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| {{for|Legendre's Diophantine equation|Legendre's equation}}
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| {{selfref|[[Associated Legendre polynomials]] are the most general solution to Legendre's Equation and '''Legendre polynomials''' are solutions that are azimuthally symmetric.}}
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| In [[mathematics]], '''Legendre functions''' are solutions to '''Legendre's differential equation''':
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| :<math>{d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0.</math>
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| They are named after [[Adrien-Marie Legendre]]. This [[differential equation|ordinary differential equation]] is frequently encountered in [[physics]] and other technical fields. In particular, it occurs when solving [[Laplace's equation]] (and related [[partial differential equation]]s) in [[spherical coordinates]].
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| The Legendre differential equation may be solved using the standard [[power series]] method. The equation has [[regular singular point]]s at ''x'' = ±1 so, in general, a series solution about the origin will only converge for |''x''| < 1. When ''n'' is an integer, the solution ''P''<sub>''n''</sub>(''x'') that is regular at ''x'' = 1 is also regular at ''x'' = −1, and the series for this solution terminates (i.e. it is a polynomial).
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| These solutions for ''n'' = 0, 1, 2, ... (with the normalization ''P<sub>n</sub>''(1) = 1) form a [[polynomial sequence]] of [[orthogonal polynomials]] called the '''Legendre polynomials'''. Each Legendre polynomial ''P''<sub>''n''</sub>(''x'') is an ''n''th-degree polynomial. It may be expressed using [[Rodrigues' formula]]:
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| :<math>P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right]. </math>
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| That these polynomials satisfy the Legendre differential equation ({{EquationNote|1}}) follows by differentiating (''n''+1) times both sides of the identity
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| :<math>(x^2-1)\frac{d}{dx}(x^2-1)^n = 2nx(x^2-1)^n</math>
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| and employing the [[general Leibniz rule]] for repeated differentiation.<ref>{{harvnb|Courant|Hilbert|1953|loc=II, §8}}</ref> The ''P''<sub>''n''</sub> can also be defined as the coefficients in a [[Taylor series]] expansion:<ref name="arfken">{{Citation |author=[[George B. Arfken]], Hans J. Weber |title=Mathematical Methods for Physicists |publisher=Elsevier Academic Press |year=2005 |page=743 |isbn=0-12-059876-0}}</ref>
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| :<math>\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x) t^n.\qquad (1)</math>
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| In physics, this [[generating function]] is the basis for [[multipole expansion]]s.
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| == Recursive definition ==
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| Expanding the Taylor series in equation (1) for the first two terms gives
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| :<math>P_0(x) = 1,\quad P_1(x) = x</math>
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| for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (1) is differentiated with respect to t on both sides and rearranged to obtain
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| :<math>\frac{x-t}{\sqrt{1-2xt+t^2}} = (1-2xt+t^2) \sum_{n=1}^\infty n P_n(x) t^{n-1}.</math>
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| Replacing the quotient of the square root with its definition in (1), and [[equating the coefficients]] of powers of t in the resulting expansion gives ''Bonnet’s recursion formula''
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| :<math> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x).\,</math>
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| This relation, along with the first two polynomials ''P''<sub>0</sub> and ''P''<sub>1</sub>, allows the Legendre Polynomials to be generated recursively.
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| Explicit representations include
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| :<math>P_n(x)= \frac 1 {2^n} \sum_{k=0}^n {n\choose k}^2 (x-1)^{n-k}(x+1)^k=\sum_{k=0}^n {n\choose k} {-n-1\choose k} \left( \frac{1-x}{2} \right)^k= 2^n\cdot \sum_{k=0}^n x^k {n \choose k}{\frac{n+k-1}2\choose n},</math>
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| where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the [[binomial coefficient]].
