50 BELL SYSTEM TECHNICAL JOURNAL 



Then the required function hit) is given by 



h{t) = fhiit - y)h2{y)dy (10) 



by Borel's Theorem (Bromwich, Theory of Infinite Series, p. 280).^ 



As a final example of the foregoing discussion we shall consider a 

 specific problem of some practical interest in itself and which involves 

 Heaviside's so-called " fractional differentiation " and his resulting 

 asymptotic solutions. The physical problem is as follows: a " unit- 

 voltage " (zero before, unity after time / = 0) is applied through a 

 terminal condenser Co to an infinitely long cable of resistance i? and 

 capacity C per unit length. Required the Voltage V at the cable 

 terminals. 



The operational formula of this problem is easily deduced; it is 



V = ^Pl^__ where 1/V^ = Co \/RfC. 



1 + yJplOL 



Consequently the integral equation can be written 



1 yJJJa 



L 



e-^'V {l)dt = 



P 1 + Vp/a 



1 1 



P 1 + vWF 



Taking the last form of l/pH{p), expanding asymptotically and 

 recognizing that 



l/pn + i = f e-*H''/n\dt 



up. VP -fe-'- (2„ - 1) [zf- 3) . . . 1 7=i 

 the resulting series solution can be recognized and summed as 



The last expansion by repeated integration by parts leads to the 



asymptotic series given by Heaviside. It is easy to show, also, that 



•This formula is quite useful; it is applied in the solution of the last example of 

 thie present paper. 



