### multipole expansion

(series expansion of a function over the surface of a sphere)

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 single linear dimension.

A commonly-used type 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**.
The first few terms are often useful as an approximation of the
function, providing a more tractable way to find useful answers.
Terms such as **octupole level** are often used as a description of
a method or calculation to assert that it produces an accuracy
equivalent to using all the series terms to and including
the octupole term.

Applications include description and analysis of electromagnetic fields,
gravitational fields, the cosmic microwave background, and potentially
any interesting property of the sky based on direction from Earth,
such as any cosmic background radiation.

(*mathematics,gravity,field,CMB,EMR*)
**Further reading:**

http://en.wikipedia.org/wiki/Multipole_expansion

**Referenced by pages:**

angular power spectrum

CMB anisotropies

dipole

Fourier series

Legendre polynomials

Love number

magnetic field

normal mode

perturbation theory

window function

Index