  The so-called Weyl transform is a linear map from a commutative algebra of
functions to a noncommutative algebra of linear operators, characterized by an
action on Cartesian coordinate functions of the form $(x, y) \mapsto (X, Y)$
such that $XY -YX = i\epsilon I$, i.e. the defining relation for the Heisenberg
Lie algebra. Study of this transform has been expansive. We summarize many
important results from the literature. The primary goal of this work is to
prove the final result: the realization of the polar transformation $(\rho,
e^{i\theta}) \mapsto (R, e^{i\Theta})$ in terms of explicit orthogonal function
expansions, while starting from elementary principles and utilizing minimal
machinery. Our results are not strictly original but their presentation here is
intended to simplify introduction to these subjects in a novel manner.
