  A new method of computing the Madelung constants for hypercubic crystal
structures in any dimension $n\geq 2$ is given. It is shown for $n\geq 3$ that
the Madelung constant may be obtained in a simple, efficient and unambiguous
way as the Hadamard finite part of the integral representation of the potential
within the crystal which is divergent at any point charge location. Such a
regularization method fails in the bidimensional case due to the logarithmic
nature of singularities for the potential. In that case, a specific approach is
proposed taking in account the scale invariance of the Poisson equation and the
existence of a finite horizon for each point charge in the plane. Since a
closed-form exact solution for the 2D electrostatic potential may be derived,
one shows that the Madelung constant may be defined via an appropriate limit
calculation as the mean value of potential energies of charges composing the
unit cell.
