  In this work, we present straightforward and concrete computations of the
unitary irreducible representations of the Euclidean motion group $M(2)$
employing the methods of deformation quantization. Deformation quantization is
a quantization method of classical mechanics and is an autonomous approach to
quantum mechanics, arising from the Wigner quasiprobability distributions and
Weyl correspondence. We advertise the utility and power of deformation theory
in Lie group representations. In implementing this idea, many aspects of the
method of orbits is also learned, thus further adding to the mathematical
toolkit of the beginning graduate student of physics. Furthermore, the
essential unity of many topics in mathematics and physics (such as Lie groups
and Lie algebras, quantization, functional analysis and symplectic geometry) is
witnessed, an aspect seldom encountered in textbooks, in an elementary way.
