Isotropic material fitted by Ornstein-Zernike form

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alan
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I have known what Ornstein-Zernike equation is. I try to plug in the form as follow to the isotropic materials:
Equation S(k vector) = N_c divided by (1 + (k l_c)^2), where S is a function of the wave vector k, N_c is in the numerator and the denominator is 1 plus the square of the product k l_c.

Still, I cannot show the pair correlation function as follow.
Equation: g(r) minus n-bar equals one over (2 pi) cubed times the integral of d k-vector S(k-vector) exp(minus i k-vector dot r-vector), equals N_c times e to the minus r over l_c, divided by 4 pi l_c squared times r.


Can anyone know what I have missed?
 

Discussion

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.
 
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