Wang 



^a37ri/4 



- Ai[{-iKio) e §] \ .^j/4 e Ai[(-iKui) ^](^/x) 



Jooe 



1-3 

 [g^{iT) +B|g^(o-)] da| . (108) 



We note that -when t is in D| the, Airy function Ai[('^iKco) t] is 

 exponentially large and AiU-iKto)'^ e" '^] is exponentially small 

 Therefore, in order that f| ^ tend to zero as | t | — ► oo in D| , we 

 require that the coefficient AiK-iKui^^^i] be zero. This deter- 

 mines the constant ^| , or B| , 



— C -iKoicr^ 1/3 -27ri/3.. ^ _ 1/4 , ^ / 



B, = j e Ai[(-iKco) e i]{t/x) g^^^y) d^/ 



s 



s 



-iKwo-^ 1/3 -27ri/3'>', •?- /'- 1/4 . , _, ...., 



e Ai[(-lKco) e §](§/x) g^(o-) da. (109) 



With "S^i given by (109) i Z(t) becomes exponentially small when 

 T is in D|. We shall not attempt to evaluate "K| explicitly in this 

 paper, however, in view of (91) and (92) we may conclude that 



B, = 0[(Kco)^] . (110) 



Since L4 is outside the circular arc |(r| = R| , we may substitute 

 the expansions given in (97), (98) and (99) into the integral along L^ 

 to obtain 



4ir(Kco) T ^'ooe 



X cr^/4sinh[|e"*/^^(e"''-T'^] dcr. 



(Ill) 



Now, If we apply the method of integration by parts to the integral 

 along L4, we can show that for t in Dj , with Z(t) being expo- 

 netially small. 



^(w)^ eB,i -ia>t^-l ^ ^^^2) 



47r(Ka)) 



Summing up all the results obtained in (105), (107) and (112), we 



212 



