  We obtain an asymptotic solution for $\ep \to 0$ of the Cauchy problem for
linear first-order symmetric hyperbolic systems with oscillatory initial values
written in the eikonal form of geometric optics with frequency $1/\ep$, but
with complex phases. For the most common linear wave propagation models, this
kind on Cauchy problems are well-known in the applied literature and their
asymptotic theory, referred to as complex geometric optics, is attracting
interest for applications. In this work, which is the first of a series of
papers dedicated to complex geometric optics for nonlinear symmetric hyperbolic
systems, we develop a rigorous linear theory and set the basis for the
subsequent nonlinear analysis.
