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The short--wave asymptotics (WKB) of spiral density waves in self-gravitatingstellar discs is well suited for the study of the dynamics of tightly--woundwavepackets. But the textbook WKB theory is not well adapted to the study ofthe linear eigenmodes in a collisionless self-gravitating disc because of thetranscendental nature of the dispersion relation. We present a modified WKB ofspiral density waves, for collisionless discs in the epicyclic limit, in whichthe perturbed gravitational potential is related to the perturbed surfacedensity by the Poisson integral in Kalnaj's logarithmic spiral form. Anintegral equation is obtained for the surface density perturbation, which isseen to also reduce to the standard WKB dispersion relation. We specialize to alow mass (or Keplerian) self-gravitating disc around a massive black hole, andderive an integral equation governing the eigenspectra and eigenfunctions ofslow precessional modes. For a prograde disc, the integral kernel turns out bereal and symmetric, implying that all slow modes are stable. We apply the slowmode integral equation to two unperturbed disc profiles, the Jalali--Tremaineannular discs, and the Kuzmin disc. We determine eigenvalues and eigenfunctionsfor both $m = 1$ and $m = 2$ slow modes for these profiles and discuss theirproperties. Our results compare well with those of Jalali--Tremaine.

A modified WKB formulation for linear eigenmodes of a collisionless self-gravitating disc in the epicyclic approximation