  A non-perturbative scheme, based on the functional generalization of the
Callan-Symanzik equation is developed to treat the Coulomb interaction in an
electron gas. The one-particle irreducible vertex functions are shown to
satisfy an evolution equation whose initial condition is given by means of the
classical action and the final point corresponds to the physical system. This
equation is truncated by expanding it in momenta and excitation energies,
leaving the electric charge as an arbitrary, not necesseraly small parameter.
Exact coupled partial differential equations up to first order in the
frequencies and excitation energies are derived. The numerical integration of
these eqations is left to a later stage. Nevertheless, in order to demonstrate
the relation with the perturbation expansion the one-loop Lindhard function and
screening are reproduced in the independent mode approximation of the evolution
equation.
