The potential within a crystal lattice

The generalised Madelung problem, that of finding the spatial electric potential within a crystal lattice, is often approached via analytic function theory. But new results can be achieved through careful application of standard theorems from classical electro-statics. In this way we show that the celebrated Madelung constant for the NaCl crystal can be rigorously bounded through symmetry arguments devoid of summations. We derive new expansions for Madelung constants of general crystals: an absolutely convergent 'sine' series which has the advantage of non-alternating summands, and an 'exponential' series which agrees in the simple cubic cases with the 'cosech' expansions of previous authors. Finally, we show how the NaCl Coulomb singularity may be removed to yield a regular power series expansion for the potential well at the origin.