134 BELL SYSTEM TECHNICAL JOURNAL 



indefinitely? From (18) and (19) 



sit) = lim ■;^. f f( t-^, - 7^) e^'dz. (29) 



We may write 



s{t) = :r^. f [F/(l - w)>^'(/5 - lim ^. f [Fw"+'/(l-w)>^'(/z (30) 



provided these functions exist. Let them be called go(^) and lim g„(/) 



n — »-oo 



respectively. Then 



qn{t) = r qo{t-\)0i\)d\. (31) 



^ — 00 



where 



d(X) =^. f w^+h'^dz. (32) 



I TTl J,+ 



By the methods used under the discussion of convergence above it can 

 then be shown that this expression exists and approaches zero as w 

 increases indefinitely provided qo(t) exists and is equal to zero for ^ < 0. 

 Equation (29) may therefore be written, subject to these conditions 



sit) = :^ r LF/il - ■w)-]e^'dz. (33) 



In the first place the integral is zero for negative values of / because 

 the integrand approaches zero faster than the path of integration 

 increases. Moreover, 



X 



[F/(l - w)~\e''dz (34) 



exists for all values of / and approaches zero for large values of t if 

 \ — w does not equal zero on the imaginary axis. Moreover, the 

 integral 



[7^/(1 - w)']e''dz (35) 



X 



exists because 



1. Since Fand iv are both analytic within the cur\e the integrand does 



not have any essential singularity there, 



2. The poles, if any, lie within a finite distance of the origin because 



w -^ as 1 s I increases, and 



3. These two statements insure that the total number of poles is 



finite. 



