Assuming that the correlation function (gleft(boldsymbol{r}_{i}, boldsymbol{r}_{j} ight)) is a function only of the distance (r=left|boldsymbol{r}_{j}-boldsymbol{r}_{i} ight|), show that (g(r)) for (r eq 0) satisfies

Assuming that the correlation function \(g\left(\boldsymbol{r}_{i}, \boldsymbol{r}_{j}\right)\) is a function only of the distance \(r=\left|\boldsymbol{r}_{j}-\boldsymbol{r}_{i}\right|\), show that \(g(r)\) for \(r eq 0\) satisfies the differential equation

\[
\frac{d^{2} g}{d r^{2}}+\frac{d-1}{r} \frac{d g}{d r}-\frac{1}{\xi^{2}} g=0
\]

Check that expression (12.11.27) for \(g(r)\) satisfies this equation in the regime \(r \gg \xi\), while expression (12.11.28) does so in the regime \(r \ll \xi\).