(3) 

 The integral limit of g, in Equation (D.31) is transformed from the variable 9 



that is between and 9 = cos"-"- {l/[4 (mK)-*"' ^] } . If 1/ [4(mK)''"''^] is larger than 1, 



(3) 

 9 becomes zero. Therefore, when K = 0, g, becomes zero. By adding Equations 



(D.36) and (D.37), g^ is given by 



- f H (-)- Y [2+ sign x] Y ( — 

 AoVm/ 4 ° o \ ra / 



(D.38) 



Finally, from Equation (114), G_(x,0,0) is given 



;3(x,0,0) =11^ 



\ (m) +(2+ sign X) Y^ [—) 



(D.39) 



2. Pure Oscillation, m = 



(1) 



Because R, , Equation (D.18), is zero, g, becomes zero. When m = 0, k.. = k_ = 



(2) 

 and k„ = k, = 0. Therefore, g, , Equation (D.22), is zero when x < 0. When 



X > 0, as the complex argument of the exponential function becomes infinitive in the 



(2) 

 first and second terms in Equation (D.29), g, becomes zero. The upper limit of 



(3) 

 the integral for g, in Equation (D,33) origined from 9 which becomes zero in this 



case. Therefore, there is no contribution from g, to g„ when m = 0. 



2 2 fl) 



With R = K, m = K + v , and m = 0, g^ "^ is 



(1) ^ _K 

 '3 ,3/2 



^2m^^ -— - 



TT J 1 



dv 



C 



2,1 J 



2 2 

 o (K +v ) 



1/2 



dv 



K_ 



27T 



(t^+1) 



Yjj exp(-Kt |x| ) dt 



-T [H (K X ) - Y (K X )] 



4 O O 



(D.40) 



72 



