Alternatively, Y,(s) can be regarded as the full-range transform of the 

 generalized function y(x) H(x), 



Y (s) = 1 y(x) H(x) e;.p isx dx 



(2.11) 



and correspondingly 



J-00 



Y_(s) =1 y(x) H(-x) exp isx dx , 



(2.12) 



H(x) denoting the Heaviside unit function, H(x) = 1 for x > 0, H(x) • 

 for x < 0. 



The FOURIER INVERSION THEOREMS run as follows:- if C + is any contour 

 from -°°io + 00 lying within the domain R and such that Y (s) exp(-isx) 

 is integrable over this contour, then 



*U 



(s) exp(-isx) ds = y(x) (x > 0) 



both of which are contained in 



y(x) H(x) 



- f v 



(x < 0) 



s) exp(-isx) ds 



(2.13) 



Tht fact that the integral vanishes for x < follows from Cauchy's 



theorem applied to a closed contour Y consisting of the contour C with 



its ends joined by a large circular arc in R, . Since the integrand is 



+ 



analytic everywhere in R 



Y (a) exp (-isx) ds = 



and the integral alonp, the circular arc vanishes if x < 



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