Editing Legendre polynomials (section)

===Expanding an inverse distance potential===
{{main|Laplace expansion (potential)}}

The Legendre polynomials were first introduced in 1782 by [[Adrien-Marie Legendre]]<ref>{{cite book |first1=A.-M. |last1=Legendre |chapter=Recherches sur l'attraction des sphéroïdes homogènes |title=Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées |volume=X |pages=411–435 |location=Paris |date=1785 |orig-year=1782 |language=fr |chapter-url=http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf |url-status=dead |archive-url=https://web.archive.org/web/20090920070434/http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf |archive-date=2009-09-20 }}</ref> as the coefficients in the expansion of the [[Newtonian potential]]
<math display="block">\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma),</math>
where {{math|''r''}} and {{math|''r''′}} are the lengths of the vectors {{math|'''x'''}} and {{math|'''x'''′}} respectively and {{math|''γ''}} is the angle between those two vectors. The series converges when {{math|''r'' > ''r''′}}. The expression gives the [[gravitational potential]] associated to a [[point mass]] or the [[Coulomb potential]] associated to a [[point charge]]. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.

Legendre polynomials occur in the solution of [[Laplace's equation]] of the static [[electric potential|potential]], {{math|1=∇<sup>2</sup> Φ('''x''') = 0}}, in a charge-free region of space, using the method of [[separation of variables]], where the [[boundary conditions]] have axial symmetry (no dependence on an [[azimuth|azimuthal angle]]). Where {{math|'''ẑ'''}} is the axis of symmetry and {{math|''θ''}} is the angle between the position of the observer and the {{math|'''ẑ'''}} axis (the zenith angle), the solution for the potential will be
<math display="block">\Phi(r,\theta) = \sum_{\ell=0}^\infty \left( A_\ell r^\ell + B_\ell r^{-(\ell+1)} \right) P_\ell(\cos\theta) \,.</math>

{{math|''A<sub>l</sub>''}} and {{math|''B<sub>l</sub>''}} are to be determined according to the boundary condition of each problem.<ref>{{cite book|last=Jackson |first=J. D. |title=Classical Electrodynamics |url=https://archive.org/details/classicalelectro00jack_449 |url-access=limited |edition= 3rd |location=Wiley & Sons |date=1999 |page=[https://archive.org/details/classicalelectro00jack_449/page/n102 103] |isbn=978-0-471-30932-1}}</ref>

They also appear when solving the [[Schrödinger equation]] in three dimensions for a central force.