third integral, we evaluate the complex integral in the fourth quadrant. The ex- 

 pression for g_ similar to Equation (D.5) is given by 



tt/2 



K exp(vx cos 9) dv 



t^S-i 



'3 2 1 2 



2u J J . r.„a/2^., .1/2 ./ 

 o o iv-[(K) +i(m) V cos 8J 



TT/ 2 °° T — 



r C -v t cx\ A 1 r K expC-iu^x cos G) 



+ _1^ f de iK expCvx cos 9) dv ^_1 ^ 1 ^^ 



■)!?■] J M-) 1/9 2 IT J , 1/2 



o o iv+[(K) '^+i(m) ' v cos 9] o [4(mK) ' cos 9-1] 



-■[ 



■'T/2 V f ■ Q\ 



K exp(-iu X cos 9) 



+ f I ^ de (D.6) 



'9 [l-ACmK)-"-^^ cos 9] 



T 



For g,, the same procedure can be applied. But, the numerator, K, should be 



1/2 1/2 



replaced by -i[mu cos 9+2 (mK) ] or i[mu cos 9-2 (mK) ] with appropriate variables 



and poles. 



If we let 



'3=^ 1/2 (°-7> 



A, = -i[mu cos 9+2 (mK)'] 



1/2 



B, = i[mu cos 6-2 (mK)'] 



we can express g for n = 3 and 4 as follows 



tt/2 <^ 



A^(-iu,9) exp(-ux cos 6) 



du 



■--t^\/'[~I. 



2 

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



Tr/2 «> T — — 



r r -^ (iu,9) exp(-ux cos 9) du C k (u ,9) exp(-iu x cos 9) 



"" 2? 1 '^ I iu-[(K)l/2_ 1/2^^^^ - 7 J 1/2 1/2 '^ 



o [4(mK) cos 9-1] 



(D.8) 

 (cont.) 



62 



