  We present a computation of the coherent state path integral for a generic
linear system using ``functional methods'' (as opposed to discrete time
approaches). The Gaussian phase space path integral is formally given by a
determinant built from a first-order differential operator with coherent state
boundary conditions. We show how this determinant can be expressed in terms of
the symplectic transformation generated by the (in general, time-dependent)
quadratic Hamiltonian for the system. We briefly discuss the conditions under
which the coherent state path integral for a linear system actually exists. A
necessary -- but not sufficient -- condition for existence of the path integral
is that the symplectic transformation generated by the Hamiltonian is
(unitarily) implementable on the Fock space for the system.
