Sum of Fermi-Dirac integrals with opposite chemical potentials: closed form (Le Bellac eq. 1.13)

Question

I am trying to reproduce the result of eq. (1.13) in Le Bellac's Thermal Field Theory book to compute the grand canonical potential of a gas of massless fermions:

$$ Ω = - \frac{V T^4}{6 π^2} \int_0^\infty dk\ k^3 \left[ \frac{1}{e^{\beta(k - \mu)} +1 } + \frac{1}{e^{\beta(k + μ)} + 1} \right] = -\frac{V}{6π^2} \left[ \frac{7 π^4 T^4}{60} + \frac{\mu^2 \pi^2 T^2}{2} + \frac{\mu^4}{4} \right]. $$

I know how to demonstrate that the Fermi-Dirac integral can be expressed in terms of $\Gamma$ function and polylogarithms if $\mu = 0$, see e.g. this answer for the Einstein-Bose statistics, showing that $$ \int_0^\infty dx \frac{x^{s-1}}{e^x - 1} = \zeta(s) \Gamma(s), $$

but I am not able to derive a similar relation for the case with $\mu \neq 0$.

Nevertheless this relation exists, see e.g. the DLMF:

$$ F_s = \frac{1}{\Gamma(s+1)} \int_0^\infty dt \frac{t^s}{e^{t-x} +1 } = - Li_{s+1}(-e^x). $$

Answer 1

As commented, this is an instance of the reflection formula $$-\operatorname{Li}_n(-e^{a})-(-1)^n\operatorname{Li}_n(-e^{-a})=\frac{(2\pi i)^n}{n!}B_n\left(\frac12+\frac{a}{2\pi i}\right)\qquad(|\Im a|<\pi)$$ mentioned also here, with a proof sketch of a more general result.

Perhaps more elementarily, we get a recurrence for $I_n(a)=\int_{-\infty}^\infty z^n f(a,z)\,dz$, where $$f(a,z)=\frac1{e^{z+a}+1}+\frac1{e^{z-a}+1}-\frac2{e^z+1}\qquad(a\in\mathbb{R})$$ (note that $f(a,z)=-f(a,-z)=f(a,z+2\pi i)$, so that $I_n(a)=0$ for even $n$), using $$\int_{-\infty}^\infty\big(z^n-(z+2\pi i)^n\big)f(a,z)\,dz=\pi i\left((\pi i)^n-\frac12(\pi i+a)^n-\frac12(\pi i-a)^n\right),$$ which is an application of the residue theorem. (But not a closed form this way.)