Since we are interested in obtaining 
for
small values of 
, it is reasonable to assume that 
also admits a complete set of
eigenfunctions, at least for sufficiently small 
. The
adjoint operator 
will
then admit a complete set of eigenfunctions that are biorthogonal
with those of 
:
These eigenfunctions may be used as a basis set for finding the solution
function 
, as in equation
(
).
The index j takes values from a set 
, which is isomorphic to
the natural numbers, as the spectrum is assumed discrete.
Let the index 0 denote the eigenvalue of 
with largest real
value,
i.e. 
for every
other 
. Then, we define a projection operator 
, which
projects out the hydrodynamic (long time) part of the Fourier
transform of a phase space distribution 
by
The k-space density function can now be split into a hydrodynamic part
and a non-hydrodynamic part
The Taylor series coefficients of 
are denoted by
These can be computed from the Taylor coefficients 
,
, which can be computed from the eigenfunctions of
by means of a recursion method (see Appendix ).
Substituting 
into (), the time dependent transport coefficients become
