Friedel density tail from a derivative singularity ($g^{\prime \prime }(q)\sim C/(q-Q)\Rightarrow {\rho }_{\mathrm{ind}}(r)\sim ZeC\pi \cos (Qr)/r^3$)

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 ${\rho }_{\mathrm{ind}}=Ze\int g^{\prime \prime }(q)\sin (qr)dq/r^3$. 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, $Q=2k_F$.