<p>We investigate the zero-temperature electron–phonon memory function of a three-dimensional Holstein-like system with the free electron-like bands. The memory function is a complex function of the incoming photon energy, and it describes the effect of electron–phonon scattering in the dynamical conductivity. In the presented approach, memory function is of the second order in electron–phonon interaction with the amplitude connected to the internal energy scales of the electron–phonon system and with imaginary and real part being connected by Kramers-Kronig relations. A closed-form expression is provided for the imaginary part of the memory function in the case when finite bandwidth, Fermi energy, and optical phonon energy are comparable to one another with several of its features closely investigated. Similarly, we analyze the properties of the real part of the memory function in a closed form whose main features are traced back to the properties of the imaginary part.</p>