Many physical phenomena — from electrical conductivity to optical properties of materials — can be modeled mathematically by a differential operator acting on a periodic structure. A key quantity associated with such operators is their density of states (DOS): a measure that describes how eigenvalues are distributed along the real line. Computing this quantity accurately is surprisingly challenging. The spectrum is continuous, the underlying domain is infinite, and naive approaches either require heavy computational resources or introduce systematic errors that are hard to control. In this talk, we compare several numerical methods for tackling this problem, from classical strategies to a recently proposed approach exploiting complex analysis — the Brillouin zone complex deformation — which achieves exponential convergence under the right conditions. We identify precisely when each method shines and when it fails, offering a practical guide for numerical computation of densities of states.