Wu 



under the assumption that G vanishes sufficiently rapidly at upstream infinity, 

 (81a) is transformed to 



__ 2/3-- (k^-a^csc^cp) 



G = S(z) 



(84) 



where 



Hence 



with 



k sin cp , k = k cos 



P _ _ 1 /3z-m I z 

 2m 



m(k,(p) = (k^- a2 csc\ + /3^) 



1/2 



(85) 



(86a) 

 (86b) 



the positive branch of m (so that m -> k as k -♦ + co) being understood. Regarding cp 

 to be real and k complex, m has two branch points at 



+ b , b = (a^ csc^cp 



■) 



(87) 



Hence b is real for a> /3. For a </3, is real or imaginary according as 

 sin^cp < or >aV/3^. Ordinary, a >> /? in most of the natural circumstances. We 

 shall only consider the case a> /3. A branch cut may be introduced between 

 k = -b and +b along the real k-axis. By inversion, we have 



G = 



-L e"^ Pe d(p e^*^^ ^^^^ 



4^' Jo Jo. 



cp -e)-m I z I kdk 



(88) 



where 



X = r cos 



y = r sin 



(89) 



and the path r is taken along the positive real k-axis except that it lies below 

 the branch cut from k = to k = b; this choice of r can be justified by the large 



time limit of a corresponding initial value 

 problem (see Wu (18)). The path of integra- 

 tion r may be further deformed for -(tt- 5) 

 < cp < 6* to proceed along the entire negative 

 imaginary k-axis from k - -iO to -ioo. For 

 61 < cp < (7/+ (9), r may be deformed to cir- 

 cumvent the branch cut from k = i around 

 the point k = btok=+iO, and then along the 

 positive imaginary axis to k = + ico (see 

 Fig. 7). 



( fl< ^<ir + e ) 



(r-e)<^<8 



Fig, 7 - Path of inte- 

 gration r 



At large distances from the singularity 

 (ar >> l), the asymptotic values of these re- 

 sulting integrals can be evaluated by the 

 method of stationary phase or by that of 



570 



