Revisiting the Maxwell multipoles for vectorized angular functions

Abstract: Across many areas of physics, multipole expansions are used to simplify problems, solve differential equations, calculate integrals, and process experimental data. Spherical harmonics are the commonly used basis functions for a multipole expansion. However, they are not the preferred basis when the expression to be expanded is written as an explicit function of the unit vector on the sphere. Here, we revisit a different set of basis functions that are well-suited for multipole expansions of such vectorized angular functions. These basis functions are known in the literature by a variety of different names, including Maxwell multipoles, harmonic tensors, symmetric trace-free (STF) tensors, and Sachs-Pirani harmonics, but they do not seem be well-known among physicists. We provide a novel derivation of the Maxwell multipoles that highlights their analogy with the Legendre polynomials. We also rederive their key properties, and compile a list of further properties for pratical use, since this list seems to be missing in the current literature. We show the equivalence between the Maxwell multipoles and the spherical harmonics, derive conversion formulas between the two, and motivate when it is preferable to use either formalism. Since vectorized functions occur naturally in many physical problems, we expect that the method presented in this article can simplify calculations for physicists in many different fields.