THE HEAVISIDE OPERATIONAL CALCULUS 51 



the series is truly asymptotic in the sense that the error is less than 

 the last term included. 



Another mode of procedure, however, suggests itself, which, by 

 the aid of equation (10) gives the solution directly without series 

 expansion. We have 



1 _ 1 1_ \a 



pH{p) p - a p - a\ p' 



where 



and 



1 r* 



— i_ = / e-i"hi{t)dt 

 p — a Jo 



r 

 p - a "S p 

 Consequently hi(/) = e*^', and since 



— \l^=f e-^'h,{i)dt. 



- a y p ^0 



h-I .-^'Vi/v/ dt 



Vp Jo 

 it follows at once from (10) that 



The solution h{t) = hi{i) + hiit) agrees with the preceding derived 

 from the asymptotic expansion, and is considerably more direct and 

 simple. 



It is interesting to compare this solution with Heaviside's own 

 operational solution (Electromagnetic Theory Vol. II, p. 40) w^hich 

 amounts to the following. The operational formula is written 



p — a p — a 

 The first term is discarded altogether and the second written as 



F=(l-^)-V^. 

 Identifying V /> with 1/ V tt / and p" with d"/dt'' the expansion becomes 



-=0-Ui-,) + (-2)(l)Q)----)NlS 



