to his opera- 

 tional calculus, it is less generally understood and applied than its 

 value warrants. The writer has had occasion to apply Heaviside's 

 methods quite extensively in electrical problems and in the course 

 of his study was led to a general formula which to him at least, has 

 proved useful in interpreting and rationalizing the operational cal- 



43 



