  We present two alternative complete sets of static modes of a homogeneous
dielectric sphere, for their use in the resonant-state expansion (RSE), a
rigorous perturbative method in electrodynamics. Physically, these modes are
needed to correctly describe the static electric field of a charge
redistribution within the optical system due to a perturbation of the
permittivity. We demonstrate the convergence of the RSE towards the exact
result for a perturbation describing a size reduction of the basis sphere. We
then revisit the quarter-sphere perturbation treated in [Doost {\it et al.},
Phys. Rev. A {\bf 90}, 013834 (2014)], where only a single static mode per each
angular momentum was introduced, and show that using a complete set of static
modes leads to a small, though non-negligible correction of the RSE result,
improving the agreement with finite-element simulations. As another example of
applying the RSE with a complete set of static modes, we calculate the resonant
states of a dielectric cylinder, also comparing the result with a
finite-element simulation.
