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

The mutual influence of singularities of the dielectric permittivity ε (q, ω) in a Coulomb system in two limiting cases ω $\rightarrow$ 0, q $\rightarrow$ 0 and q $\rightarrow$ 0, ω $\rightarrow$ 0 is established. It is shown that the dielectric permittivity ε (q $\rightarrow$ 0, ω) 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.