for large values of s. If one knows Y(s), one has then to decompose 

 Y(s) into Y (s) + Y_(s) and then look at Y (s) for large values of s, a 

 procedure explained In detail in [14]. 



In a similar way, the asymptoticsof y(x) as y ■* 0- are determined 

 by those of Y (-iv) as v -»■ + <*>, using Watson's lemma. 



Now in our problem we assuiae that the deflection y in the reflected 

 or transmitted wave is finite as x ■* Q±. Then the leading order term 

 in the expansion of Y (iu) is 



(Const) J exp(-ux) dx » (Const)u" 1 (3.10) 



and similarly for Y (-iv), so that Y (x) are each 0(s -1 ) at infinity in 

 R , respectively. K (s) each tend to 1 at infinity, while G (s) are 

 each 0(s -1 ). It then follows from (3.6) that E(s), which is the 

 polynomial P(s), is 0(s -1 ) at infinity in R , and from (3.7) that it 

 is 0(s -1 ) at infinity in R , and hence, because R have a common strip, 

 P(s) is a polynomial which vanishes everywhere at infinity like e' 4 . 

 Therefore P(s) must be identically zero, and the solution subject to 

 the condition of finiteness at the junction x •= is 



V 8 > " G + (s)/K + (s) 



Y_(s) - G_(s) K_(s) 

 and in explicit form thia gives 



(3.11) 



17 



i 

 i 



