Transport matrix of a solenoid with linear fringe field

The problem of the transport matrix through a solenoid with a linear fringe field is solved exactly. The mathematical problem is simplified by the introduction of a new set of functions that generalize the ordinary hyperbolic functions. The results are used to characterize the physical properties of a solenoid exploited as magnetic Ions. In particular it is shown that the role of a solenoid, with a linear fringe, in the transport of an electron beam is equivalent to the combined effect of a thin lens, a drift section and a beam expander.