  Orientational expansions, which are widely used in the natural sciences,
exist in angular and Cartesian form. Although these expansions are orderwise
equivalent, it is difficult to relate them in practice. In this article, both
types of expansions and their relations are explained in detail. We give
explicit formulas for the conversion between angular and Cartesian expansion
coefficients for functions depending on one, two, and three angles in two and
three spatial dimensions. These formulas are useful, e.g., for comparing
theoretical and experimental results in liquid crystal physics. The application
of the expansions in the definition of orientational order parameters is also
discussed.
