Eggers 



A. Gxx+ Gyy+ ^zz' provided the corresponding v integrals exist, which is 

 guaranteed for I X - <f ! > ; 



B. 70G2+ Gxx= for Z= 0; 



C. Gy= for Y= ±T; 



D. G^-* as z->-oo; 



F. G = 0(X- <f)" ^ as X->+co, G= 0(1) as X-»-co, G^ = 0(X- ^")' ^ as X-»+co , 

 Gx= 0(1) as X--CO. 



Since the structure of G is symmetric, corresponding relations can be obtained 

 under exchange of x, Y, and z with t , -n, and i. 



It remains to be shown that (a) the expressions for G and G^ match in a 

 continuous way at x = ^ and (b) for \y - r]\ <t, Z<0, and C < 0, G and G^ be- 

 come singular only for x= <f , Y = ^7, and Z- i, , and that the functions 



G = G + l/r 



and 



with 



Gx+ (l/r)<. 



(A3a) 



(A3b) 



,- 1/2 



r = [(X-^)2 + (Y-7i)2 + (Z- O']' 



remain finite here (convergence of the series for ^ ^ Z is shown in Ref. 6. 



Statement (a) is evident for the function G. Assume for simplicity <f = 0, 

 Then it is sufficient to show that 



K K^ro(z+o 1 



-rr- e = Iim — 



M., v^ r^ W 



•Kyn X 



V COS (Vy^Z) - U^ sin(V>oZ) 



- U^ sin (V/oO 



dV 



J U^ + V 

 for arbitrary u^ > and z^ i with 



U = yV^77J7 , M, = /mUJ , K^=(l + M,)/2 



On the right-hand side we may substitute 



V cos (V7oO 



(A4a) 

 (A4b) 



670 