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| The first few Legendre polynomials are:
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| <center><table style="background:white;">
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| <tr>
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| <td width="20%" align="center">'''n'''</td>
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| <td align="center"><math>P_n(x)\,</math></td>
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| </tr>
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| <tr>
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| <td align="center">0</td>
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| <td align="center"><math>1\,</math></td>
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| </tr>
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| <tr>
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| <td align="center">1</td>
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| <td align="center"><math>x\,</math></td>
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| </tr>
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| <tr>
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| <td align="center">2</td>
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| <td align="center"><math>\begin{matrix}\frac12\end{matrix} (3x^2-1) \,</math></td>
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| </tr>
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| <tr>
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| <td align="center">3</td>
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| <td align="center"><math>\begin{matrix}\frac12\end{matrix} (5x^3-3x) \,</math></td>
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| </tr>
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| <tr>
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| <td align="center">4</td>
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| <td align="center"><math>\begin{matrix}\frac18\end{matrix} (35x^4-30x^2+3)\,</math>
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| </tr>
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| <tr>
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| <td align="center">5</td>
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| <td align="center"><math>\begin{matrix}\frac18\end{matrix} (63x^5-70x^3+15x)\,</math>
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| </tr>
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| <tr>
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| <td align="center">6</td>
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| <td align="center"><math>\begin{matrix}\frac1{16}\end{matrix} (231x^6-315x^4+105x^2-5)\,</math>
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| </tr>
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| <tr>
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| <td align="center">7</td>
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| <td align="center"><math>\begin{matrix}\frac1{16}\end{matrix} (429x^7-693x^5+315x^3-35x)\,</math>
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| </tr>
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| <tr>
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| <td align="center">8</td>
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| <td align="center"><math>\begin{matrix}\frac1{128}\end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35)\,</math>
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| </tr>
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| <tr>
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| <td align="center">9</td>
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| <td align="center"><math>\begin{matrix}\frac1{128}\end{matrix} (12155x^9-25740x^7+18018x^5-4620x^3+315x)\,</math>
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| </tr>
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| <tr>
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| <td align="center">10</td>
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| <td align="center"><math>\begin{matrix}\frac1{256}\end{matrix} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)\,</math>
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| </tr>
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| </table>
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| </center>
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| The graphs of these polynomials (up to ''n'' = 5) are shown below:
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| [[File:Legendrepolynomials6.svg|640px|center]]
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| == Orthogonality ==
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| An important property of the Legendre polynomials is that they are [[orthogonal]] with respect to the [[Lp space|L<sup>2</sup> inner product]] on the interval −1 ≤ ''x'' ≤ 1:
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| :<math>\int_{-1}^{1} P_m(x) P_n(x)\,dx = {2 \over {2n + 1}} \delta_{mn}</math>
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| (where δ<sub>''mn''</sub> denotes the [[Kronecker delta]], equal to 1 if ''m'' = ''n'' and to 0 otherwise).
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| In fact, an alternative derivation of the Legendre polynomials is by carrying out the [[Gram-Schmidt process]] on the polynomials {1, ''x'', ''x''<sup>2</sup>, ...} with respect to this inner product. The reason for this orthogonality property is that the Legendre differential equation can be viewed as a [[Sturm–Liouville theory|Sturm–Liouville problem]], where the Legendre polynomials are [[eigenfunction]]s of a [[hermitian operator|Hermitian]] [[differential operator]]: | |
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| :<math>{d \over dx} \left[ (1-x^2) {d \over dx} P(x) \right] = -\lambda P(x),</math>
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| where the eigenvalue λ corresponds to ''n''(''n'' + 1).
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| ==Applications of Legendre polynomials in physics==
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| The Legendre polynomials were first introduced in 1782 by [[Adrien-Marie Legendre]]<ref>M. Le Gendre, "Recherches sur l'attraction des sphéroïdes homogènes," ''Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées'', Tome X, pp. 411-435 (Paris, 1785). [Note: Legendre submitted his findings to the Academy in 1782, but they were published in 1785.] Available on-line (in French) at: http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf .</ref> as the coefficients in the expansion of the [[Newtonian potential]]
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| :<math>
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| \frac{1}{\left| \mathbf{x}-\mathbf{x}^\prime \right|} = \frac{1}{\sqrt{r^2+r^{\prime 2}-2rr'\cos\gamma}} = \sum_{\ell=0}^{\infty} \frac{r^{\prime \ell}}{r^{\ell+1}} P_{\ell}(\cos \gamma)
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| </math>
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| where <math>r</math> and <math>r'</math> are the lengths of the vectors <math>\mathbf{x}</math> and <math>\mathbf{x}^\prime</math> respectively and <math>\gamma</math> is the angle between those two vectors. The series converges when <math>r>r'</math>. 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.
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| Legendre polynomials occur in the solution of [[Laplace equation]] of the [[electric potential|potential]], <math>\nabla^2 \Phi(\mathbf{x})=0</math>, 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>\widehat{\mathbf{z}}</math> is the axis of symmetry and <math>\theta</math> is the angle between the position of the observer and the <math>\widehat{\mathbf{z}}</math> axis (the zenith angle), the solution for the potential will be
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| :<math>
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| \Phi(r,\theta)=\sum_{\ell=0}^{\infty} \left[ A_\ell r^\ell + B_\ell r^{-(\ell+1)} \right] P_\ell(\cos\theta).
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| </math>
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| <math>A_\ell</math> and <math>B_\ell</math> are to be determined according to the boundary condition of each problem.<ref>Jackson, J.D. ''Classical Electrodynamics'', 3rd edition, Wiley & Sons, 1999. page 103</ref>
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| They also appear when solving Schrödinger equation in three dimensions for a central force.
