Softened potentials and the multipolar expansion

When the gravitational potential is developed in a multipolar series, each multipole is well defined and corresponds to a finite sum of terms in the series. In order to use the gravitational potential in numerical simulations, however, a multipolar expansion is usually applied to a softened Newtonian potential. It turns out that the commonly used multipolar expansion in this case no longer isolates each multipole as in the former case; instead, each multipole is spilled over an infinity of terms. In this paper we show how to recover the complete multipoles. Fortunately, the overall effect of using incomplete multipoles instead of complete ones turns out to be negligible in the cases of interest, for example, in its use in tree codes.