It's a weird question to me, too. Nevertheless, please refer to below formula.
$V(r_1,r_2)=\langle\int\varepsilon_\nu(R_1)\varepsilon_\nu(R_2)\frac{e^{i\omega|R_1-r_1|}}{|R_1-r_1|}\frac{e^{i\omega|R_2-r_2|}}{|R_2-r_2|}dS_1dS_2\rangle$
$V(r_1,r_2)=\int\langle|\varepsilon_\nu(R)|^2\rangle|R|^2\frac{e^{i\omega|R-r_1|}}{|R-r_1|}\frac{e^{i\omega|R-r_2|}}{|R-r_2|}dS$
These are derivation process of radio interferometry spatial coherence function.
It's the radio astronomical problem, but I thought it needed a closer approach to math.
$\varepsilon_\nu$ be an amplitude of electric field having a complex quantity.
You can consider $\varepsilon_\nu$ as the ergodic, stationary random process.
It seems to make some rough sense because the correlation $\langle\varepsilon_\nu(R_1)\varepsilon_\nu(R_2)\rangle$ will be a delta function since to uncorrelation $\langle\varepsilon_\nu(R_1)\varepsilon_\nu(R_2)\rangle=0$ where $R_1\neq R_2$
However, I don't think that have a infinite value where $R_1=R_2$.
And then, how can derive (1-4) from (1-3)?
Or am I wrong about anything?