Approximate solution for the gravitational potential of thin disks

We are interested in the derivation of reliable formulae for the gravitational potential of disks, under the assumption of axial symmetry. As a consequence of the Newton's law, the formula contains a diverging kernel which is always difficult to manage. In the particular case of vertically homogeneous bodies, we have built an equivalent kernel which is free of singularity, and therefore, well suited to numerical computations. From this new expression, in the case of geometrically thin disks, we have derived a good approximation. This formula reproduces the potential inside the body with a relative error as low as 10^{-3} typically. We will see, through various torus shapes, that our approximation can be used for thin and relatively thick disks