  We extend the self-consistent Hartree-Fock-Popov calculations by Nikuni et
al. [Phys. Rev. Lett. 84, 5868 (2000)] concerning the Bose-Eistein condensation
of magnons in TlCuCl3 to include a realistic dispersion of the excitations. The
result for the critical field as a function of temperature behaves as
Hc(T)-Hc(0) T^{3/2} below 2K but deviates from this simple power-law at higher
temperature and is in very good agreement with the experimental results. The
specific heat is computed as a function of temperature for different values of
the magnetic field. It shows a lambda-like shape at the transition and is in
good qualitative agreement with the results of Oosawa et al. [Phys. Rev. B 63,
134416 (2001)].
