  Path integral derivations are presented for two recently developed complex
trajectory techniques for the propagation of wave packets, Complex WKB and
BOMCA. Complex WKB is derived using a standard saddle point approximation of
the path integral, but taking into account the hbar dependence of both the
amplitude and the phase of the intial wave function, thus giving rise to the
need for complex classical trajectories. BOMCA is derived using a modification
of the saddle point technique, in which the path integral is approximated by
expanding around a near-classical path, chosen so that up to some predetermined
order there is no need to add any correction terms to the leading order
approximation. Both Complex WKB and BOMCA give the same leading order
approximation; in Complex WKB higher accuracy is achieved by adding correction
terms, while in BOMCA no additional terms are ever added -higher accuracy is
achieved by changing the path along which the original approximation is
computed. The path integral derivation of the methods explains the need to
incorporate contributions from more than one trajectory, as observed in
previous numerical work. On the other hand, it emerges that the methods provide
efficient schemes for computing the higher order terms in the asymptotic
evaluation of path integrals. The understanding we develop of BOMCA suggests
that there should exist near-classical trajectories that give exact quantum
dynamical results when used in the computation of the path integral keeping
just the leading order term. We also apply our path integral techniques to give
a compact derivation of the semiclassical approximation to the coherent state
propagator.
