On the fast multipole method for computing the energy of periodic assemblies of charged and dipolar particles

In two dimensions, it is convenient to represent the coordinates (x, y) of particles as complex numbers z = x + iy. The energy of interaction of two point charges q1 and q 2 at points represented by the complex numbers z1 and z2 is then since the natural logarithm is the singular part of the Greens function for the two-dimensional Laplace equation. In performing molecular dynamics and Monte Carlo simulations of neutral systems of charged particles or of point dipoles, it is necessary to compute the energies and forces of an infinite periodic system in which the N charges or dipoles at the points z1 ... zn resident in the primary (usually square) simulation cell are replicated everywhere in the plane