  A unified numerically solvable framework for dispersion relations with
arbitrary number of species drifting at arbitrary directions and with Krook
collision is derived for linear uniform/homogenous kinetic plasma, which
largely extended the standard one [say, T. Stix, {\em Waves in Plasmas}, AIP
Press, 1992]. The purpose of this work is to provide a kinetic plasma
dispersion relation tool not only the physical model but also the numerical
approach be as general/powerful as possible. As a very general application
example, we give the final dispersion relations which assume further the
equilibrium distribution function be bi-Maxwellian and including parallel
drift, two directions of perpendicular drift (i.e., drift across magnetic
field), ring beam and loss-cone. Both electromagnetic and electrostatic
versions are provided, with also the Darwin (a.k.a., magnetoinductive or
magnetostatic) version. The species can be treated either magnetized or
unmagnetized. Later, the equations are transformed to the matrix form be
solvable by using the powerful matrix algorithm [H. S. Xie and Y. Xiao, Plasma
Science and Technology, 18, 2, 97, 2016], which is the first approach can give
all the important solutions of a linear kinetic plasma system without requiring
initial guess for root finding and thus can be extremely useful to the
community. To our knowledge, the present model would be the most comprehensive
one in literature for the distribution function constructed bases on
Maxwellian, which thus can be applied widely for study waves and instabilities
in space, astrophysics, fusion and laser plasma. We limit the present work to
non-relativistic case.
