A modified WKB formulation for linear eigenmodes of a collisionless
  self-gravitating disc in the epicyclic approximation
  The short--wave asymptotics (WKB) of spiral density waves in self-gravitating
stellar discs is well suited for the study of the dynamics of tightly--wound
wavepackets. But the textbook WKB theory is not well adapted to the study of
the linear eigenmodes in a collisionless self-gravitating disc because of the
transcendental nature of the dispersion relation. We present a modified WKB of
spiral density waves, for collisionless discs in the epicyclic limit, in which
the perturbed gravitational potential is related to the perturbed surface
density by the Poisson integral in Kalnaj's logarithmic spiral form. An
integral equation is obtained for the surface density perturbation, which is
seen to also reduce to the standard WKB dispersion relation. We specialize to a
low mass (or Keplerian) self-gravitating disc around a massive black hole, and
derive an integral equation governing the eigenspectra and eigenfunctions of
slow precessional modes. For a prograde disc, the integral kernel turns out be
real and symmetric, implying that all slow modes are stable. We apply the slow
mode integral equation to two unperturbed disc profiles, the Jalali--Tremaine
annular discs, and the Kuzmin disc. We determine eigenvalues and eigenfunctions
for both $m = 1$ and $m = 2$ slow modes for these profiles and discuss their
properties. Our results compare well with those of Jalali--Tremaine.
