Now add (7.31) and (7.32) defining the difference 



to get 



2 + <s) - Z + ( 3 ) - Y_(-e) , 



Ms) 



it* 



Pjt) dt 



o w f e f it; at 

 tG.(-s) -H + (s) ] + ^y + jL j K(t)(t+8) » . 



while if we subtract (7.31) from (7.32) ar.d define 

 S + (s) - Z + (s) + Y_(-s) 



we get 



S^(s) 



it*. 



S^(t) dt 



[-G_(-s) -H + (e)l + ^ - — J K _ (t)(t „ s) - 



so that (7.33) and (7.34) are a pair of similar uncoupled integral 

 equations. 



Let us now look at the forcing functions in these equations. 

 The additive split of G(s) in (7.21) is simple, giving 



G + (s) 



(s + k ) K (-k ) 

 v o - o 



G (8) - 



s + k K (s) K (-k ) 



For the function H(s) in (7.26) we have to use the Cauchy integrals, 

 which give 



-it* 



if i 



H + (8) " 2ui J (t + k ) K,(t)(t - s) dt 

 o + 



(7.33) 



(7.34) 



(7.35) 



(7.36) 



We cannot complete the contour with a large semicircle in R , because the 

 factor exp(-it£) is exponentially large there. Instead we complete the 



45 



,iii»»-j--L^ 1 l^i«:'»-:--ai.;T.vA J .a»v--....-....---^.... -j.i.. J -.. ; ^.,-..^--^^.^ 



