Isotropic material fitted by Ornstein-Zernike form

[alan]

I have known what Ornstein-Zernike equation is. I try to plug in the form as follow to the isotropic materials:

Still, I cannot show the pair correlation function as follow.

Can anyone know what I have missed?

 

[Fred Wright]

I don't know if this will help but I get the result you are looking for by evaluating the integral,$$\frac{N_c}{4\pi^3r}\int_{-\infty}^{\infty}dk\frac{e^{-ikr}}{1+k^2l_c^2}$$
by closing the infinite semi-circle in the lower half of the complex plane and using the residue theorem.