Editing Fermi gas (section)

===Exact solution for power-law density-of-states===
{{Cite check section|date=November 2020|reason=equations are inconsistent with other parts of the article.}}

Many systems of interest have a total density of states with the power-law form:
<math display="block">D(\varepsilon) = V g(\varepsilon) = \frac{V g_0}{\Gamma(\alpha)} (\varepsilon - \varepsilon_0)^{\alpha-1}, \qquad \varepsilon \geq \varepsilon_0</math>
for some values of {{math|''g''<sub>0</sub>}}, {{math|''α''}}, {{math|''ε''<sub>0</sub>}}. The results of preceding sections generalize to {{math|''d''}} dimensions, giving a power law with:
* {{math|1=''α'' = ''d''/2}} for non-relativistic particles in a {{math|''d''}}-dimensional box,
* {{math|1=''α'' = ''d''}} for non-relativistic particles in a {{math|''d''}}-dimensional harmonic potential well,
* {{math|1=''α'' = ''d''}} for hyper-relativistic particles in a {{math|''d''}}-dimensional box.
For such a power-law density of states, the grand potential integral evaluates exactly to:<ref>{{cite book |last1=Blundell |title=Concepts in Thermal Physics |date=2006 |publisher=Oxford University Press |isbn=9780198567707 |chapter=Chapter 30: Quantum gases and condensates}}</ref>
<math display="block">\Omega(T,V,\mu) = - V g_0 (k_{\rm B}T)^{\alpha+1} F_{\alpha} \left( \frac{\mu - \varepsilon_0}{k_{\rm B}T} \right),</math>
where <math>F_{\alpha}(x)</math> is the [[complete Fermi–Dirac integral]] (related to the [[polylogarithm]]). From this grand potential and its derivatives, all thermodynamic quantities of interest can be recovered.