For a 2-dimensional gas of $N$ nonrelativistic, non-interacting, spinless bosons, find the density of states $g(\varepsilon)$ in the neighbourhood of energy $\varepsilon$. [Hint: consider the gas in a box of size $L \times L$ which has periodic boundary conditions. Work in the thermodynamic limit $N \rightarrow \infty, L \rightarrow \infty$, with $N / L^{2}$ held finite.]

Calculate the number of particles per unit area at a given temperature and chemical potential.

Explain why Bose-Einstein condensation does not occur in this gas at any temperature.

[Recall that

$$\left.\frac{1}{\Gamma(n)} \int_{0}^{\infty} \frac{x^{n-1} d x}{z^{-1} e^{x}-1}=\sum_{\ell=1}^{\infty} \frac{z^{\ell}}{\ell^{n}}\right]$$