NOTES ON THE HEAVISIDE OPERATIONAL CALCULUS 151 



reducing the differential equations to an algebraic form, the formal 

 solution of which we shall write 



Here h = hit) is the variable with whose determination we are con- 

 cerned and H{p) is the Heaviside function, derived as stated from 

 the differential equations of the problem. This equation is as yet 

 purely symbolic, and its conversion into an explicit solution for h, 

 as a function of t, constitutes the Heaviside problem. 



Bromwich ^ formulates the problem as the infinite integral 



hit) =^. TWrr^^P- (2) 



The writer's formulation of the problem is, that h is uniquely 

 determined by the integral equation ^ 





^W^"'* = WW) <'^ 



This equation is valid for all values of p, for which its real part is 

 greater than some finite constant c; c must be at least large enough 

 to make the infinite integral converge. In the majority of physical 

 problems this constant may be taken as 0; in some, however, the 

 equation is valid only when c is greater than some finite constant. 



The equivalence of (2) and (3) is very easily established in a num- 

 ber of ways; perhaps the simplest is to show, following March,^ that 

 (2) is the formal solution of (3). Either can be deduced from the 

 other. The Bromwich solution can, of course, be derived directly 

 from the Heaviside problem, as shown below. 



I 



One of the most interesting and perhaps the least generally under- 

 stood of Heaviside's methods of solving the operational equation is 

 the process whereby he derives a series solution, usually divergent 

 and asymptotic, in inverse fractional powers of t. What I have termed 

 the Heaviside Rule ^ for deriving this type of solution may be formu- 

 lated as follows: 



* "The Heaviside Operational Calculus," B. S. T. J., 1922; Bulletin Amer. Math. 

 Soc, 1926. 



*"The Heaviside Operational Calculus," Bulletin Amer. Math. Soc, 1927. 



* In terming this process the Heaviside Rule I do not in any sense imply that 

 Heaviside himself would have applied it incorrectly. In fact in one case he adds 

 an extra term which contributes to numerical accuracy although the series itself is 



