122 BELL SYSTEM TECHNICAL JOURNAL 



is 



—iuB 



F{iu) = — - 



Consequently, the low frequency output is 



A^{R) = -J- I e~""'Jn{uR)u'Uu 

 2tt J-oo 



where the path of integration is indented downwards at the origin. When 

 B > R the value of the integral is zero since then the path of integration 

 may be closed in the lower half plane by an infinite semi-circle This value 

 also follows at once from the physics of the problem. When —R<B<R 

 we may integrate by parts and get 



Ao{R) = ^ [ e-'^'^iiBJoiuR) + RJr{uR)]u~^ du 



B 1 r°° 



= —- + -/ [5 sin uBJo{uR) + R cos iiBJ i{uR)]u~^ du 



2 IT Jq 



B .B . B ,1 /- 



— - + - arc sm - + - V^" — B- 



Z IT K TT 



(4.3-14) 



R 



This hypergeometric function turns up again in equation (4.7-6). Also 

 in the range —R<B<R, 



dR xV R'' 



When B is negative and R < —B, the path of integration may be closed 

 by an infinite semicircle in the upper half plane and the value of the integral 

 is proportional to the residue of the pole at the origin: 



■■(4) 



Ao{R) = 2wi[ -— i-iB) 



= -B 



Thus, to summarize, the low frequency output for our linear rectifier is, 

 for B > 0, {R is always positive) 



AoiR) = 0, R < B 



B B B I / (4.3-15) 



AoiR) = -^ + - arc sin ^ + -^ VR^ - B\ B < R 



2 IT R IT 



