Adds logarithm branch cut

paper.tex

@@ -107,6 +107,7 @@ Using the form described by Solyom\cite{SolyomV3},
 	\epsilon_{\mathrm{Lindhard}}(\vec{q}, \omega) = 1 + \frac{q_{TF}^2}{q^2}\frac{\displaystyle 1 + \frac{\omega + \flatfrac{i}{\tau}}{2 \vf q} \ln(\frac{\omega - \vf q + \flatfrac{i}{\tau}}{\omega + \vf q + \flatfrac{i}{\tau}})}{\displaystyle 1 + \frac{\flatfrac{i}{\tau}}{2 \vf q} \ln(\frac{\omega - \vf q + \flatfrac{i}{\tau}}{\omega + \vf q + \flatfrac{i}{\tau}})}. \label{eq:lindhardsolyom}
 \end{equation}
 Here, $q_{TF}$ is the Thomas-Fermi wavevector $q_{TF}^2 = 3 \flatfrac{\omega_p^2}{\vf^2}$, $\omega_p$ is the plasma frequency $\sqrt{\flatfrac{4 \pi n e^2}{m}}$, $\tau$ is the collision time and $\vf$ is the Fermi velocity.
+The branch cut for the logarithm is chosen here such that their imaginary parts lie between $\pm i \pi$.
 This can be shown to reduce to the Drude dielectric function in the $q \rightarrow 0$ limit.
 
 The Lindhard dielectric function reflects the important property of having an imaginary part that vanishes for $q$ such that $\abs{\varepsilon_q - \omega} >  q\vf$.