Internal gravity waves propagate in density-stratified fluids owing to the restoring effect of buoyancy. They are anisotropic and dispersive, with a propagation equation that is elliptic in the frequency range of evanescent waves, and hyperbolic in the range of propagating waves. A Green's function approach of their generation is proposed, that separates the problem into two distinct ones: the representation of the generation process by a distribution of singularities, obtained by the boundary integral method; and the calculation of the waves radiated by this distribution, using Fourier analysis. The first problem is the most difficult of the two, involving anisotropic coordinate stretching in the evanescent range followed by causal analytic continuation into the propagating range. The approach is implemented for two-dimensional elliptical forcing and three-dimensional spheroidal forcing. The interpretation of the results in terms of added mass is discussed, together with the implications of added mass for the calculation of the conversion rate of barotropic forcing into baroclinic internal tides at the ocean bottom. The application of the analysis to the recent discovery of internal wave focusing by ring forcing is presented, and compared with novel experiments involving oscillating tori at LEGI and an annular wave generator at the physics laboratory of ENS de Lyon.