  We use the analytical solution of the quantum Rabi model to obtain absolutely
convergent series expressions of the exact eigenstates and their scalar
products with Fock states. This enables us to calculate the numerically exact
time evolution of <\sigma_x(t)> and <\sigma_z(t)> for all regimes of the
coupling strength, without truncation of the Hilbert space. We find a
qualitatively different behavior of both observables which can be related to
their representations in the invariant parity subspaces.
