Deriving relations for a hard sphere phase diagram

[Sat D]

Homework Statement

In Torquato's Random Heterogeneous Materials, he has written
$\frac{p}{\rho kT} = 1+2^{d-1}\eta g_2 (D^{+}) = 1+2^{d-1}\eta [c(D^+)-c(D^-)]$
which he arrives to using the Ornstein-Zernike equation.
How does he reach this conclusion?

Relevant Equations

where $g_2(D^+)$ is the contact value from the right-side of the radial distribution function, and $\eta$ is a dimensionless reduced density.

Ornstein-Zernike states that
$h(r_{12}) = c(r_{12}) + \rho \int d\mathbf{r}_3 c(r_{13})h(r_{32})$
which after a Fourier transform becomes
$\hat{C} (\mathbf{k}) = \frac{\hat{H}(\mathbf{k})}{1+\rho \hat{H}(\mathbf{k})}$

However, I don't see how to simplify this to the second equation he has. I would appreciate any advice you have.

 

[NateD]

The second equation is obtained by taking the inverse Fourier transform of the first equation. This process involves expressing the integrand in terms of a convolution. Thus, we obtain the second equation:$h(r_{12}) = c(r_{12}) + \rho \int d\mathbf{r}_3 \hat{C}(\mathbf{k}) * c(r_{13})h(r_{32})$