# Legendre polynomials

﻿
Legendre polynomials

:"Note: People sometimes refer to the more general associated Legendre polynomials as simply "Legendre polynomials"."

In mathematics, Legendre functions are solutions to Legendre's differential equation:

:$\left\{d over dx\right\} left \left[ \left(1-x^2\right) \left\{d over dx\right\} P_n\left(x\right) ight\right] + n\left(n+1\right)P_n\left(x\right) = 0.$

They are named after Adrien-Marie Legendre. This 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 equations) in spherical coordinates.

The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at "x"= &plusmn; 1 so, in general, a series solution about the origin will only converge for |"x"| &lt; 1. When "n" is an integer, the solution Pn(x) that is regular at "x"=1 is also regular at "x"=-1, and the series for this solution terminates (i.e. is a polynomial).

These solutions for "n" = 0, 1, 2,... (with the normalization "Pn"(1)=1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial P"n"("x") is an "n"th-degree polynomial. It may be expressed using Rodrigues' formula:

:$P_n\left(x\right) = \left\{1 over 2^n n!\right\} \left\{d^n over dx^n \right\} left \left[ \left(x^2 -1\right)^n ight\right] .$

The orthogonality property

An important property of the Legendre polynomials is that they are orthogonal with respect to the L2 inner product on the interval −1 &le; "x" &le; 1:

:$int_\left\{-1\right\}^\left\{1\right\} P_m\left(x\right) P_n\left(x\right),dx = \left\{2 over \left\{2n + 1 delta_\left\{mn\right\}$

(where &delta;"mn" denotes the Kronecker delta, equal to 1 if "m" = "n" and to 0 otherwise). In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1, "x", "x"2, ...} 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 problem

:$\left\{d over dx\right\} left \left[ \left(1-x^2\right) \left\{d over dx\right\} P\left(x\right) ight\right] = -lambda P\left(x\right),$

where the eigenvalue &lambda; corresponds to "n"("n"+1).

Examples of Legendre polynomials

These are the first few Legendre polynomials:

 n $P_n\left(x\right),$ 0 $1,$ 1 $x,$ 2 3 4 5 6 7 8 9 10

The graphs of these polynomials (up to "n" = 5) are shown below:

Applications of Legendre polynomials in physics

Legendre polynomials are useful in expanding functions like

:$frac\left\{1\right\}\left\{left| mathbf\left\{x\right\}-mathbf\left\{x\right\}^prime ight = frac\left\{1\right\}\left\{sqrt\left\{r^2+r^\left\{prime 2\right\}-2rr\text{'}cosgamma = sum_\left\{ell=0\right\}^\left\{infty\right\} frac\left\{r^\left\{prime ell\left\{r^\left\{ell+1 P_\left\{ell\right\}\left(cos gamma\right)$

where $r$ and $r\text{'}$ are the lengths of the vectors $mathbf\left\{x\right\}$ and $mathbf\left\{x\right\}^prime$ respectively and $gamma$ is the angle between those two vectors. This expansion holds where $r>r\text{'}$.This expression is used, for example, to obtain the potential of a point charge, felt at point $mathbf\left\{x\right\}$ while the charge is located at point $mathbf\left\{x\right\}\text{'}$. The expansion using Legendre polynomials might be useful when integrating this expression over a continuous charge distribution.

Legendre polynomials occur in the solution of Laplace equation of the potential, $abla^2 Phi\left(mathbf\left\{x\right\}\right)=0$, 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 azimuthal angle). Where $widehat\left\{mathbf\left\{z$ is the axis of symmetry and $heta$ is the angle between the position of the observer and the $widehat\left\{mathbf\left\{z$ axis (the zenith angle), the solution for the potential will be

:$Phi\left(r, heta\right)=sum_\left\{ell=0\right\}^\left\{infty\right\} left \left[ A_ell r^ell + B_ell r^\left\{-\left(ell+1\right)\right\} ight\right] P_ell\left(cos heta\right).$

$A_ell$ and $B_ell$ are to be determined according to the boundary condition of each problem [Jackson, J.D. "Classical Electrodynamics", 3rd edition, Wiley & Sons, 1999. page 103] .

Legendre polynomials in multipole expansions Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently)::$frac\left\{1\right\}\left\{sqrt\left\{1 + eta^\left\{2\right\} - 2eta x = sum_\left\{k=0\right\}^\left\{infty\right\} eta^\left\{k\right\} P_\left\{k\right\}\left(x\right)$

which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.

As an example, the electric potential $Phi\left(r, heta\right)$ (in spherical coordinates) due to a point charge located on the "z"-axis at $z=a$ (Fig. 2) varies like

:$Phi \left(r, heta \right) propto frac\left\{1\right\}\left\{R\right\} = frac\left\{1\right\}\left\{sqrt\left\{r^\left\{2\right\} + a^\left\{2\right\} - 2ar cos heta.$

If the radius "r" of the observation point P isgreater than "a", the potential may be expanded in the Legendre polynomials

:$Phi\left(r, heta\right) proptofrac\left\{1\right\}\left\{r\right\} sum_\left\{k=0\right\}^\left\{infty\right\} left\left( frac\left\{a\right\}\left\{r\right\} ight\right)^\left\{k\right\} P_\left\{k\right\}\left(cos heta\right)$

where we have defined $eta = a/r < 1$ and $x = cos heta$. This expansion is used to develop the normal multipole expansion.

