Kramers-Kronig relations for the dielectric function and the static conductivity of Coulomb systems

The mutual influence of singularities of the dielectric permittivity e(q, ¿) in a Coulomb system in two limiting cases ¿¿0, q¿0 and q¿0, ¿¿0 is established. It is shown that the dielectric permittivity e(q¿0, ¿) satisfies the Kramers-Kronig relations, which possesses the singularity due to a finite value of the static conductivity. This singularity is associated with the "long tails" of the time correlation functions.