Multipole expansion

A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex in general. Multipole expansions are very frequently used in the study of electromagnetic, gravitational fields and aeroacoustics, 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.