A solution method has been developed for stress concentration problems of fibre-reinforced compounds based on the method of complex-valued stress functions combined with conformal mappings. Using this method, stress and displacement fields of anisotropic plates with a finite boundary and an elastic inclusion can be calculated. The inclusion is considered to be a full elliptical disk, also made of an anisotropic material. It is connected to the notched plate by the conditions of continuous normal stresses and displacements at the boundary. In this way, the solution is analytically continued from the inner boundary to the rest of the plate. The remaining unknowns are obtained semi-analytically using a combination of boundary collocation method and least squares method. As the computation examples demonstrate, this analytical approach is markedly superior to a finite element analysis.