726 BELL SYSTEM TECHNICAL JOURNAL 



Now define an auxiliary function £;(/) l)y the operational ciiuation 

 1 



Then 



1 



kprf'^'^'- 



pi hip) pILiip) Jo 

 and by Borel's llieoreni 



£,(/)= f ll,{T)h;{t~T)dT 



"'(it 



= ffhirVnit- 



■V/7 



From this equation it follows that s{o)=o, and hence coniparing the 

 ojierational equations for /; and », we ha\'c In' aid of Tlu-orcm III 



HD^git) 



and hence 



m4Jjnir)hU-r)dr 



= U"^-^'^"^^'-'^'' 



This theorem is extremely important, although not stated or 

 employed b}' Heaviside himself. We shall make use of it in estab- 

 lishing two important general theorems and shall have frequent 

 occasion to employ it in specific problems occurring in connection 

 with the subsequent discussion of transmission theor\'. 



Theorem V 



If li = li{t) and g = g(t) are defined by the operational equations 



*" i/(/>+X) 

 where X is a positive real parameter, then 



g{t) = {l+\£dt)c-^'h{l). 



