= Friedel density tail from a derivative singularity
{c}
{title2=$g''(q)\sim C/(q-Q)\quad\Rightarrow\quad\rho_{\rm ind}(r)\sim ZeC\pi\cos(Qr)/r^3$}
For the <radial induced-charge screening kernel>, integrate the sine transform twice by parts, with $g(0)=g(\infty)=0$ and vanishing remaining endpoint terms. This gives $\rho_{\rm ind}=Ze\int g''(q)\sin(qr)dq/r^3$. A <Cauchy principal value> singularity at $Q>0$ contributes $C\pi\cos(Qr)$ to that integral: set $s=q-Q$ and use $\int_{-\infty}^{\infty}\sin(sr)ds/s=\pi$ for $r>0$. Smoothly localizing around $Q$ justifies the large-$r$ extension of the limits. In a three-dimensional isotropic <Fermi gas>, $Q=2k_F$.
Back to article page
