Multipole expansion

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,

R

3

{\displaystyle \mathbb {R} ^{3}}

. Multipole expansions are useful because, similar to Taylor series, often times 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

R

3

{\displaystyle \mathbb {R} ^{3}}

, or less often on

R

n

{\displaystyle \mathbb {R} ^{n}}

for some other

n

{\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.

The multipole expansion is expressed as a sum of terms with progressively finer angular features (moments). The first (the zeroth-order) term is called the monopole moment, the second (the first-order) term is called the dipole moment, the third (the second-order) the quadrupole moment, the fourth (third-order) term is called the octupole moment, the fifth (fourth-order) term is called the hexadecapole moment, and so on. Given the limitation of Greek numeral prefixes, terms of higher order are conventionally named by adding "-pole" to the number of poles—e.g., 32-pole (rarely dotriacontapole or triacontadipole) and 64-pole (rarely tetrahexacontapole or hexacontatetrapole). A multipole moment usually involves powers (or inverse powers) of the distance to the origin, as well as some angular dependence.