IDEAL FILTERS 221 



where a Is a constant which determines the slope of the phase curve. 

 But it is well known that 



'^^"2«-2a (' _f\(' _J^' ^^^^ 



If, then, in (4) we choose 



fi = oi, ji = la, ' • ■ , fA= Aa, 



so that P becomes identical with the first A terms of (12), and if in 

 addition we choose our unit of frequency so that ^* (A -\- l)a = 1, 

 we readily see that in the transmitted range Q must equal 



(^-^^)(^-(rTWO 



Q = - r- \ / 7^ — ^ ' /<^' (1^^) 



1 --^-4 — ^ 111- -^ 



(l+a)VV' (l+3a)2 

 while by (9) and (11) in the attenuated range it must be given by 



1 „ inf \ 2V/ \ {A-r)'a\ 



a 



A'^a^ 



Expressed in terms of Gamma functions, (13) and (14) become ^^ 



r.(>)r(i^/)r(i±/) 



a / \ 2a / \ 2q: 

 and 



r.(±)p(ifi)r(X+J 



" This means that we express all frequencies in terms of the first critical frequency 

 of (12) which falls in the transition interval. 



" The necessary transformations may be found in Whittaker & Watson, "Modern 

 Analysis," 12.13, 12.15, and 12.33. 



