The Heaviside Operational Calculus

The art of electrical communication owes a great and increasingly 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 inductance. His operational methods of solving the differential equations 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 character, therefore deals with a subject of practical importance to the communication engineer. Without attempting to give any adequate idea of the striking originality and ingenuity of Heaviside'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 operator -jj- These differential equations may be reduced formally to algebraic equations by replacing the differential operator by the symbol p and by this expedient a purely symbolic solution is obtained. This symbolic solution is called the operational formula of the problem. In order to interpret the purely symbolic operational formula, Heaviside proceeded as follows: By direct comparison of the operational formula of specific problems with their known explicit solutions he was led to assign a definite significance to the operator p. Thereupon, he obtained by induction generalized specific criteria or rules for solving the operational formula.