On exact summations in long-range interactions

In this work, simple exact results are presented for summations in two-particle potential with long-range interactions. Polygamma function is used to evaluate summations. Results are found when a periodic media is consider. Periodic boundary conditions are applied by symmetric repetitions of a central cell. Contributions over all space are used in the problem. The potential depends strongly on the size of the system; however, forces are convergents anywhere.