4. INVERSION OF THE FOURIER INTEGRALS 



From (3.12), the full-range transform of the scattered field y(x) 

 Is given by 



Y(s) - Y + (a) + Y_(a) , 

 and 



y(x) * ^ J Y(s) exp(-isx) ds 



where C runB from -»to +«°in the strip D. Since Y + (s) behave 

 algebraically at infinity in R +> the convergence of the integral is 

 dictated by the exp(-isx) factor. 



For x > 0, close the contour C with a large semi-circle ia the 

 lower half-plane. The contribution from this •semi-circle vanishes 

 as the radius becomes infinite because exp(-isx) is exponentially small 

 for x > and Im s < 0. (Again, a more careful proof of this should 

 really be given.) Further, inside the closed contour consisting of C 

 and the large semi-circle, Y_(s) ■>• analytic and by Cauchy's theorem 

 makes zero contribution to the integral. Y (s) does have singularities 

 in R , but these consist in fact of just a simple pole at s ■ -k . 

 Noting that the contour is described clockwise rather than counter- 

 clockwise, we have 



y(x>0) - — (-2iri) (Residue of Y + (s) ats- -kj) expU^x) 

 2ik 



- Texp(ikjX) 

 where the transmission coefficient is 



20 



