  The power-law disks are a family of infinitesimally thin, axisymmetric
stellar disks of infinite extent. The rotation curve can be rising, falling or
flat. The self-consistent power-law disks are scale-free, so that all physical
quantities vary as a power of radius. They possess simple equilibrium
distribution functions depending on the two classical integrals, energy and
angular momentum. While maintaining the scale-free equilibrium force law, the
power-law disks can be transformed into cut-out disks by preventing stars close
to the origin (and sometimes also at large radii) from participating in any
disturbance. This paper derives the homogeneous Fredholm integral equation for
the in-plane normal modes in the self-consistent and the cut-out power-law
disks. This is done by linearising the collisionless Boltzmann equation to find
the response density corresponding to any imposed density and potential. The
normal modes -- that is, the self-consistent modes of oscillation -- are found
by requiring the imposed density to equal the response density. In practice,
this scheme is implemented in Fourier space, by decomposing both imposed and
response densities in logarithmic spirals. The Fredholm integral equation then
relates the transform of the imposed density to the transform of the response
density. Numerical strategies to solve the integral equation and to isolate the
growth rates and the pattern speeds of the normal modes are discussed.
