Editing Multipole expansion (section)

==General mathematical properties==
Multipole moments in [[mathematics]] and [[mathematical physics]] form an [[orthogonal basis]] for the decomposition of a function, based on the response of a [[field (physics)|field]] to point sources that are brought infinitely close to each other. These can be thought of as arranged in various geometrical shapes, or, in the sense of [[Distribution (mathematics)|distribution theory]], as [[directional derivative]]s.

Multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated [[differential equation]]s. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of [[group representation|irreducible representations]] of the rotational [[symmetry group]], which leads to spherical harmonics and related sets of [[orthogonal]] functions. One uses the technique of [[separation of variables]] to extract the corresponding solutions for the radial dependencies.

In practice, many fields can be well approximated with a finite number of multipole moments (although an infinite number may be required to reconstruct a field exactly). A typical application is to approximate the field of a localized charge distribution by its [[Monopole (mathematics)|monopole]] and [[dipole]] terms. Problems solved once for a given order of multipole moment may be [[linear combination|linearly combined]] to create a final approximate solution for a given source.