Rigorous proof of those two expressions is presented in Appendix A.1. For particular cases of d = 1, 2, 3 (corresponding to typical 1D/2D/3D systems, see Fig. ), explicit formulations of the ( ) and ( ) are presented in Table . Visually, the response Lindhard functions are depicted in Fig. . It is worth noting that for very localized defects/impurities, the resultant Friedel oscillations are of identical shape like Lindhard functions.