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| '''Legendre polynomials in multipole expansions'''
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| [[File:Point axial multipole.svg|frame|right|Figure 2]]
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| Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
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| :<math>
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| \frac{1}{\sqrt{1 + \eta^{2} - 2\eta x}} = \sum_{k=0}^{\infty} \eta^{k} P_{k}(x)
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| </math>
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| which arise naturally in [[multipole expansion]]s. The left-hand side of the equation is the [[generating function]] for the Legendre polynomials.
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| As an example, the [[electric potential]] <math>\Phi(r, \theta)</math> (in [[spherical coordinates]]) due to a [[point charge]] located on the ''z''-axis at <math>z=a</math> (Figure 2) varies like
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| :<math>
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| \Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^{2} + a^{2} - 2ar \cos\theta}}.
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| </math>
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| If the radius ''r'' of the observation point '''P''' is
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| greater than ''a'', the potential may be expanded in the Legendre polynomials
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| :<math>
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| \Phi(r, \theta) \propto
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| \frac{1}{r} \sum_{k=0}^{\infty} \left( \frac{a}{r} \right)^{k}
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| P_{k}(\cos \theta)
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| </math>
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| where we have defined ''η'' = ''a''/''r'' < 1 and ''x'' = cos ''θ''. This expansion is used to develop the normal [[multipole expansion]].
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| Conversely, if the radius ''r'' of the observation point '''P''' is
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| smaller than ''a'', the potential may still be expanded in the
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| Legendre polynomials as above, but with ''a'' and ''r'' exchanged.
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| This expansion is the basis of [[interior multipole expansion]].
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| == Additional properties of Legendre polynomials ==
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| Legendre polynomials are symmetric or antisymmetric, that is
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| :<math>P_n(-x) = (-1)^n P_n(x). \,</math><ref name="arfken">George B. Arfken, Hans J. Weber, ''Mathematical Methods for Physicists'', Elsevier Academic Press, 2005, pg. 753.</ref>
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| Since the differential equation and the orthogonality property are
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| independent of scaling, the Legendre polynomials' definitions are
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| "standardized" (sometimes called "normalization", but note that the
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| actual norm is not unity) by being scaled so that
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| :<math>P_n(1) = 1. \,</math>
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| The derivative at the end point is given by
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| :<math>P_n'(1) = \frac{n(n+1)}{2}. \, </math>
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| As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursion formula
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| :<math> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)\,</math>
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| and
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| :<math> {x^2-1 \over n} {d \over dx} P_n(x) = xP_n(x) - P_{n-1}(x).</math>
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| Useful for the integration of Legendre polynomials is
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| :<math>(2n+1) P_n(x) = {d \over dx} \left[ P_{n+1}(x) - P_{n-1}(x) \right].</math>
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| From the above one can see also that
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| :<math>{d \over dx} P_{n+1}(x) = (2n+1) P_n(x) + (2(n-2)+1) P_{n-2}(x) + (2(n-4)+1) P_{n-4}(x) + \ldots</math>
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| or equivalently
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| :<math>{d \over dx} P_{n+1}(x) = {2 P_n(x) \over \| P_n(x) \|^2} + {2 P_{n-2}(x) \over \| P_{n-2}(x) \|^2}+\ldots</math>
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| where <math>\| P_n(x) \|</math> is the norm over the interval −1 ≤ x ≤ 1
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| :<math>\| P_n(x) \| = \sqrt{\int _{- 1}^{1}(P_n(x))^2 \,dx} = \sqrt{\frac{2}{2 n + 1}}.</math>
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| From Bonnet’s recursion formula one obtains by induction the explicit representation
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| :<math>P_n(x) = \sum_{k=0}^n (-1)^k \begin{pmatrix} n \\ k \end{pmatrix}^2 \left( \frac{1+x}{2} \right)^{n-k} \left( \frac{1-x}{2} \right)^k.</math>
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| The [[Askey–Gasper inequality]] for Legendre polynomials reads
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| :<math>\sum_{j=0}^n P_j(x)\ge 0\qquad (x\ge -1).</math>
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| == Shifted Legendre polynomials ==
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| The '''shifted Legendre polynomials''' are defined as <math>\tilde{P_n}(x) = P_n(2x-1)</math>. Here the "shifting" function <math>x\mapsto 2x-1</math> (in fact, it is an [[affine transformation]]) is chosen such that it [[bijection|bijectively maps]] the interval [0, 1] to the interval [−1, 1], implying that the polynomials <math>\tilde{P_n}(x)</math> are orthogonal on [0, 1]:
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| :<math>\int_{0}^{1} \tilde{P_m}(x) \tilde{P_n}(x)\,dx = {1 \over {2n + 1}} \delta_{mn}.</math>
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| An explicit expression for the shifted Legendre polynomials is given by
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| :<math>\tilde{P_n}(x) = (-1)^n \sum_{k=0}^n {n \choose k} {n+k \choose k} (-x)^k.</math>
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| The analogue of [[Rodrigues' formula]] for the shifted Legendre polynomials is
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| :<math>\tilde{P_n}(x) = \frac{1}{n!} {d^n \over dx^n } \left[ (x^2 -x)^n \right].\, </math>
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| The first few shifted Legendre polynomials are:
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| <center>
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| {| class="wikitable"
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| |'''''n'''''
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| | align=center | <math>\tilde{P_n}(x)</math>
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| |-
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| | 0
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| | 1
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| |-
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| | 1
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| | <math>2x-1</math>
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| |-
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| | 2
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| | <math>6x^2-6x+1</math>
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| |-
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| | 3
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| | <math>20x^3-30x^2+12x-1</math>
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| |-
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| |4
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| | <math>70x^4-140x^3+90x^2-20x+1</math>
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| |}
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| </center>
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| ==Legendre functions==
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| As well as polynomial solutions, the Legendre equation has non-polynomial solutions represented by infinite series. These are the ''Legendre functions of the second kind'', denoted by <math>Q_n(x)</math>.
