Notes on calculating temperature of the superconducting transition: interacting fermi gas and phonon mechanisms in non-adiabatic regime

We analyze the mathematical structure of equations for temperature Tc of the superconductivity transition in a gas of interacting Fermi particles or at the phonon-mediated pairing in a metal in case of non-adiabatic conditions, i. e., when the characteristic phonon frequency is comparable or larger than the Fermi energy . The integral equations for Tc are derived in the logarithmical approximation. The equations contain no divergent terms in the anti-adiabatic limit.