THE HEAVISIDE OPERATIONAL CALCULUS 49 



where a = (!//>// (p) )p = oo. 



d 



ldpHip)X^o. 

 ' = LHip)] 



Ak = 



1 



pkH'ip.y 



and Pi . . . p„ are the roots of n{p) = 0. 

 By virtue of this expansion ^ the solution is 



h{t) = aP + 6 + c/ + V 



gpkt 



^PkH'iPkY 



where P denotes a " pulse " at the origin / = 0; that is, 



P = 00 at / = 0, 

 = for / >0, 



/oo 

 Pdt = 1. 



In the usual case where a = c = and h = \/H{G), this reduces to- 



gpkt 



h{t) = l/H{0) +2^ 



pkH'iPkY 



which will be recognized as the celebrated Heaviside Expansion 

 Solution. 



As illustrating the flexibility of the integral identity (9), another 

 form of solution will be given which is often of value in practical 

 problems where an explicit solution cannot be obtained. Suppose 

 that l/pH(p) can be written as 



1 1 1 



pH(,p) pH,{p) pH^ip) 



and that functions hi{t) and h2{t) can be found which satisfy the 

 equations 





'The terms a + c/p^ in this expansion were suggested by Dr. O. J. Zobel and 

 must be included in a number of important problems in electric circuit theory. 



