## 1 Introduction
This regularized method can also be derived before the imposition of boundary conditions, i.e., directly on spectral functional equations (that is, before obtaining the GWHEs of the problem) by reversing the classical order of imposing boundary conditions and then applying Fredholm regularization to obtain the same effectiveness. We call this new methodology _direct Fredholm factorization_. We observe that the impossibility of mapping GWHEs to CWHEs in arbitrary linear media is similar to the impossibility of defining a unique angular complex plane for the SM, KL, and WH methods; however, the new WH methodology proposed in this study overcomes this obstacle by resorting to direct Fredholm factorization applied to GWHEs. From the solution of the GWHEs inherent to the angular region problem, we obtain the spectral representation of the field components along the faces delimiting homogeneous angular regions. The complete spectral analysis of the diffraction problems is then obtained by resorting again to spectral functional equations written for an arbitrary azimuthal direction. Finally, spectral inversion yields field components in the physical domain for any point in the angular regions. An alternative method for obtaining the field is also proposed; because of the linearity, it is based on the use of superposition on spectral representations prior to spectral inversion by identifying the spectral contributions of the faces of the angular regions through the equivalence theorem. All of the theoretical properties of the mathematical statements are fully described in the text, although completely rigorous mathematical proofs are sometimes limited. On the other hand, validation of the proposed novel theoretical package through examples is reported, starting by demonstrating the effectiveness of direct Fredholm factorization applied to GWHEs in the scattering from a PEC wedge immersed in an isotropic medium and ending with validation of functional equations of angular regions in arbitrary linear media with the analysis of a PEC half-plane immersed in particular anisotropic media. While implementing the method, we observe that the main difficulty resides in correctly estimating the kernel functions in the GWHEs and the corresponding FIE formulations for the presence of multivalued functions that need particular attention in their definition and calculation. The following sections highlight all multivalued functions and their correct estimations and assumptions. In summary, we highlight in brief the main novelties of this work with respect to the state of the art reported in the introduction:* Development of a novel spectral method capable of handling scattering in arbitrary linear media with multiple propagation constants. * Introduction of a novel solution procedure for GWHEs, particularly for those with multiple propagation constants, which we call direct Fredholm factorization. * A description of spectral functional equations and related integral representations for angular regions filled by arbitrary linear media in terms of the network interpretation. * Computation of the field at each point within the angular region by resorting to the equivalence theorem and using Kirchhoff representations in the spectral domain;
* Improving the quality of the approximate spectral solutions by reimposing GWHEs, referred to as iteration. It is important to highlight that the applicability of the proposed WH technique to arbitrary linear media resides in its formulation directly in the Laplace domain while avoiding other complex planes; notably, other techniques such as SM use complex angular planes based on Sommerfeld representations, which are applicable only to isotropic media or special cases of anisotropic media. Moreover, although SM also uses Fredholm integral equations in the complex angular plane for approximate solutions [2; 6], it is limited to isotropic media; again, the proposed WH method is extended to arbitrary linear media with direct Fredholm factorization because it is directly formulated in the Laplace domain.