154 BELL SYSTEM TECHNICAL JOURNAL 



Perhaps the simplest way of establishing the Heaviside Rule for 

 the asymptotic solution of the operational equation h = l/H(p) and 

 the conditions under which it is valid, is as follows: We start with the 

 integral equation 



r h(t)e~p'dt = l/pH(p) (15) 



Jo 



and specify that the singularities of llpH(p) and its derivatives are 

 all confined to the left hand side of the complex plane, except at the 

 point ^ = 0, in the neighborhood of which 



ao 



- + ai + a•2^1p + a^p + a^p^fp + • • • . (16) 



pH{p) ^jp 



In other words, l/pH{p) admits of expansion in powers of V/? 



Now since 



•M p-pt 1 



we have from (15) 



r h^dt= ^ (17) 



Jo 



r( 



"-^)'-"-pm-w <'«^ 



By virtue of the restrictions imposed on l/pH{p), equation (18) is 

 valid at ^ = 0, whence by (16) 



r("-s)*="- ''"' 



Now differentiate (18) with respect to ^; we get 



Now add H ^ ^ dt to the left of (20) and its value a2/2-yJp to 

 Jo 2 ^j^^t 



the right hand side; we have 



