Hybrid and asymptotic methods of diffraction theory used when the electric size of a scatterer is large are based on the use of a solution for the radiation pattern of a cylindrical wave, which appears when a plane wave is scattered by an edge with a given boundary condition. The expression of the radiation pattern of a wave scattered by the edge of a half-plane with generalized semitransparent boundary conditions has not been obtained using the Sommerfeld integral technique. The use of the solution for the semitransparent case allows us to expand the asymptotic methods to scatterers with semitransparent edges. In this article, based on solving the problem of the scattering of a plane wave by a half-plane with the generalized semitransparent boundary conditions using the Sommerfeld integral technique, we derive a solution for the far-field of the cylindrical wave scattered by the semitransparent edge. Based on the obtained solution, we derive an expression for the calculation of an amendment to the physical optics approximation, which allows us to extend the physical theory of diffraction method to scatterers with semitransparent edges. We apply the uniform asymptotic theory of diffraction method to the problem of cylindrical-wave scattering by the semitransparent half-plane.