  The Dbar dressing method is extended to study the focusing/defocusing
nonlinear Schr\"odinger (NLS) equation with nonzero boundary condition. A
special type of complex function is considered. The function is meromorphic
outside an annulus with center 0 and satisfies a local Dbar problem inside the
annulus. The theory of such function is extended to construct the Lax pair of
the NLS equation with nonzero boundary condition. In this procedure, the
relation between the NLS potential and the solution of the Dbar problem is
established. A certain distribution for the Dbar problem is introduced to
obtain the focusing/defocusing NLS equation and the conservation laws. The
explicit solutions of the focusing/defocusing NLS equation with nonzero
boundary condition are given from special distributions.
