The singularities of G(a) and [G(a)]-1 are called structural singularities, and they always have an important physical meaning. To illustrate this point, let us consider some typical W-H geometries, as shown in Fig. 1 of the Preface. These geometries may be considered as the junction of two (or more) waveguides, or as a single waveguide in which geometrical discontinuities are present. The singularities of G(a) and of its inverse [G(a)]-1 are related to the propagation constants of the modes of the involved waveguides. For instance, in the halfplane problem we are dealing with free space with propagation constant k. This implies that in the continuous modes have propagation constants (k2 - a2)1/2. In this case the singularitiesof G(a) and [G(a)]-1 are the branch points a 1/4 +k. Singularities constituted by poles are instead present when we deal with closed waveguides.It is difficult to obtain additive decomposition in the scalar and matrix it can be solved by using Cauchy integration approach.Wiener-Hopf equations