Editing Multipole expansion (section)

==Expansion in spherical harmonics==
Most commonly, the series is written as a sum of [[spherical harmonics]]. Thus, we might write a function <math>f(\theta,\varphi)</math> as the sum
<math display="block">f(\theta,\varphi) = \sum_{\ell=0}^\infty\, \sum_{m=-\ell}^\ell\, C^m_\ell\, Y^m_\ell(\theta,\varphi)</math>
where <math>Y^m_\ell(\theta,\varphi)</math> are the standard spherical harmonics, and <math>C^m_\ell</math> are constant coefficients which depend on the function. The term <math>C^0_0</math> represents the monopole; <math>C^{-1}_1,C^0_1,C^1_1</math> represent the dipole; and so on. Equivalently, the series is also frequently written<ref>{{cite book | last=Thompson | first=William J. | title=Angular Momentum | publisher=John Wiley & Sons, Inc.}}</ref> as
<math display="block">f(\theta,\varphi) = C + C_i n^i + C_{ij}n^i n^j + C_{ijk}n^i n^j n^k + C_{ijk\ell}n^i n^j n^k n^\ell + \cdots</math>
where the <math>n^i</math> represent the components of a [[unit vector]] in the direction given by the angles <math>\theta</math> and <math>\varphi</math>, and indices are [[Einstein summation convention|implicitly summed]]. Here, the term <math>C</math> is the monopole; <math>C_i</math> is a set of three numbers representing the dipole; and so on.

In the above expansions, the coefficients may be [[real number|real]] or [[complex number|complex]]. If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties. In the spherical harmonic expansion, we must have
<math display="block">C_\ell^{-m} = (-1)^m C^{m\ast}_\ell \, .</math>
In the multi-vector expansion, each coefficient must be real:
<math display="block">C = C^\ast;\ C_i = C_i^\ast;\ C_{ij} = C_{ij}^\ast;\ C_{ijk} = C_{ijk}^\ast;\ \ldots</math>

While expansions of [[scalar (mathematics)|scalar]] functions are by far the most common application of multipole expansions, they may also be generalized to describe [[tensor]]s of arbitrary rank.<ref>{{cite journal | last=Thorne | first=Kip S. | journal=Reviews of Modern Physics | title=Multipole Expansions of Gravitational Radiation |date=April 1980 | volume=52 | issue=2 | pages=299–339 | doi=10.1103/RevModPhys.52.299 | bibcode=1980RvMP...52..299T| url=https://authors.library.caltech.edu/11159/1/THOrmp80a.pdf }}</ref> This finds use in multipole expansions of the [[vector potential]] in electromagnetism, or the metric perturbation in the description of [[gravitational wave]]s.

For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, <math>r</math>—most frequently, as a [[Laurent series]] in powers of <math>r</math>. For example, to describe the electromagnetic potential, <math>V</math>, from a source in a small region near the origin, the coefficients may be written as:
<math display="block">V(r,\theta,\varphi) = \sum_{\ell=0}^\infty\, \sum_{m=-\ell}^\ell C^m_\ell(r)\, Y^m_\ell(\theta,\varphi)= \sum_{j=1}^\infty\, \sum_{\ell=0}^\infty\, \sum_{m=-\ell}^\ell \frac{D^m_{\ell,j}}{r^j}\, Y^m_\ell(\theta,\varphi) .</math>