multipole expansion

A multipole expansion is a type of mathematical series expansion for functions whose domain is the surface of a sphere. It is analogous to a Fourier series expansion of a periodic function, but is for a function on two dimensions over directions from a point, or equivalently, the surface of a sphere, rather than a function of a single linear dimension.

A commonly-used type of multipole expansion is based on spherical harmonics (harmonics over divisions of a sphere rather than over divisions of a line segment as is used for description of waves such as sound waves). The function can be real or complex, and can also be a function of the radius (as a third dimension), e.g., representing a field surrounding a specific point. The first term of the series has the name monopole, the second term is the dipole, the third, the quadrupole (or quadrapole), the fourth, octupole (or octopole: folks in different areas of application often have a preference), with further terms more often identified by number, such as 16-pole, 32-pole. Each such term-type is characterized by formula (itself representing a function over the sphere), that includes values specific to the function being expanded (coefficients). The first few series terms often serve as a useful approximation of the (expanded) function, providing a tractable way to find useful approximations of the function's values. Phrases such as octupole level are often used to describe a particular method or calculation to assert that it produces an accuracy equivalent to using all the series terms to and including the octupole term. Applications of multipole expansions include description and analysis of electromagnetic fields, gravitational fields, the cosmic microwave background (CMB), and potentially any interesting property of the sky based on direction from Earth, such as any cosmic background radiation.