Landau damping of collective excitation in a homogeneous Bose-Einstein condensate (BEC) is studied by the Hartree-Fock-Bogoliubov (HFB) mean-field theory. The rigorous derivation is carried out without using the two approximations of quasi-particle resonance transition and collective excitation energy. The Landau damping as a function of temperature is given in a wide parameter range, and two limits of absolute zero and critical temperature of phase transition are highlighted. The contribution of quasi-particle transitions with different energies to damping is analyzed by the error function. In addition, the above two approximations are used for derivation, and the applicable range of the approximations is analyzed by the results of the error function.