134 



BELL SYSTEM TECHNICAL JOURNAL 



The value of (57), then reduces to 



AJP\m i=n ^m—\ 



Ro J ti (m - 1) ! 



(59) 



If now we substitute n = t/2D, P ^ juD; RoP = j^L (59) reduces 

 to the integral 



R-J ^wri ~~' ^^^' 



(w - 1) ! 

 X 



=(vfe)"^5^r'"^"^"" 



t[i.ii<Lc)+Mdt. (60) 

 If we integrate (60) by parts, successively, there results the series 





1 + jcoVLC 



-Ul/ViC+J'^) 



+ 



VVIc/ 



1 



(I+jcoVLC)' 



(l+icoVLC)'«-i 



+ 



Vlc 



(m- 1)!(1+7coVLC) 



(61) 



The complete solution of (55) is then obtained by adding terms of 

 the kind given in (61). For example the steady state term is given 

 by the series 



ml' 



+ 



2! 



(1 + jo^^LCy^ (1 + >a/LC)— 1 (1 + coVLC) 



+ (-1)^ 



2- _ m2-^-\\ +jo:^LC) + '^^''\, ^^ 2--2(l + JcoVLC:)^ 



2! 



+ ••• + (- 1)"'(1 +JcoVLC)' 



(1 + jc^i^Lcy^ 



This is readily seen to be the binomial expansion of 

 2 - (1 +iwVIC) 



[^ 



+ joi-^LC 



_ [ 1 -jWLC I' 



[r 



(62) 



The other terms given in equation (38) follow directly by addition 

 and reference to equations (55) and (61). 



