Show that expression (8.2.20) for the paramagnetic susceptibility of an ideal Fermi gas can be written in the form

\[\frac{\chi}{V}=\frac{N}{V} \frac{\mu^{* 2} \mu_{0}}{k T} \frac{f_{1 / 2}(z)}{f_{3 / 2}(z)}\]

Using this result, verify equations (8.2.24) and (8.2.27).