Quadratic density response function of a two dimensional electron gas

A retarded quadratic density response function of a two dimensional non-relativistic homogeneous electron gas at zero temperature is obtained in a closed-form written in terms of a non-analytic complex function. A careful analysis is made of the mathematical mechanism of the 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.