Wu 



1 -1 



Aq(s) = -s f v(x,Oi,s) dx = s J e^C^+DWCx.O.s) dx . (29c) 



Thus, ¥ is known except for an additive constant term Ao(s). Furthermore, 

 from (20) it follows that 



$(x,Oi,s) = ReF(x + iO,s) = 0, (|x| >1) . (30) 



This Riemann-Hilbert problem, specified by (29), (30) and conditions (21) and 

 (22) can be readily solved, giving 



in which the function (z - 1) ^'^ (z + 1) *^^ is defined with a branch cut from 

 z = -ltoz=lso that this function tends to 1 as | z| ^ ». The leading- edge 

 singularity can be separated out in the above solution by suitable integrations 

 while using (29a), giving 



i /z - i\''' 1 r' /z2 - n''' ^,U'^) 



F(z,s)=.vs)--a,(s)(-^j ^-J^irrii) yrr^^' (3ia) 



where 



- /-^ W (^ s) 

 S-„(5) = A„(s) t i J — !— df . (31b) 



-. (,-^.)'" 



Now, substituting the value of $(x, 0,s) [for x < -1, which can be readily deduced 

 from (31a)] into the second- integral representation of (29c), then after some ap- 

 propriate integrations by parts, using the identity 



)^ ^ - X Bx ^ - X 



we determine the coefficient 3^(3) as 



ao(s) = ^ ( [^ - G(s)(l + f)] ^^^''^ d^ , (32a) 



"J 1/2 



where 



1182 



