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':
:$\{d\; over\; dx\}\; left[\; (1-x^2)\; \{d\; over\; dx\}\; P(x)$
ight] + 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 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 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:
:$P\_n(x)\; =\; \{1\; over\; 2^n\; n!\}\; \{d^n\; over\; dx^n\; \}\; left[\; (x^2\; -1)^n$
ight].

Contents

The orthogonality property

Examples of Legendre polynomials

Applications of Legendre polynomials in physics

Additional properties of Legendre polynomials

Shifted Legendre polynomials

Legendre polynomials of fractional order

See also

External links

References

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'', ''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
:$\{d\; over\; dx\}\; left[\; (1-x^2)\; \{d\; over\; dx\}\; P(x)$
ight] = -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
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
:$$
rac{1}{left| mathbf{x}-mathbf{x}^prime
ight|} = rac{1}{sqrt{r^2+r^{prime 2}-2rr'cosgamma}} = sum_{ell=0}^{infty} rac{r^{prime ell}}{r^{ell+1}} P_{ell}(cos gamma)

where $r$ and $r\text{'}$ 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 $r>r\text{'}$.
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 $mathbf\{x\}\text{'}$. 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, $$
abla^2 Phi(mathbf{x}), in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where $widehat\{mathbf\{z\}\}$ is the axis of symmetry and $heta$ is the angle between the position of the observer and the $widehat\{mathbf\{z\}\}$ axis, the solution for the potential will be
:$$
Phi(r, heta)=sum_{ell=0}^{infty} left[ A_ell r^ell + B_ell r^{-(ell+1)}
ight] P_ell(cos heta).

$A\_ell$ 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):
:$$
rac{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 $Phi(r,\; heta)$ (in spherical coordinates) due to a point charge located on the ''z''-axis at $z=a$ (Fig. 2) varies like
:$$
Phi (r, heta ) propto rac{1}{R} = rac{1}{sqrt{r^{2} + a^{2} - 2ar cos heta}}.

If the radius ''r'' of the observation point 'P' is
greater than ''a'', the potential may be expanded in the Legendre polynomials
:$$
Phi(r, heta) propto
rac{1}{r} sum_{k=0}^{infty} left( rac{a}{r}
ight)^{k}
P_{k}(cos heta)

where we have defined 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.

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
:
Legendre polynomials can be constructed using the three term recurrence relations
:$(n+1)\; P\_\{n+1\}\; =\; (2n+1)\; x\; P\_n\; -\; n\; P\_\{n-1\},$
and
:$\{x^2-1\; over\; n\}\; \{d\; over\; dx\}\; P\_n\; =\; xP\_n\; -\; P\_\{n-1\}.$
Useful for the integration of Legendre polynomials is
:$(2n+1)\; P\_n\; =\; \{d\; over\; dx\}\; left[\; P\_\{n+1\}\; -\; P\_\{n-1\}$
ight].

Shifted Legendre polynomials

The shifted Legendre polynomials $ilde\{P\_n\}(x)$ are defined as being orthogonal on the unit interval [0,1]
:$int\_\{0\}^\{1\}\; ilde\{P\_m\}(x)\; ilde\{P\_n\}(x),dx\; =\; \{1\; over\; \{2n\; +\; 1\}\}\; delta\_\{mn\}.$
An explicit expression for these polynomials is given by
:$ilde\{P\_n\}(x)=(-1)^n\; sum\_\{k=0\}^n\; \{n\; choose\; k\}\; \{n+k\; choose\; k\}\; (-x)^k.$
The analogue of Rodrigues' formula for the shifted Legendre polynomials is:
:$ilde\{P\_n\}(x)\; =\; (\; n!)^\{-1\}\; \{d^n\; over\; dx^n\; \}\; left[\; (x^2\; -x)^n$
ight].,
The first few shifted Legendre polynomials are:

{| class="wikitable"
|'n'
| align=center | $ilde\{P\_n\}(x)$
|-
| 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.

See also

★ Gaussian quadrature

★ Associated Legendre polynomials

★ Legendre rational functions

External links

★ A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen

★ Wolfram MathWorld entry on Legendre polynomials

★ Module for Legendre Polynomials by John H. Mathews

★ Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics

References

1. Jackson, J.D. ''Classical Electrodynamics'', 3rd edition, Wiley & Sons, 1999. page 103


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