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analytic antenna applied array of parallel-plate asymptotic behavior bifurcated waveguide boundary conditions branch cut Chapter complex a-plane consider contour convergence cosh Cramer's rule cylindrical derive determined dielectric dielectric loading Diffraction discussed in Section E+(a edge condition entire function evaluated excitation expression F+(a factor finite formulation Fourier transform Fresnel integral G+(a given Green's function Green's identity IEEE Trans incident field infinite array infinite set integral equation integrand lower half a-plane matrix Maxwell's equations meromorphic function Mittra mode mode-matching technique modified residue-calculus method Note obtain open-ended waveguide parallel-plate waveguides propagating reflection coefficient regular residue S. W. Lee saddle point saddle-point method satisfies scattered field Section 1-6 semi-infinite set of equations shown in Figure singularities sinh yx solution solved space harmonics Theorem unknowns upper half a-plane wave equation Wiener-Hopf equation Wiener-Hopf technique