condensed matter - Yet another question on the Lindhard function

Here's another question concerning the Lindhard function as used in the physical description of metals.

First we define the general Lindhard function in the Random Phase approximation as $\chi(q,\omega)=\frac{\chi_{0}}{1-\frac{4\pi e^2}{q^2}\chi_{0}}$ where $\chi_{0}$ denotes the "bare" Lindhard function" excluding any feedback effects of electrons. A reference is Kittel's book "Quantum Theory of solids" Chapter 6, however he is using the dielectric function instead of the Lindhard function.

By writing $\chi_{0}(q,\omega)=\chi_{01}+i\chi_{02}$ as a complex number and using the relation $\frac{1}{z-i\eta} = P(1/z) + i\pi\delta(z)$ in the limit that $\eta$ goes to zero from above and P(..) denotes the principal value it is straight forward to see that the imaginary part of the lindhard function is zero except of the case when we have electron-hole excitations of the form $\hbar\omega=\epsilon_{k+q}-\epsilon_{k}$.

Now let's assume we study plasma excitations using the actual Lindhard function (not the bare one). These plasma excitations appear for small q outside of the electron hole continuum. Therefor we have a vanishing imaginary part of $\chi_{0}$ in this case and consequently since also a vanishing part of $\chi$ itself, since its a function of $\chi_{0}$ which is "real" except of the "$\chi_{0}$"-part.

However when deriving the plasma excitations using a small q expansion we obtain a none vanishing imaginary part of $\chi$ as given in equation (3.48) of the following document: http://www.itp.phys.ethz.ch/education/lectures_fs10/Solid/Notes.pdf

I don't see why this is consistent.

I'd be more than happy if you could help me.

Thanks in advance.