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(x)\right]+n(n+1)P(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 solution is finite (i.e. the series converges) provided |x| < 1. Furthermore, it is finite at x = ± 1 provided n is a non-negative integer, i.e. n = 0, 1, 2,... . In this case, the solutions 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)=(2^{n}n!)^{-1}{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}\right]P(x)=-\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	${\begin{matrix}{\frac {1}{2}}\end{matrix}}(3x^{2}-1)\,$
3	${\begin{matrix}{\frac {1}{2}}\end{matrix}}(5x^{3}-3x)\,$
4	${\begin{matrix}{\frac {1}{8}}\end{matrix}}(35x^{4}-30x^{2}+3)\,$
5	${\begin{matrix}{\frac {1}{8}}\end{matrix}}(63x^{5}-70x^{3}+15x)\,$
6	${\begin{matrix}{\frac {1}{16}}\end{matrix}}(231x^{6}-315x^{4}+105x^{2}-5)\,$

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'\cos \gamma }}}=\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 hold 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, $\nabla ^{2}\Phi (\mathbf {x} )$ , 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 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 }(\cos \theta ).

$A_{\ell }$ and $B_{\ell }$ are to be determined according to the boundary condition of each problem^[1].

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}-2\eta 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\cos \theta }}}

If the radius r of the observation point P is much 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 taken $\eta \equiv a/r<1$ and $x\equiv \cos \theta$ . This expansion is used to develop the normal multipole expansion.

Conversely, if the radius r of the observation point P is much 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}=(2n+1)xP_{n}-nP_{n-1}

and

{x^{2}-1 \over n}{d \over dx}P_{n}=xP_{n}-P_{n-1}.

Useful for the integration of Legendre polynomials is

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

Shifted Legendre polynomials

The shifted Legendre polynomials ${\tilde {P_{n}}}(x)$ are defined as being orthogonal on the unit interval [0,1]

\int _{0}^{1}{\tilde {P_{m}}}(x){\tilde {P_{n}}}(x)\,dx={1 \over {2n+1}}\delta _{mn}.

An explicit expression for these 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	$20x^{3}-30x^{2}+12x-1$