144 BELL SYSTEM TECHNICAL JOURNAL 



and that the path of integration C runs along the real axis from — oo to oo 

 with a downward indentation at the origin. The integral (4.9-6) for hnk 

 becomes 



•n+k—v—l /• 



= -^^ r(. + 1) u'—'jn{Pu)i 



ZTT J C 



:n-\rk—v—\ 



ink = — ~ 1 ir -r J.; ; 11 Jn\i"ii)e " " du 



(I \(i'-A;)/2 

 - i , \ . ^,J^±^.,n+V,-.) (4.10-5) 



P' 



where the integration has been performed by expanding J„(Pii) in powers 

 of u and using 



l 



e "" u'^ ^ du = ie ^^"^ a ^ sin Xxr(X) 



^ (1 - e-'-'nviX) (4.10-6) 



tire 



a^T{l - X) 



it being understood that arg w = on the positive portion of C. 

 From (4.9-9), the dc component of / is 



P-p'-i-'r--) 



hoo= ^/ ' \ (^") i/^i(-^;l; --t) (4.10-7) 



2r 



which reduces to the expression (4.2-3) when v = 1 for the Unear rectifier 

 (aside from the factor a). 



When the input (sine wave plus noise) is confined to a relatively narrow 

 band, and when we are interested in the low frequency output, consideration 

 of the modulation products suggests that we consider the difference products 

 from the products of order (0, 0), (0, 2), (0, 4), • • • (1, 1), (1, 3), • • • (2, 0), 

 (2, 2), • • • etc. where the typical product is of order (n, k). The orders 

 (0, 0) and (2, 0) give the dc and second harmonic and hence are not con- 

 sidered in the computation of Wdf). Of the remaining terms, either (0, 2) 

 or (1, 1) gives the greatest contribution to the series (4.9-12) and (4.9-10) 

 for Wcif) and "^dr). The remaining terms contribute less and less as n and 



