Multipole expansion in the context of Taylor series

A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, ${\displaystyle \mathbb {R} ^{3}}$. Multipole expansions are useful because, similar to Taylor series, oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on ${\displaystyle \mathbb {R} ^{3}}$, or less often on ${\displaystyle \mathbb {R} ^{n}}$ for some other ${\displaystyle n}$.

Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space.