  It has long been known that the ellipse normally used to model the phase
space extension of a beam in linear dynamics may be represented by a complex
number which can be interpreted similarly to a complex impedance in electrical
circuits, so that classical electrical methods might be used for the design of
such beam transport lines. However, this method has never been fully developed,
and only the transport transformation of single particular elements, like drift
spaces or quadrupoles, has been presented in the past. In this paper, we
complete the complex formalism of linear beam dynamics by obtaining a general
differential equation and solving it, to show that the general transformation
of a linear beam line is a complex Moebius transformation. This result opens
the possibility of studying the effect of the beam line on complete regions of
the complex plane and not only on a single point. Taking advantage of this
capability of the formalism, we also obtain an important result in the theory
of the transport through a periodic line, proving that the invariant points of
the transformation are only a special case of a more general structure of the
solution, which are the invariant circles of the one-period transformation.
Among other advantages, this provides a new description of the betatron
functions beating in case of a mismatched injection in a circular accelerator.
