CIRCUIT THEORY AND OrERATIOS'AL CALCULUS 753 



vnltaRe in the non-inductive cable in response to a "unit e.m.f. It 

 will be recalletl that in a preceding chapter we derived the operational 

 formula 



r = r^«» (124) 



where u = .v^/?C for the voltage at distance .v from the terminal of a non- 

 inductive cable of distributed resistance R and capacity C, in response 

 to a "unit e.m.f." impressed at point .x; = 0. Heaviside's solution of 

 this operational equation proceeds as follows: 



Expansion of the exponential function in the usual power series gives 



1! 2! 3! "•" 4! 



which ma>' lie rearranged as 



Hcaviside then discards the series in integral powers of p entirely, 

 replaces y/p by 1/\/t/ and p" by d'/dl" in the first series, and then gets 



or 



This solution is correct, as will be shown subsequently. 



A rather remarkable feature of this solution — a point on which 

 Heaviside makes no comment — is that it is absolutely convergent. 

 In other words, a process of expansion which in other problems leads 

 to a divergent or asymptotic solution, here results in a convergent 

 series expansion. 



To verify this solution we start with the corresponding integral 

 equation of the problem 



[e-V^= f V(l)e-"dt. (128) 



P Jn 



id the 

 V{t)=f^<l>{t)dt 



P 

 It follows from this fcjrmula and theorem (\ '; that 



