  We derive an analytical trace formula for the level density of the
two-dimensional elliptic billiard using an improved stationary phase method.
The result is a continuous function of the deformation parameter (eccentricity)
through all bifurcation points of the short diameter orbit and its repetitions,
and possesses the correct limit of the circular billiard at zero eccentricity.
Away from the circular limit and the bifurcations, it reduces to the usual
(extended) Gutzwiller trace formula which for the leading-order families of
periodic orbits is identical to the result of Berry and Tabor. We show that the
circular disk limit of the diameter-orbit contribution is also reached through
contributions from closed (periodic and non-periodic) orbits of hyperbolic type
with an even number of reflections from the boundary. We obtain the Maslov
indices depending on deformation and energy in terms of the phases of the
complex error and Airy functions. We find enhancement of the amplitudes near
the common bifurcation points of both short-diameter and hyperbolic orbits. The
calculated semiclassical level densities and shell energies are in good
agreement with the quantum mechanical ones.
