The existence of a characteristic coherence length in FEL SASE Physics determines the independent lasing of different portions, namely the slices, of the electron bunch. Each slice may be characterized by different phase space properties (not necessa
rily equal emittances and Twiss coefficients). This fact opens new questions on the concept of beam matching and how the various portions of the beam contribute to the performances of the output radiation, including those associated with the transverse coherence.
We consider the matrix representation of the Eisenstein numbers and in this context we discuss the theory of the Pseudo Hyperbolic Functions. We develop a geometrical interpretation and show the usefulness of the method in Physical problems related to the anomalous scattering of light by crystals
