Equation (29) is the potential function of a unit source at the origin where K = 



2 

 6J /g. With a change in the integral path, the Green function can be expressed as 



Vy-^^-^\ 4^^^^ "« 



The contour of the integral path is around the pole from beneath. With a change in 

 the variables Equation (30) can be written as 



G^^ =- ^ Re (e^^+i^^E^ (Kz+iKy) } - ie^^+i^l^l (31) 



The derivation of Equation (31) is given in Appendix A. The exponential integral is 

 E (u) and is defined as 



00 



dt (32) 



The asymptotic properties of the two-dimensional Green function can be obtained from 

 the corresponding approximation of the exponential integral. Eor small values of Kr, 

 Equation (31) can be expressed as (see Appendix B) 



^2D = u [T+^n(Kr)-i^] (33) 



where y = 0.577... which is Euler's constant and (r,9) are polar coordinates such 

 that y = r sin G and z = -r cos 0. For large values of K|y| the asymptotic approxi- 

 mation of Equation (31) represents the outgoing two-dimensional plane waves in the 

 form 



„ . Kz+iK y ^ 1^1 I -.-^ 1 /-o/N 



G„ = -le '^' for K|y| » 1 (34) 



