710 BELL SYSTEM TECHNICAL JOURNAL 



is the asymptotic expansion of l/II{p). Comparing the two ex- 

 pansions and making a term by term identification, we see that 

 hn = a„ and 



hit) =ao+ail/\\+a-2t-/2\+ .... 



which agrees witli tiie Heaviside formula. 



This procedure, howe\er, while giving the correct result has serious 

 defects from a mathematical point of \iew. For example, the asym- 

 totic expansion of \/II{p) has usualh- only a limited region of con- 

 vergence, and it is only in this region that term by term integration 

 is legitimate. Furthermore we have assumed the possibility of ex- 

 panding h{t) in a power series: an assumption to which there are 

 serious theoretical objections, and which, furthermore, is not always 

 justified. A more satisfactory derivation, and one which establishes 

 the condition for the existence of a jwwer series expansion, proceeds 

 as follows : — 



F-et \/II(p) be a function which admits of the formal as\mptotic 

 expansion 



'^ajp" 



and let it include no component which is as\mptoticall>' represcntable 

 by a series all of whose terms are zero, that is a function 4>{p) such 

 that the limit, as p—x, of p"4>{p) is zero for every value of n. Such 

 a function is e'''. With this restriction understood, start with the 

 integral equation, and integrate by parts: we get 



]-=h{o)+ f" e-t"h^'\t)dt 



mp) 



where /»'"*(/) denotes d"/dl"h{t). Now let p approach infinity: in the 

 limit the integral vanishes and by virtue of the asymptotic expansion 



l/n(p)o<,\^a„'p", (36) 



\/II{p) a|)pr()achcs the limit cio- ("onse<iueiUly 



li{o) =ao. 

 ,\()\v integrate .i^ain b\- |)arts: we gel 



/>(l,7/{/>)-a„)=//<'>(o)-|-y c-'"li'-'{l)dl. 



