content_image/332312.jpg content_image/332311.jpg content_image/332310.jpg content_image/332309.jpg content_image/332305.jpg content_image/332306.jpg content_image/332307.jpg content_image/332308.jpg content_image/332304.jpg content_image/332303.jpg content_image/332302.jpg content_image/332301.jpg
We can derive the asymptotic expression as x → −∞ ( y → −∞ ) by using the following formula for the asymptotic behaviour of the hypergeometric function [26] 

<img src='content_image/332312.jpg'>

and obtain: 

where 

and A, B, C, D are given by: 

<img src='content_image/332311.jpg'>

<img src='content_image/332310.jpg'>

<img src='content_image/332309.jpg'>

<img src='content_image/332305.jpg'>

<img src='content_image/332306.jpg'>

<img src='content_image/332307.jpg'>

<img src='content_image/332308.jpg'>

Similarly we can derive the asymptotic form of lower component χ ( x ) from Eq. (5a): 

<img src='content_image/332304.jpg'>

Similarly for the solution in the positive region we have from Eq. (24): 

<img src='content_image/332303.jpg'>

Now we recall that z → 0 when x → ∞ and imposing the boundary condition of the scattering problem that in the ( x > 0 region) we only have a wave travelling to the right (only the transmitted wave) we find: 

and χ$_{R}$ ( x ) is found again through Eq. (5a) in terms of φ$_{R}$ : 

<img src='content_image/332302.jpg'>

<img src='content_image/332301.jpg'>

