222 BELL SYSTEM TECHNICAL JOURNAL 



Asymptotic Series for P and Q 

 If we now take the logarithm of (15), and apply Stirling's formula, 

 we obtain the asymptotic series 



, ^ , 4^/"r^(i/2«) , 1, ,. „, 



" (- 1)^(2^- - i)^,-a^'-i r 1 1 



+ h 2ri2r - 1) [ (1 + fY"' (1 " ff-' 



(17) 



where the 5's are Bernoulli numbers. Since Stirling's formula holds 

 only for s > 0, this expansion is valid, as inspection of equation (15) 

 will show, only for/ < 1, but as the unit of frequency was so chosen 

 that /i < 1 < Jb, this includes the entire wanted band, and none of 

 the attenuating range. 



If we apply a similar process to (16) we are again led to (17), except 

 that now the range of validity is/ > 1. But this includes the entire 

 attenuating range, and none of the wanted band. 



That is, the single formula (17) represents the lacunar ^^ function 

 Q in both ranges in which it is well defined. 



We shall now determine the transition factors ga+i, 0^+2, • • • , a<; by 

 comparison of (8) with (17). If we adopt the notation 



/a+1 = 1 + Ci, /4+2 =1+^2, • • • , /c = 1 + Cm, (18) 



SO that the c's measure, not the critical frequencies themselves, but 

 their displacement from unit frequency, each factor of (8) has the 

 characteristic form 



(1 -^af (1 +c,f\ ' I -f/\ '1 +//' 



whence (8) becomes 



1+.^. l+r4-. •••^l+T^.^l+ ' 



^ ~ ^"f ; — w "r — ^ — ; — ~7 — w ~r — \^^ - ^• 



1 -fJ\ 1 +// V 1 -//\ 1 +f) 



(19) 



where X" is a constant multiplier which depends on the c's. We will 

 neglect it in this analysis since it may be readily determined later from 

 the condition that Q = I when / = 0. 



'* A lacunar function is one which is well defined in several regions, but not capable 

 of analytic continuation from one to the other. 



