A new analytical technique to study the complex resonances of a rectangular patch in a multilayered medium is introduced. The problem is formulated as an electric field integral equation (EFIE) in the spectral domain and discretized by means of products of Chebyshev polynomials of first and second kind multiplied by their orthogonal weights in a Galerkin's scheme. The method is fast convergent, i.e., few expansion functions are needed to achieve accurate results, but leads to the numerical evaluation of infinite double integrals of oscillating and slowly decaying functions. To overcome this problem, suitable half-space contributions are pulled out of the kernels of such integrals in order to obtain exponentially decaying integrands. Moreover, the slowly converging integrals of the extracted contributions are expressed as combinations of quickly converging integrals by means of algebraic manipulations and an appropriate integration procedure in the complex plane.
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