These formulae treat the Debye model at all temperatures. The more elementary formulae given further down give the asymptotic behavior in the limit of low and high temperatures. The essential reason for the exactness at low and high energies is, respectively, that the Debye model gives the exact dispersion relation E ( ν ν ) at low frequencies, and corresponds to the exact density of states ( ∫ ∫ g ( ν ν ) d ν ν ≡ ≡ 3 N ) at high temperatures, concerning the number of vibrations per frequency interval.