  We derive an exact formula for the complex frequency in spatio-temporal
stability analysis that is valid for arbitrary complex wave numbers. The
usefulness of the formula lies in the fact that it depends only on purely
temporal quantities, which are easily calculated. We apply the formula to two
model dispersion relations: the linearized complex Ginzburg--Landau equation,
and a model of wake instability. In the first case, a quadratic truncation of
the exact formula applies; in the second, the same quadratic truncation yields
an estimate of the parameter values at which the transition to absolute
instability occurs; the error in the estimate decreases upon increasing the
order of the truncation. We outline ways in which the formula can be used to
characterize stability results obtained from purely numerical calculations, and
point to a further application in global stability analyses.
