  We investigate the behavior of the anomalous correlation function in two
dimensional Bose gas. In the local case, we find that this quantity has a
finite value in the limit of weak interactions at zero temperature. The effects
of the anomalous density on some thermodynamic quantities are also considered.
These effects can modify in particular the chemical potential, the ground sate
energy, the depletion and the superfluid fraction. Our predictions are in good
agreement with recent analytical and numerical calculations. We show also that
the anomalous density presents a significant importance compared to the
non-condensed one at zero temperature. The single-particle anomalous
correlation function is expressed in two dimensional homogenous Bose gases by
using the density-phase fluctuation. We then confirm that the anomalous average
accompanies in analogous manner the true condensate at zero temperature while
it does not exist at finite temperature.
