A (u ,0) exp(-iu X cos 9) . . B (u, ,9) exp(iu,x cos 9) 



. r A \u ,<3) exp^.-lu X cos d; . . 

 ij — ,„ 1/2 ^^-i j 



1/2 "^"^ "J 1/2 "'■''^ 



[l-4(inK) ' cos 0] o [l+4(mK) ' cos 0] 



tt/2 

 f. B (u„,9) exp(iu_x cos 0) 



^ -i^-^ ^ --J— d0 for x > (D.8) 



^ J 1/2 ^/2 



o [l+4(mK) ' cos 9] 



A (iu,9) exp(ux cos 9) 



i^^\ 



2tt^ J J . r,^a/2^.. .1/2 „T^ 



o o iu-[(K) +i(m) u cos 0J 



tt/2 °° 



B (-iu,9) exp(ux cos 9) 



du 



+ -^ d9 ^ 2<^" 



o o iu+[(K) +i(m) u cos 9] 



T — — Tr/2 



A (u, ,9) expC-iu^x cos 9) . r k {u^ ,Q) expC-iu^x cos 9) 



n 1 1 ,„ . X [ n 1 1 ^g ^j^_g^ 



^ r k (u, ,9) expC-iu^x cos 9) . r 



■n J T ,„ 1/2 TT J 



1/2 "^'^ " -' 1/2 """^^ 



[4(mK) ' cos 9-1] 9 [l-4(mK) ' cos 9] 



for X < 



The first two double- integral terms in Equations (D.8) and (D.9) can be reduced 

 with substitution of u cos 9 = v (Reference 17) to: 

 1, n = 3 and x > for A„ = B = K 



63 



