  By using various properties of the complete elliptic integrals, we have
derived an alternative expression for the gravitational potential of axially
symmetric bodies, which is free of singular kernel in contrast with the
classical form. This is mainly a radial integral of the local surface density
weighted by a regular "mean Green function" which depends explicitly on the
body's vertical thickness. Rigorously, this result stands for a wide variety of
configurations, as soon as the density structure is vertically homogeneous.
Nevertheless, the sensitivity to vertical stratification | the Gaussian profile
has been considered | appears weak provided that the surface density is
conserved. For bodies with small aspect ratio (i.e. geometrically thin discs),
a first-order Taylor expansion furnishes an excellent approximation for this
mean Green function, the absolute error being of the fourth order in the aspect
ratio. This formula is therefore well suited to studying the structure of
self-gravitating discs and rings in the spirit of the "standard model of thin
discs" where the vertical structure is often ignored, but it remains accurate
for discs and tori of finite thickness. This approximation which perfectly
saves the properties of Newton's law everywhere (in particular at large
separations), is also very useful for dynamical studies where the body is just
a source of gravity acting on external test particles.
