  The two-dimensional Helmholtz equation separates in elliptic coordinates
based on two distinct foci, a limit case of which includes polar coordinate
systems when the two foci coalesce. This equation is invariant under the
Euclidean group of translations and orthogonal transformations; we replace the
latter by the discrete dihedral group of N discrete rotations and reflections.
The separation of variables in polar and elliptic coordinates is then used to
define discrete Bessel and Mathieu functions, as approximants to the well-known
continuous Bessel and Mathieu functions, as N-point Fourier transforms
approximate the Fourier transform over the circle, with integrals replaced by
finite sums. We find that these 'discrete' functions approximate the numerical
values of their continuous counterparts very closely and preserve some key
special function relations.
