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Alternative representations are obtained directly by the method of characteristic Green's functions for the fields radiated by arbitrarily placed scalar and vector (electromagnetic) point sources in regions bounded by surfaces which are describable in terms of single coordinates in a spherical coordinate system. Such surfaces are spheres, cones, and planes, or a combination of these. The representations are distinguished from one another by different convergence properties for various ranges of values of the location of source and observation points and of the boundary surface geometry. Considered in particular are the diffraction by a sphere, a wedge, and a wedge with a spherical dome for both the scalar and vector cases.