Putting the results of §§2.1-2.3 together,
we can derive the Laplace transform of and using
equations (5)--(8) to get the
distribution function of the dressed particle.
First, for a particular spherical harmonic, the action-angle transform
of 
is
where 
is a some general well-behaved function of time.
The quantity 
is shorthand for W with
. Because the motion of a star on a regular orbit
is quasi-periodic, we will consider terms with pure
sinusoidal dependence: 
.
Substituting this into equation (9) and Laplace
transforming gives the desired result:
where the Laplace transform of
is