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

Legendre polynomials are symmetric or antisymmetric, that is:$P_k\left(-x\right) = \left(-1\right)^k P_k\left(x\right). ,$

Since the differential equation and the orthogonality property areindependent of scaling, the Legendre polynomials' definitions are"standardized" (sometimes called "normalization", but note that theactual norm is not unity) by being scaled so that:$P_k\left(1\right) = 1. ,$

The derivative at the end point is given by:$P_k\text{'}\left(1\right) = frac\left\{k\left(k+1\right)\right\}\left\{2\right\}. ,$

Legendre polynomials can be constructed using the three term recurrence relations:$\left(n+1\right) P_\left\{n+1\right\}\left(x\right) = \left(2n+1\right) x P_n\left(x\right) - n P_\left\{n-1\right\}\left(x\right),$

and

:$\left\{x^2-1 over n\right\} \left\{d over dx\right\} P_n\left(x\right) = xP_n\left(x\right) - P_\left\{n-1\right\}\left(x\right).$

Useful for the integration of Legendre polynomials is:$\left(2n+1\right) P_n\left(x\right) = \left\{d over dx\right\} left \left[ P_\left\{n+1\right\}\left(x\right) - P_\left\{n-1\right\}\left(x\right) ight\right] .$

Shifted Legendre polynomials

The shifted Legendre polynomials are defined as $ilde\left\{P_n\right\}\left(x\right) = P_n\left(2x-1\right)$. Here the "shifting" function $xmapsto 2x-1$ (in fact, it is an affine transformation) is chosen such that it bijectively maps the interval " [0,1] " to the interval " [−1,1] ", implying that the polynomials $ilde\left\{P_n\right\}\left(x\right)$ are orthogonal on " [0,1] ":

:$int_\left\{0\right\}^\left\{1\right\} ilde\left\{P_m\right\}\left(x\right) ilde\left\{P_n\right\}\left(x\right),dx = \left\{1 over \left\{2n + 1 delta_\left\{mn\right\}.$

An explicit expression for the shifted Legendre polynomials is given by:$ilde\left\{P_n\right\}\left(x\right) = \left(-1\right)^n sum_\left\{k=0\right\}^n \left\{n choose k\right\} \left\{n+k choose k\right\} \left(-x\right)^k.$

The analogue of Rodrigues' formula for the shifted Legendre polynomials is:

:$ilde\left\{P_n\right\}\left(x\right) = \left( n!\right)^\left\{-1\right\} \left\{d^n over dx^n \right\} left \left[ \left(x^2 -x\right)^n ight\right] .,$

The first few shifted Legendre polynomials are:

Legendre polynomials of fractional order

Legendre polynomials of fractional order exist and follow from insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the gamma function) into the Rodrigues' formula. The exponents of course become fractional exponents which represent roots.

ee also

* Associated Legendre functions
* Legendre rational functions
* Turán's inequalities

* [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]
* [http://mathworld.wolfram.com/LegendrePolynomial.html Wolfram MathWorld entry on Legendre polynomials]
* [http://math.fullerton.edu/mathews/n2003/LegendrePolyMod.html Module for Legendre Polynomials by John H. Mathews]
* [http://www.du.edu/~jcalvert/math/legendre.htm Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics]

References

*
* Belousov, S. L. (1962), "Tables of normalized associated Legendre polynomials", Mathematical tables series Vol. 18, Pergamon Press, 379p.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Legendre wavelet — Legendre wavelets: spherical harmonic wavelets = Compactly supported wavelets derived from Legendre polynomials are termed spherical harmonic or Legendre wavelets [1] . Legendre functions have widespread applications in which spherical coordinate …   Wikipedia

• Legendre-Polynome — Die Legendre Polynome, auch zonale Kugelfunktionen genannt, sind die partikulären Lösungen der legendreschen Differentialgleichung. Sie sind spezielle reelle oder komplexe Polynome, die ein orthogonales Funktionensystem bilden. Benannt sind sie… …   Deutsch Wikipedia

• Legendre-Polynom — Die Legendre Polynome (nach Adrien Marie Legendre), auch zonale Kugelfunktionen genannt, sind spezielle Polynome, die auf dem Intervall [ 1,1] ein orthogonales Funktionensystem bilden. Sie sind die partikulären Lösungen der legendreschen… …   Deutsch Wikipedia

• Legendre rational functions — In mathematics the Legendre rational functions are a sequence of functions which are both rational and orthogonal. A rational Legendre function of degree n is defined as::R n(x) = frac{sqrt{2{x+1},L nleft(frac{x 1}{x+1} ight)where L n(x) is a… …   Wikipedia

• Legendre, Adrien-Marie — (1752 1833)    mathematician    Born in Paris, Adrien Marie Legendre was commissioned by the Convention at the time of the revolution of 1789 to work on geodesics and, in doing so, enriched the study of trigonometry, developing a method, for… …   France. A reference guide from Renaissance to the Present

• Classical orthogonal polynomials — In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical… …   Wikipedia

• Associated Legendre function — Note: This article describes a very general class of functions. An important subclass of these functions mdash;those with integer ell and m mdash;are commonly called associated Legendre polynomials , even though they are not polynomials when m is …   Wikipedia

• Orthogonal polynomials — In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the… …   Wikipedia

• Polinomios de Legendre — Saltar a navegación, búsqueda En matemáticas al resolver la formula de Rodrigues, las Funciones de Legendre son las soluciones a las Ecuaciones Diferenciales de Legendre: llamadas así por el matemático francés Adrien Marie Legendre. Estas… …   Wikipedia Español

• Adrien-Marie Legendre — Infobox Scientist name = Adrien Marie Legendre caption = Adrien Marie Legendre birth date = birth date|1752|9|18|mf=y birth place = Paris, France death date = death date and age|1833|1|10|1752|9|18|mf=y death place = Paris, France residence =… …   Wikipedia