The anisotropy of composite plates often poses difficulties for stress field analysis in the presence of notches. The most common methods for these analyses are: (i) analytical means (AM), (ii) finite element analysis (FEA), and (iii) semi-analytical means (SAM). In industry, FEA has been especially popular for the determination of stresses in small to medium size parts but can require a considerable amount of computing power and time. For faster analyses, one can use AM. The available solutions for a given problem, however, can be quite limited. Additionally, AM implemented in commercial computer software are scarce and difficult to find. Due to this, these methods are not widespread and SAM were proposed. SAM combine the (easy) implementation of complex problems from FEA and the computational efficiency from AM to reduce the difficulty on mathematical operation and increase computational speed with respect to FEA. AM, however, are still the fastest and most accurate way to determine the stress field in a given problem. Complex problems, however, e.g., finite width plates with multiple loaded/unloaded notches, require a significant amount of mathematical involvement which quickly discourages, even seasoned, scientists, and engineers. To encourage the use of AM, this paper gives a brief review of the mathematical basis of AM followed by a historic perspective on the expansions originating from this mathematical basis. Specifically the case of a two-dimensional anisotropic plate with unloaded cut-outs subjected to in-plane static load is presented.