On holomorphic and piecewise holomorphic signals

The properties of holomorphic and piecewise holomorphic signals during propagation in a dispersing medium are compared. It is shown that the status of the signal (holomorphic or piecewise holomorphic) cannot be changed by any physically realizable (that is, not violating the principle of causality) filter. It is shown that the properties of holomorphic and piecewise holomorphic signals are not only different, but usually directly opposite. For example, piecewise holomorphic signals (unlike holomorphic ones) necessarily have precursors, fade out in an absorbing medium according to a hyperbolic (rather than exponential) law, have an anthropogenic (rather than natural) origin, and transfer information (unlike holomorphic ones). Holomorphic signals (unlike piecewise holomorphic ones) are capable of propagating as a whole with a group velocity (and any one – sublight, superluminal, negative and complex).