Segal-Bargmann transform and Paley-Wiener theorems on $M(2)\mathrm{.}$

We study the Segal-Bargmann transform on $M(2)\mathrm{.}$ 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 $M(2)\mathrm{.}$ We also prove a Paley-Wiener theorem for the inverse Fourier transform