Obtain an analytical expression for the finite-temperature long-wavelength quadratic response

Derive an explicit analytical expression for the finite-temperature long-wavelength limit of the ideal quadratic response function, represented by the chemical-potential integral in Eq. (\ref{eq:k1_0_limit}), instead of evaluating the integral numerically.

Background

The paper evaluates the ideal quadratic density response function of three-dimensional non-interacting fermions at arbitrary temperature, frequency, and wave vector. In the long-wavelength limit, the authors use the Maldague method to transform the finite-temperature response into an integral over zero-temperature results weighted by the derivative of the Fermi–Dirac distribution.

For the limit in which one wave vector tends to zero, the resulting expression involves a one-dimensional integral over an auxiliary chemical-potential variable and the logarithmic factor characteristic of the zero-temperature response. The authors state that no explicit analytical expression for this finite-temperature integral is available and therefore evaluate it numerically. Finding a closed-form representation would provide a more analytic characterization of the zeroth-harmonic and long-wavelength response.