Many systems of interest have a total density of states with the power-law form:
for some values of , , . The results of preceding sections generalize to  dimensions, giving a power law with:
*  for non-relativistic particles in a -dimensional box,
*  for non-relativistic particles in a -dimensional harmonic potential well,
*  for hyper-relativistic particles in a -dimensional box.
For such a power-law density of states, the grand potential integral evaluates exactly to:
where  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.