  Semiclassical methods are extremely important in the subjects of wave packet
and coherent state dynamics. Unfortunately, these essentially saddle point
approximations are considered nearly impossible to carry out in detail for
systems with multiple degrees of freedom due to the difficulties of solving the
resulting two-point boundary value problems. However, recent developments have
extended the applicability to a broader range of systems and circumstances. The
most important advances are first to generate a set of real reference
trajectories using appropriately reduced dimensional spaces of initial
conditions, and second to feed that set into a Newton-Raphson search scheme to
locate the $exposed$ complex saddle trajectories. The arguments for this
approach were based mostly on intuition and numerical verification. In this
paper, the methods are put on a firmer theoretical foundation and then extended
to incorporate saddles $hidden$ from Newton-Raphson searches initiated with
real trajectories. This hidden class of saddles is relevant to tunneling-type
processes, but a hidden saddle can sometimes contribute just as much as or more
than an exposed one. The distinctions between hidden and exposed saddles
clarifies the interpretation of what constitutes tunneling for wave packets and
coherent states in the time domain.
