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We present analytical expressions for the polarizability $P_\mu(q_x,\omega)$of graphene modeled by the hexagonal tight-binding model for small wave number$q_x$, but arbitrary chemical potential $\mu$. Generally, we find$P_\mu(q_x,\omega)=P_\mu^<(\omega/\omega_q)+q_x^2P_\mu^>(\omega)$ with$\omega_q=v_Fq_x$ the Dirac energy, where the first term is due to intra-bandand the second due to inter-band transitions. Explicitly, we derive theanalytical expression for the imaginary part of the polarizability includingintra-band contributions and recover the result obtained from the Dirac coneapproximation for $\mu\rightarrow0$. For $\mu<\sqrt{3}t$, there is asquare-root singularity at $\omega_q=v_Fq_x$ independent of $\mu$. For dopinglevels close to the van Hove singularity, $\mu=t\pm\delta\mu$,$ImP_\mu(q_x,\omega)$ is constant for $\delta\mu/t<\omega/\omega_q\ll1$.

Analytical expressions for the polarizability of the honeycomb lattice