The proposed complexification scheme can be extended to more general boundary conditions by leveraging the inherent flexibility of the boundary integral equation (BIE) framework. In the context of transmission problems, where different media with distinct properties are present, the boundary conditions can be formulated to account for the continuity of the field and its normal derivative across the interface. By defining appropriate double layer potentials that respect these continuity conditions, the complexification approach can be adapted to handle the complexities introduced by varying material properties.
For layered media, the scheme can be extended by considering the layered structure as a series of interfaces, each governed by its own set of boundary conditions. The complexification can be applied to each layer's boundary, allowing for the integration of the effects of multiple layers into a unified framework. This involves modifying the contour of integration in the complex plane to reflect the geometry of the layered media, ensuring that the analytic continuation of the solution remains valid across the different layers. The robustness of the analytic continuation in the complex plane allows for the treatment of more complex geometries and boundary conditions, making the method versatile for various applications in wave propagation and scattering problems.