  We study the Segal-Bargmann transform on $M(2).$ The range of this transform
is characterized as a weighted Bergman space. In a similar fashion Poisson
integrals are studied. Using a Gutzmer type formula we characterize the range
as a class of functions extending holomorphically to an appropriate domain in
the complexification of $M(2).$ We also prove a Paley-Wiener theorem for the
inverse Fourier transform
