Quadratic Density Response Function of a Two Dimensional Electron Gas

A closed-form expression of the retarded quadratic density response function of a two dimensional non-relativistic electron gas at zero temperature is written in terms of a non-analytic complex function. A careful analysis is made of the mathematical mechanism of exact static features such as vanishing, discontinuity and peaks. Special attention is paid to an ambiguity in treating the phase of the non-analytic complex function.