1. (a) Show that an ideal gas of spinless fermions, the pressure is given by
$\beta P = \frac{1}{\lambda^3} f_{5/2}(z)$,
where $z = e^{\beta \mu}$, $\lambda = \sqrt{\frac{2\pi \beta \hbar^2}{m}}$, $m$ is the mass of the particle,
$f_{5/2}(z) = \frac{4}{\sqrt{\pi}} \int_0^{\infty} dx x^2 \ln(1 + ze^{-x^2}) = \sum_{\ell=1}^{\infty} (-1)^{\ell+1} z^\ell/\ell^{5/2}$,
And the chemical potential is related to the average density $\rho = \langle N \rangle/V$
by $\rho \lambda^3 = f_{3/2}(z) = \sum_{\ell=1}^{\infty} (-1)^{\ell+1} z^\ell/\ell^{3/2}$
(b) Similarly, show that the internal energy, $\langle U \rangle$, obeys the relation
$\langle E \rangle = \frac{3}{2} PV$
