  We investigate the finite lifetime effects on first-order correlation
functions of dissipative Bose-Einstein condensates. By taking into account the
phase fluctuations up to all orders, we show that the finite lifetime effects
are neglible for the spatial first-order correlation functions, but have an
important effect on the temporal correlations. As an application, we calculate
the one-particle density matrix of a quasi-condensate of photons. Finally, we
also consider the photons in the normal state and we demonstrate that the
finite lifetime effects decrease both the spatial and temporal first-order
correlation functions.
