We can compute the density and potential response corresponding to the
dressed particle by integrating the perturbed distribution function
from equation (3) over velocities:
The Laplace transformed expansion coefficients of the biorthogonal basis
are then
This computation is easily performed by noting that the Jacobian of
the canonical transform of 
is unity and we are free to
choose any set. This procedure is the motivation behind the
biorthogonal expansion. If we choose 
and use
equation (7), we can do the angle integration trivially.
Then, noting that 
, we can do
the integral in 
using the orthogonality of the rotation
matrices:
(Edmonds 1960). We get:
Equation (15) has the form 
and describes the response of the
stellar system, 
, to the perturbation, 
.
The self-gravitating response, then, is the solution to 
:
If 
, the equation for the response becomes an eigenvalue
problem. The eigenvalues s are the zeros of the dispersion relation
. For the general stable spherical
stellar system, there will be no modes for 
and only in
special cases with restricted phase space does one find oscillatory
modes, 
To evaluate the coefficients as a function of time, we perform the
inverse Laplace transform, deforming the contour to the
:
Assuming that the background is stable with no oscillatory modes,
has no poles in the half plane
. The final term gives a pole at 
. Although
there may be poles in the half plane 
, these will vanish for
relative to the pure imaginary contribution. To
perform the integral, one may take the s integration into the
phase-space integral for the elements of 
. In
addition to , the s-dependence is in two
simple poles and one finds an integral of the form
For large values of t, this expression oscillates rapidly and we may
extract the dominant coherent contribution. There are two cases:
without and with a resonance in the phase space. The existence of a
resonance in phase space is defined by 
for 
. For the
non-resonant case, the integrand has no singularity and we can
consider each term separately. The first term yields a contribution
in phase with the perturbation while the second term in the brackets
oscillates incoherently and makes no net contribution. The second
term, therefore, can be ignored. For the resonant case, the
contribution at large t has a sharp peak about as . Expanding
about 
and retaining only
dominant terms, one finds the contribution near the resonance is
We will adopt the latter asymptotic form here and see in the final
computation that the 
will cancel leaving only the
delta functions. Putting both cases together yields a simple
expression
To simplify notation, we have explicitly noted in equations
(21) and (22) that the solution takes the the
form 
. The integrals in the matrix
elements of 
may be analytically continued using the
Landau prescription (e.g. Krall & Trivelpiece 1973)
after a conformal mapping of the discrete interval in E. Notice
that this expression takes both resonant and non-resonant cases into
account; without a resonance, the delta function does not contribute
and principal value of a non-singular integrand is the integrand
itself. One recovers the non-self-gravitating but global response by
setting 
.