Degenerate-plasma dielectric response ($ϵ=1+\frac{3{\omega }_p^2}{k^2{v}_F^2}\left[1-\frac{s}{2}\log \frac{s+1}{s-1}\right],s=\frac{\omega }{k{v}_F}>1$)

For an isotropic completely degenerate nonrelativistic Fermi gas in classical-trajectory collisionless kinetic theory, the parallel marginal distribution is $F(u)=3n(v_F^2-u^2)1_{|u|<v_F}/(4v_F^3)$. Integrating $F^{\prime }(u)/(u-\omega /k)$ yields the electron dielectric function in the title for fixed ions. Its real roots outside the occupied interval have no Landau damping. Quantum recoil, collisions and finite-temperature tails require an enlarged model.

The short-wavelength root of the degenerate-plasma dielectric response, approaching the edge of the occupied parallel velocity interval from outside. Writing $s=1+\delta$ makes its logarithmic singularity give the exponentially small positive correction in the title. It is the high-wavenumber limit of the charged Langmuir wave branch in this approximation; neutral zero sound has a different interaction and long-wavelength limit.