The 1/r Coulomb potential is calculated for a two dimensional system with
periodic boundary conditions. Using polynomial splines in real space and a
summation in reciprocal space we obtain numerically optimized potentials which
allow us efficient calculations of any periodic (long-ranged) potential up to
high precision. We discuss the parameter space of the optimized potential for
the periodic Coulomb potential. Compared to the analytic Ewald potential, the
optimized potentials can reach higher precisions by up to several orders of
magnitude. We explicitly give simple expressions for fast calculations of the
periodic Coulomb potential where the summation in reciprocal space is reduced
to a few terms