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In this paper, we develop novel closed-form representations for the diffraction integrals associated with the classical problem of plane wave diffraction by a two-dimensional aperture in a ground plane. After approximating the aperture field by the incident field, spectral-domain techniques are used to represent the diffracted fields as inverse Fourier transforms, which are often referred to as the angular spectrum. The resulting inverse Fourier transforms are integrated analytically by using contour deformation techniques, thereby yielding closed-form representations for the diffracted fields that only involve rapidly-computable special functions. The diffracted fields that are computed using the closed-form representations are validated by comparing with results obtained by direct numerical integration of the diffraction integrals.