Electrostatic energy of polygonal charge distributions

An asymptotic series for the electrostatic energy epsilon(1)(N) of an N-gonal charge distribution, i.e., a set of unit charges occupying vertices of a regular N-gon with a unit circumradius, is derived. Application of Pade approximants to truncations of this expansion produces compact approximate formulae capable of estimating epsilon(1)(N) with great accuracy. A closed-form expression for the energy of electrostatic interaction of two polygonal charge distributions is obtained from the respective Fourier series. The availability of this expression allows for a rapid calculation of the relevant energy with computational effort independent of the numbers of particles involved.