  We derive necessary and sufficient conditions for a continuous bounded
function $f: R\to C$ to be a characteristic function of a probability measure.
The Cauchy transform $K_f$ of $f$ is used as analytic continuation of $f$ to
the upper and lower half-planes in $C$. The conditions depend on the behavior
of $K_f(z)$ and its derivatives on the imaginary axis in $C$. The main results
are given in terms of completely monotonic and absolutely monotonic functions.
