  We study the problem of propagation of analytic regularity for semi-linear
symmetric hyperbolic systems. We adopt a global perspective and we prove that
if the initial datum extends to a holomorphic function in a strip of radius
(=width) \epsilon_0, the same happens for the solution u(t,.) for a certain
radius \epsilon(t), as long as the solution exists. Our focus is on precise
lower bounds on the spatial radius of analyticity \epsilon(t) as t grows. We
also get similar results for the Schroedinger equation with a real-analytic
electromagnetic potential.
