Radiation from a flanged parallel-plate waveguide is studied analytically using the Weber-Schafheitlin discontinuity integral when the waveguide is excited by TE-mode waves, TM-mode waves, or an electromagnetic line source. Electromagnetic fields are represented by the discontinuity integral in the exterior space, and in the waveguide region they are expanded by normal modes of the waveguide. Imposing the boundary conditions in the aperture gives a set of simultaneous equations for the expansion coefficients. Once the solutions of these equations are determined, quantities of interest, such as reflection coefficients, modal coefficients of higher-order modes excited at the aperture, and radiation patterns, are readily obtained without knowing the field distribution in the aperture. Numerical calculations for the radiation patterns, reflection coefficients, and modal coefficients of higher order modes are carried out, and a part of the results is compared with the asymptotic solution by Lee . The effect of truncation in solving the equations numerically is also studied, and this method is found to exhibit rapid numerical convergence.