Generation of potential/surface density pairs in flat disks

Aims.We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks. Methods.Potential/surface density pairs consist of a “homogeneous” pair (a closed form expression) corresponding to a uniform disk, and a “residual” pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical. Results.The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions (i.e. surface densities of the form $\Sigma(R) \propto R^s$), the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as $1/N^3$. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications