Show that the low-temperature specific heat of the relativistic Fermi gas, studied in Section 8.5, is given by

\[\frac{C_{V}}{N k}=\pi^{2} \frac{\left(x^{2}+1\right)^{1 / 2}}{x^{2}} \frac{k T}{m c^{2}} \quad\left(x=\frac{p_{F}}{m c}\right) .\]

Check that this formula gives correct results for the nonrelativistic case as well as for the extreme relativistic one.