Sommerfeld expansion

Remarks

Integrate by parts with $K(\varepsilon) = \int_{-\infty}^{\varepsilon} H$ to get $\int K(\varepsilon) \, (-f_T'(\varepsilon)) \, d\varepsilon$. The weight $-f_T'$ is an even bump around $\mu$ of width about $k_B T$ and total mass $1$, so Taylor expanding $K$ at $\mu$ leaves only even moments. These moments are $\int_{\R} x^{2n} \frac{e^x}{(e^x + 1)^2} \, dx = 2 (1 - 2^{1-2n}) (2n)! \, \zeta(2n)$ after the scaling $x = (\varepsilon - \mu)/k_B T$; the parts of the integral far from $\mu$ are exponentially small. The series is asymptotic and need not converge.

Example: for the free electron gas, take $H(\varepsilon) = g(\varepsilon)$, the density of states. Keeping the first correction gives the chemical potential $\mu(T) \approx E_F \left(1 - \frac{\pi^2}{12} (k_B T / E_F)^2\right)$ and the electronic heat capacity $C \approx \frac{\pi^2}{2} N k_B \, \frac{k_B T}{E_F}$, linear in $T$.

Going deeper

For a student

The expansion is the standard tool for the degenerate Fermi gas at $k_B T \ll E_F$. One applies it twice: once to the particle number, to find $\mu(T)$, and once to the energy, to find the heat capacity. The coefficients $\pi^2/6$ and $7\pi^4/360$ are $2\eta(2)$ and $2\eta(4)$, where $\eta(s) = (1 - 2^{1-s}) \zeta(s)$ is the Dirichlet eta function. The first correction is usually enough, unless $H$ varies strongly on the scale $k_B T$ around $\mu$.