Editing Legendre polynomials (section)

=== Definition via differential equation ===
A third definition is in terms of solutions to '''Legendre's differential equation''':
{{NumBlk||<math display="block">(1 - x^2) P_n''(x) - 2 x P_n'(x) + n (n + 1) P_n(x) = 0.</math>|{{EquationRef|1}}}}

This [[differential equation]] has [[regular singular point]]s at {{math|1=''x'' = ±1}} so if a solution is sought using the standard [[Frobenius method|Frobenius]] or [[power series]] method, a series about the origin will only converge for {{math|{{abs|''x''}} < 1}} in general. When {{math|''n''}} is an integer, the solution {{math|''P<sub>n</sub>''(''x'')}} that is regular at {{math|1=''x'' = 1}} is also regular at {{math|1=''x'' = −1}}, and the series for this solution terminates (i.e. it is a polynomial). The orthogonality and completeness of these solutions is best seen from the viewpoint of [[Sturm–Liouville theory]]. We rewrite the differential equation as an eigenvalue problem,
<math display="block">\frac{d}{dx} \left( \left(1-x^2\right) \frac{d}{dx} \right) P(x) = -\lambda P(x) \,,</math>
with the eigenvalue <math>\lambda</math> in lieu of <math> n(n+1)</math>. If we demand that the solution be regular at
<math>x = \pm 1</math>, the [[differential operator]] on the left is [[Hermitian]]. The eigenvalues are found to be of the form
{{math|''n''(''n'' + 1)}}, with <math>n = 0, 1, 2, \ldots</math> and the eigenfunctions are the <math>P_n(x)</math>. The orthogonality and completeness of this set of solutions follows at once from the larger framework of Sturm–Liouville theory.

The differential equation admits another, non-polynomial solution, the [[Legendre function#Legendre functions of the second kind (Qn)|Legendre functions of the second kind]] <math>Q_n</math>.
A two-parameter generalization of (Eq.&nbsp;{{EquationNote|1}}) is called Legendre's ''general'' differential equation, solved by the [[Associated Legendre polynomials]]. [[Legendre functions]] are solutions of Legendre's differential equation (generalized or not) with ''non-integer'' parameters.

In physical settings, Legendre's differential equation arises naturally whenever one solves [[Laplace's equation]] (and related [[partial differential equation]]s) by separation of variables in [[spherical coordinates]]. From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the [[spherical harmonics]], of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis. The polynomials appear as <math>P_n(\cos\theta)</math> where <math>\theta</math> is the polar angle. This approach to the Legendre polynomials provides a deep connection to rotational symmetry. Many of their properties which are found laboriously through the methods of analysis — for example the addition theorem — are more easily found using the methods of symmetry and [[group theory]], and acquire profound physical and geometrical meaning.