  The "point mass singularity" inherent in Newton's law for gravitation
represents a major difficulty in accurately determining the potential and
forces inside continuous bodies. Here we report a simple and efficient
analytical method to bypass the singular Green kernel 1/|r-r'| inside the
source without altering the nature of the interaction. We build an equivalent
kernel made up of a "cool kernel", which is fully regular (and contains the
long-range -GM/r asymptotic behavior), and the gradient of a "hyperkernel",
which is also regular. Compared to the initial kernel, these two components are
easily integrated over the source volume using standard numerical techniques.
The demonstration is presented for three-dimensional distributions in
cylindrical coordinates, which are well-suited to describing rotating bodies
(stars, discs, asteroids, etc.) as commonly found in the Universe. An example
of implementation is given. The case of axial symmetry is treated in detail,
and the accuracy is checked by considering an exact potential/surface density
pair corresponding to a flat circular disc. This framework provides new tools
to keep or even improve the physical realism of models and simulations of
self-gravitating systems, and represents, for some of them, a conclusive
alternative to softened gravity.
