On the fast multipole method for computing the energy of periodic assemblies of charged and dipolar particles (Q686623)

The terms of a series of complex numbers may be set in a two dimensional grid which extends to infinity in all directions; beginning at an origin, successive terms are placed in contiguous positions on a spiral path, leaving no unfilled places. A sequence of nested regions, which tend to include all points, may be traced in the grid (e.g. successive spiral regions including the partial sums of the original series, or concentric squares, etc.) The regions include points at which terms have been set and determine successive partial sums of a new series. If the original series is absolutely convergent, the sum of the new series is that of the old; if original convergence is conditional, the new sum depends upon the geometric configurations adopted. Successive terms of a double series may be set in a grid and treated similarly. The paper deals with a conditionally convergent double series in this way, the sum resulting from a particular ordering being derived.