Show that the velocity of sound \(w\) in an ideal Fermi gas is given by

\[w^{2}=\frac{5 k T}{3 m} \frac{f_{5 / 2}(z)}{f_{3 / 2}(z)}=\frac{5}{9}\left\langle u^{2}\rightangle\]

where \(\left\langle u^{2}\rightangle\) is the mean-square speed of the particles in the gas. Evaluate \(w\) in the limit \(z \rightarrow \infty\) and compare it with the Fermi velocity \(u_{F}\).