  We present analytical formulas for the vacuum-polarization Uehling potential
in the case where the finite size of the nucleus is modeled by a Fermi charge
distribution. Using a Sommerfeld-type development, the potential is expressed
in terms of multiple derivatives of a particular integral. The latter and its
derivatives can be evaluated exactly in terms of Bickley-Naylor functions,
which connection to the Uehling potential was already pointed out in the pure
Coulomb case, and of usual Bessel functions of the second kind. The cusp and
asymptotic expressions for the Uehling potential with a Fermi charge
distribution are also provided. Analytical results for the
higher-order-contribution K\"all\`en-Sabry potential are given.
