RESrOXSE Of KECriFlER TO SIGNAL AND NOISE 103 



which is correct for the first two terms in the Taylor series expansion near 

 G = 0. Therefore, when /*„ approaches zero as N approaches infniity, 



N 



a f ^ - 2 '-nz^i dz 



ooo.-.o = / = — w- \ Jo{PoZ)e " = i — 



IttJc z^ 



^-f [ Mr„,)e-" ■">''!! (24) 



Zir J c 2- 



'J'he contour integral cannot be replaced by a real integral directly ])ecaiise 

 the integrand goes to inhnity at the origin. However, since 



Jo{u) ^ _ Jiju) _ d^ Joju) . V 



u^ u du ti 



MPz) J,{Pz) d MPz) JiiPz) 1 d JoiPz) 



PV d{Pz) Pz PH"" P^dz 



(26) 



we can substitute (26) in the integral and perform an integration by parts 

 to give the result. 



7 = « f .-'■-^/^ \PoMPoz) ^ „.^^^^(p^,)] ^, 



7r Jo L 2 J 



by Hankel's formula.^ But it may be shown that (see Appendix II) 



iF^{hU-M)-e-'"'lo(^^ (28) 



iFi (i; 2; - u) = e-"" [h («/2) - h(u/2)] (29) 



Hence, 



7^«y/^"e----{/e(nV2irj4-|P; 



(30) 



[/c(nV2If^n) + Il{Ws/2]Vn)][ 



which is identical with the result of Section I, noting that a = \/Wn • We 

 point out that a resistance-capacity coupled amplifier will not pass this 

 component since there is no transmission at zero frequency. 



^ Watson, "Theory of Bessel Functions," p. 393. As pointed out by Watson, in a foot- 

 note, the difficulty with singularities at the origin could be avoided by expressing Hankel's 

 formula in terms of a contour integral instead of an ordinary integral along the real axis. 

 Tliis procedure would lead directly to the hypergeonietric function given in (11). 



