As a first step the concept of coupling impedance (c.i.) is extended to characterize the relationship between cause and effect relevant to distinct points. Then we can exploit a new law that relates the transverse c.i. to the lonitudinal one by means of a second derivative. When cause and effect coincide we get the standard definition of c.i. At a first sight the computation of the coupling impedance for the present model seems a trivial problem. In fact the c.i. is proportional to the Green function of the relevant model, which can be found as an open series, for instance in the treatise of Morse and Feshbach. But the authors themselves warn that this series is poorly convergent and in addition should not be differentiated. We succeeded in closing the series. The problem formally seems to be solved. But this procedure could however turn out to be sterile for numerical purposes, where we obviously need series expansions, unless one finds better alternative expansions. This is the final and possibly the most important step, where a different extremely rapidly convergent expansion is found. The use of this expansion has led to the recognition of a vast number of numerical results that are synthesized in the graphic representations of this work. The longitudinal c.i. is scarsely affected by the ellipticity and excentricity of the beam, so that, unless the beam is far off the central beam position, it behaves as in a circular cross section pipe.