Find an analytical solution of the Maldague integral for the finite-temperature quadratic response

Derive an analytical solution for the Maldague integral that expresses the finite-temperature ideal quadratic response function in closed form for arbitrary wave vectors and frequencies, rather than relying on numerical integration.

Background

The Maldague formulation represents the finite-temperature ideal quadratic response as a weighted integral of the corresponding ground-state response over an auxiliary energy or chemical-potential variable. This approach is numerically advantageous because the ground-state integrand is directly related to a physical observable and does not diverge, although it can contain sharp features.

The paper uses numerical quadrature because an analytical evaluation of this integral was not obtained. An analytical solution would extend the closed-form ground-state results to finite temperatures and could improve both the theoretical characterization and computational treatment of the response.