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:

{d over dx} left[(1-x^2) {d over dx} P_n(x) right] + n(n+1)P_n(x) = 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= ± 1 so, in general, a series solution about the origin will only converge for |x| < 1. When n is an integer, the solution P_n(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 P_n(1)=1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial P_n(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:

P_n(x) = {1 over 2^n n!} {d^n over dx^n } left[(x^2 -1)^n 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 ≤ x ≤ 1:

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

(where δ_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, x², ...} 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

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

where the eigenvalue λ corresponds to n(n+1).

Examples of Legendre polynomials

These are the first few Legendre polynomials:

n	$P_n(x),$
0	$1,$
1	$x,$
2	$begin{matrix}frac12end{matrix} (3x^2-1) ,$
3	$begin{matrix}frac12end{matrix} (5x^3-3x) ,$
4	$begin{matrix}frac18end{matrix} (35x^4-30x^2+3),$
5	$begin{matrix}frac18end{matrix} (63x^5-70x^3+15x),$
6	$begin{matrix}frac1{16}end{matrix} (231x^6-315x^4+105x^2-5),$
7	$begin{matrix}frac1{16}end{matrix} (429x^7-693x^5+315x^3-35x),$
8	$begin{matrix}frac1{128}end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35),$
9	$begin{matrix}frac1{128}end{matrix} (12155x^9-25740x^7+18018x^5-4620x^3+315x),$
10	$begin{matrix}frac1{256}end{matrix} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63),$

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

where $r$ and $r'$ are the lengths of the vectors $mathbf{x}$ and $mathbf{x}^prime$ respectively and $gamma$ is the angle between those two vectors. This expansion holds where $r>r'$ . This expression is used, for example, to obtain the potential of a point charge, felt at point $mathbf{x}$ while the charge is located at point $mathbf{x}'$ . 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, $nabla^2 Phi(mathbf{x})=0$ , 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 azimuthal angle). Where $widehat{mathbf{z}}$ is the axis of symmetry and $theta$ is the angle between the position of the observer and the $widehat{mathbf{z}}$ axis (the zenith angle), the solution for the potential will be

Phi(r,theta)=sum_{ell=0}^{infty} left[A_ell r^ell + B_ell r^{-(ell+1)} right] P_ell(costheta).

$A_ell$ and $B_ell$ are to be determined according to the boundary condition of each problem.

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{1}{sqrt{1 + eta^{2} - 2eta x}} = sum_{k=0}^{infty} eta^{k} P_{k}(x)

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 $Phi(r, theta)$ (in spherical coordinates) due to a point charge located on the z-axis at $z=a$ (Fig. 2) varies like

Phi (r, theta ) propto frac{1}{R} = frac{1}{sqrt{r^{2} + a^{2} - 2ar costheta}}.

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

Phi(r, theta) propto frac{1}{r} sum_{k=0}^{infty} left(frac{a}{r} right)^{k} P_{k}(cos theta)

where we have defined $eta = a/r < 1$ and $x = cos theta$ . 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.

Additional properties of Legendre polynomials

Legendre polynomials are symmetric or antisymmetric, that is

P_k(-x) = (-1)^k P_k(x). ,

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but note that the actual norm is not unity) by being scaled so that

P_k(1) = 1. ,

The derivative at the end point is given by

P_k'(1) = frac{k(k+1)}{2}. ,

Legendre polynomials can be constructed using the three term recurrence relations

(n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x),

and

{x^2-1 over n} {d over dx} P_n(x) = xP_n(x) - P_{n-1}(x).

Useful for the integration of Legendre polynomials is

(2n+1) P_n(x) = {d over dx} left[P_{n+1}(x) - P_{n-1}(x) right].

Shifted Legendre polynomials

The shifted Legendre polynomials are defined as $tilde{P_n}(x) = P_n(2x-1)$ . 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 $tilde{P_n}(x)$ are orthogonal on [0,1]:

int_{0}^{1} tilde{P_m}(x) tilde{P_n}(x),dx = {1 over {2n + 1}} delta_{mn}.

An explicit expression for the shifted Legendre polynomials is given by

tilde{P_n}(x) = (-1)^n sum_{k=0}^n {n choose k} {n+k choose k} (-x)^k.

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

tilde{P_n}(x) = (n!)^{-1} {d^n over dx^n } left[(x^2 -x)^n right].,

The first few shifted Legendre polynomials are:

n	$tilde{P_n}(x)$
0	1
1	$2x-1$
2	$6x^2-6x+1$
3	$20x^3-30x^2+12x-1$

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.