Editing Multipole expansion (section)

{{Short description|Mathematical series}}
A '''multipole expansion''' is a [[Series (mathematics)|mathematical series]] representing a [[Function (mathematics)|function]] that depends on [[angle]]s—usually the two angles used in the [[spherical coordinate system]] (the polar and [[Azimuth|azimuthal]] angles) for three-dimensional [[Euclidean space]], <math>\R^3</math>. Multipole expansions are useful because, similar to [[Taylor series]], oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be [[Real number|real]]- or [[complex number|complex]]-valued and is defined either on <math>\R^3</math>, or less often on <math>\R^n</math> for some other {{nowrap|<math>n</math>.}}

Multipole expansions are used frequently in the study of [[electromagnetic field|electromagnetic]] and [[gravitational field]]s, 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.<ref name=Edmonds>{{cite book | last = Edmonds | first = A. R. | title = Angular Momentum in Quantum Mechanics | year = 1960 | url = https://archive.org/details/angularmomentumi0000edmo | url-access = registration | publisher = Princeton University Press| isbn = 9780691079127 }}</ref>

The multipole expansion is expressed as a sum of terms with progressively finer angular features ([[Moment (physics)|moments]]). The first (the zeroth-order) term is called the [[Monopole (mathematics)|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, and so on. Given the limitation of [[Greek numerals|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).<ref>{{cite book|last1=Auzinsh|first1=Marcis| last2=Budker|first2=Dmitry|last3=Rochester|first3=Simon|title=Optically polarized atoms : understanding light-atom interactions| date=2010|publisher=New York|location=Oxford|isbn=9780199565122|page=100}}</ref><ref>{{cite journal|last1=Okumura|first1=Mitchio| last2=Chan|first2=Man-Chor|last3=Oka|first3=Takeshi|title=High-resolution infrared spectroscopy of solid hydrogen: The tetrahexacontapole-induced transitions|journal=Physical Review Letters|date=2 January 1989|volume=62|issue=1| pages=32–35| doi=10.1103/PhysRevLett.62.32|pmid=10039541|bibcode=1989PhRvL..62...32O|url=https://authors.library.caltech.edu/5428/1/OKUprl89.pdf }}</ref><ref>{{cite journal|last1=Ikeda|first1=Hiroaki|last2=Suzuki|first2=Michi-To|last3=Arita|first3=Ryotaro| last4=Takimoto|first4=Tetsuya|last5=Shibauchi|first5=Takasada|last6=Matsuda|first6=Yuji|title=Emergent rank-5 nematic order in URu2Si2| journal=Nature Physics|date=3 June 2012|volume=8|issue=7|pages=528–533| doi=10.1038/nphys2330| arxiv=1204.4016| bibcode=2012NatPh...8..528I|s2cid=119108102 }}</ref> A multipole moment usually involves [[Exponentiation|powers]] (or inverse powers) of the distance to the origin, as well as some angular dependence.

In principle, a multipole expansion provides an exact description of the potential, and generally [[Convergent series|converges]] under two conditions: (1) if the sources (e.g. charges) are localized close to the origin and the point at which the potential is observed is far from the origin; or (2) the reverse, i.e., if the sources are located far from the origin and the potential is observed close to the origin. In the first (more common) case, the coefficients of the series expansion are called ''exterior multipole moments'' or simply ''multipole moments'' whereas, in the second case, they are called ''interior multipole moments''.