REGENERATION THEORY 133 



CONVERGEXCE OF SERIES 



We shall next prove that the limit s{t) exists for all finite values of /. 

 It mav be stated as of incidental interest that the limit 





S^{z)e^''dz (20) 



does not necessarily exist although the limit s{t) does. Choose Mq and 

 N such that 



|/o(X)|<iVo. 0<X</. (21) 



\G{t-\)\^N. 0<X</. (22) 



We ma>' write * 



Similarly 



/:(/)= I G{t - \)f,{\)d\. (23) 



I/i(/) I < r .1/o.VJX = IfoA^. (24) 



Mt)= r G(t - \)f,{\)d\. (25) 



t/ — 00 



1/2(0 1 ^ r Af^rndt = Momyii (26) 



!/„(0|<MoiV«/"/w! (27) 



|5„(/) I < Moil + Nt+ '-- NH-ln^). (28) 



It is shown in almost any text ^ dealing with the convergence of 

 series that the series in parentheses converges to e^"'' as n increases 

 indefinitely. Consequently, Sn{t) converges absolutely as n increases 

 indefinitely. 



Relation Between s{t) and w 



Next consider what happens to s{t) as / increases. As / increases 

 indefinitely s{t) may converge to zero, indicating a condition of 

 stability, or it may go beyond any value however large, indicating a 

 runaway condition. The question which presents itself is: Referring to 

 (18) and {19), what properties of w(z) and further ivhat properties of 

 AJ(ibi) determine whether s{t) converges to zero or diverges as t increases 



^G. A. Campbell, "Fourier Integral," B. S. T. J., Oct. 1928, Pair 202. 

 ^ E.g., Whittaker and Watson, "Modern Analysis," 2d ed., p. 531. 



