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Diffraction of an electromagnetic plane wave by an infinite slit in a conducting screen of finite thickness is studied theoretically using Weber-Schafheitlin discontinuous integrals. The direction of the incident wave is assumed to be in a plane perpendicular to that of the screen, and both polarizations are treated. The problems are reduced to simulataneous equations with an infinite number of unknowns, which are truncated in actual numerical computation. Numerical results for transmission coefficients are compared with those calculated from the Mathieu function expansion, and fairly good agreement is obtained. Far-field patterns are measured experimentally for the -Polarization using a parallel-plate waveguide. The measured results confirm the validity of the theoretical prediction.