One-electron Green functions are obtained and investigated in the framework of the s-f exchange model of a ferromagnetic semiconductor, at temperatures T much lower than the Curie temperature Tc. An exact procedure is developed for working out the series expansions of the Green functions in powers of the magnon occupation numbers. Using these expansions the spectrum shift at T<<Tc and the damping of the conduction electrons due to the magnons are calculated, the ratio of the s-f exchange parameter to the band width and the value of the magnetic ion spin S being arbitrary quantities. The vanishing of the effective electron-magnon interaction as the magnon wavevector tends to zero is proved rigorously. In addition, a perturbation theory involving the formal parameter 1/2S that supports the idea of the Hamiltonian involving an effective electron-magnon interactions is developed.