Editing Legendre polynomials (section)

=== In multipole expansions ===
[[File:Point axial multipole.svg|right|Diagram for the multipole expansion of electric potential.]]
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
<math display="block">\frac{1}{\sqrt{1 + \eta^2 - 2\eta x}} = \sum_{k=0}^\infty \eta^k P_k(x),</math>
which arise naturally in [[multipole expansion]]s.  The left-hand side of the equation is the [[generating function]] for the Legendre polynomials.

As an example, the [[electric potential]] {{math|Φ(''r'',''θ'')}} (in [[spherical coordinates]]) due to a [[point charge]] located on the {{math|''z''}}-axis at {{math|1=''z'' = ''a''}} (see diagram right) varies as
<math display="block">\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^2 + a^2 - 2ar \cos\theta}}.</math>

If the radius {{math|''r''}} of the observation point {{math|P}} is greater than {{math|''a''}}, the potential may be expanded in the Legendre polynomials
<math display="block">\Phi(r, \theta) \propto \frac{1}{r} \sum_{k=0}^\infty \left( \frac{a}{r} \right)^k P_k(\cos \theta),</math>
where we have defined {{math|1=''η'' = {{sfrac|''a''|''r''}} < 1}} and {{math|1=''x'' = cos ''θ''}}. This expansion is used to develop the normal [[multipole expansion]].

Conversely, if the radius {{math|''r''}} of the observation point {{math|P}} is smaller than {{math|''a''}}, the potential may still be expanded in the Legendre polynomials as above, but with {{math|''a''}} and {{math|''r''}} exchanged. This expansion is the basis of [[interior multipole expansion]].