130 BELL SYSTEM TECHNICAL JOURNAL 



The expression on the left can be written 



1 



V;coV2X + joi 



(47) 



Noting that the square of the first factor has the form of an inductance 

 and the second the form of a resistance and inductance in series we 

 replace 



JO) -> i?Oi ( J ) where i?o, -> «= ; -Pi -> 



i?o.Pi = ico 



/ 1 _ g-2P2 \ 



and 2X + jw -> R^^ I ^ 1 where i^o^ -^ «^ ; Po -> ; and 



P02P2 = 2X + jw. We note that Pi has the form jwD where R^p = 1, 

 while P2 has the form A -j- jojD where Ro^A = 2X and Ro^ = 1. 

 Substituting these values in (47), we have 



1 



Expanding these two factors by the binomial theorem, we have 

 2 



r 1 + 1 e-;(2a,i» _^ ^/^ ^^ ^1^ g-;(J<-D) 



22 "(w!) 



r . 1 



X 



_ (2w) !g-^(2""-p) __ 1 



1 C2wVg-2n(4+;cu7)) 



2 ' ' 22"(«!)2 



At this point we make use of Stirling's theorems on factorials which 

 states that when K is large 



The typical terms of the above expression become 



(9^\2n 



