  The pseudogap phenomena in the cuprate superconductors requires a theory
beyond the mean field BCS level. A natural candidate is to include strong
pairing fluctuations, and treat the two-particle and single particle Green's
functions self-consistently. At the same time, impurities are present in even
the cleanest samples of the cuprates. Some impurity effects can help reveal
whether the pseudogap has a superconducting origin and thus test various
theories. Here we extend the pairing fluctuation theory for a clean system
[Phys. Rev. Lett. 81, 4708 (1998)] to the case with nonmagnetic impurities.
Both the pairing and the impurity $T$ matrices are included and treated
self-consistently. We obtain a set of three equations for the chemical
potential $\mu$, $T_c$, the excitation gap $\Delta(T_c)$ at $T_c$, or $\mu$,
the order parameter $\Delta_{sc}$, and the pseudogap $\Delta_{pg}$ at
temperature $T<T_c$, and study the effects of impurity scattering on the
density of states, $T_c$ and the order parameter, and the pseudogap. Both $T_c$
and the order parameter as well as the total excitation gap are suppressed,
whereas the pseudogap is not for given $T\le T_c$. Born scatterers are about
twice as effective as unitary scatterers in suppressing $T_c$ and the gap. In
the strong pseudogap regime, pair excitations contribute a new $T^{3/2}$ term
to the low $T$ superfluid density. The initial rapid drop of the zero $T$
superfluid density in the unitary limit as a function of impurity concentration
$n_i$ also agrees with experiment.
