RESPONSE OF RECTIFIER TO SIGNAL AND NOISE 



111 



f e'hix) dx = xe'ihix) - h{x)] 



f e-'Io(x) dx = xe-Vo(x) + L(x)] 



. \ (62) 



j e'hix) dx = e'[{l - x)h{x) + xh{x)] 



f e-^'hix) dx - ^-^(1 + x)Io(x) + xh(x)] 



These integrals may be derived by differentiating the right hand members, 

 and could, therefore, serve as a basis for an alternate derivation of (60). 



In addition it was noted in Eq. (11) that the constant term in the modula- 

 tion spectrum could be expressed in terms of iFi (—1/2; 1; —z); from the 

 equations given, it follows that we must have the relation: 



iFi (- 1/2; 1 ; -z) = e-'" [(1 + z) h{z/2) + z hiz/l)] (63) 



Another interesting set of formulas which can be obtained as a by-product 

 from (62) by setting x — iy is: 



/ /o(y) cosydy = >'[/o(y) cos y + /i(y) sin y\ 



I Joiy) sin ydy = y[Jo(y) sin y — Ji(y) cos y] 



I Jiiy) cos ydy = yJi{y) cos y — Jo{y){y sin y — cos y) 



I Ji(y) sin y dy = yJiiy) sin y + Jo(y)iy cos y — sin y) 



The hypergeometric notation is particularly convenient in determining 

 series expansions for the coefficients to be used for calculation when the 

 variable s is either very small or very large. For small values of z, the form 

 (54) suffices; for large values of s, we may use the general asymptotic expan- 

 sion formula for the real part of z positive : 



(64) 



iFiia; c; -z) = 



Tie) 



T{c — fl)2" 



iFoia, I + a — c; 1/z) 



r(c - a) 



r a(l + g 



L 1!2 



-c) 



(65) 



a(g+l)(l + a-c)(2 + a-c) 

 "^ 2!22 "^ 



Copson, "Functions of a Complex Variable," pp. 264-5. 