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| :<math>Q_n(x)=\frac{n!}{1.3\cdots(2n+1)}\left[x^{-(n+1)}+\frac{(n+1)(n+2)}{2(n+3)}x^{-(n+3)}+\frac{(n+1)(n+2)(n+3)(n+4)}{2.4(2n+3)(2n+5)}x^{-(n+5)}+\cdots\right]</math>
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| The differential equation
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| :<math>{d \over dx} \left[ (1-x^2) {d \over dx} f(x) \right] + n(n+1)f(x) = 0</math>
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| has the general solution
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| :<math>f(x)=AP_n(x)+BQ_n(x)</math>,
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| where ''A'' and ''B'' are constants.
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| == Legendre functions of fractional order ==
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| {{main|Legendre function}}
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| Legendre functions of fractional order exist and follow from insertion of fractional derivatives as defined by [[fractional calculus]] and non-integer [[factorial]]s (defined by the [[gamma function]]) into the [[Rodrigues' formula]]. The resulting functions continue to satisfy the Legendre differential equation throughout (−1,1), but are no longer regular at the endpoints. The fractional order Legendre function ''P''<sub>''n''</sub> agrees with the [[associated Legendre polynomial]] ''P''{{su|b=''n''|p=0}}.
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| ==See also==
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| * [[Associated Legendre function]]s
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| * [[Gaussian quadrature]]
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| * [[Gegenbauer polynomials]]
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| * [[Legendre rational functions]]
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| * [[Turán's inequalities]]
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| * [[Legendre wavelet]]
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| * [[Jacobi polynomials]]
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| * [[Spherical Harmonics]]
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| * {{Abramowitz_Stegun_ref2|8|332|22|773}}
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| * {{citation|last=Bayin|first=S.S.|year=2006|title=Mathematical Methods in Science and Engineering|publisher=Wiley}}, Chapter 2.
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| * {{citation|last=Belousov|first=S. L.|year=1962|title=Tables of normalized associated Legendre polynomials|series=Mathematical tables|volume=18|publisher=Pergamon Press}}.
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| * {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|year=1953|title=Methods of Mathematical Physics, Volume 1|publisher=Interscience Publischer, Inc|publication-place=New York}}.
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| *{{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
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| *{{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|authorlink=Tom H. Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
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| * {{Citation | author=Refaat El Attar | title= Legendre Polynomials and Functions | publisher= CreateSpace | year=2009 | isbn = 978-1-4414-9012-4}}
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| ==External links==
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| *[http://www.physics.drexel.edu/~tim/open/hydrofin A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen]
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| *{{springer|title=Legendre polynomials|id=p/l058050}}
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| *[http://mathworld.wolfram.com/LegendrePolynomial.html Wolfram MathWorld entry on Legendre polynomials]
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| *[http://math.fullerton.edu/mathews/n2003/LegendrePolyMod.html Module for Legendre Polynomials by John H. Mathews]
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| *[http://www.du.edu/~jcalvert/math/legendre.htm Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics]
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| *[http://www.morehouse.edu/facstaff/cmoore/Legendre%20Polynomials.htm The Legendre Polynomials by Carlyle E. Moore]
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| *[http://hyperphysics.phy-astr.gsu.edu/hbase/math/legend.html Legendre Polynomials from Hyperphysics]
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| {{DEFAULTSORT:Legendre Polynomials}}
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| [[Category:Special hypergeometric functions]]
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| [[Category:Orthogonal polynomials]]
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| [[Category:Polynomials]]
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