  Both the gauge-invariant fermion Green function and gauge-dependent
conventional Green function in $ 2+1 $ dimensional QED are studied in the large
$ N $ limit. In temporal gauge, the infra-red divergence of gauge-dependent
Green function is found to be regulariable, the anomalous dimension is found to
be $ \eta= \frac{64}{3 \pi^{2} N} $. This anomalous dimension was argued to be
the same as that of gauge-invariant Green function. However, in Coulomb gauge,
the infra-red divergence of the gauge-dependent Green function is found to be
un-regulariable, anomalous dimension is even not defined, but the infra-red
divergence is shown to be cancelled in any gauge-invariant physical quantities.
The gauge-invariant Green function is also studied directly in Lorentz
covariant gauge and the anomalous dimension is found to be the same as that
calculated in temporal gauge.
