Legendre polynomials
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- Note: The term Legendre polynomials is sometimes used (wrongly) to indicate the associated Legendre polynomials.
In mathematics, Legendre functions are solutions to Legendre's differential equation:
- <math>{d \over dx} \left[ (1-x^2) {d \over dx} P(x) \right] + n(n+1)P(x) = 0.</math>
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 Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:
- <math>P_n(x) = (2^n n!)^{-1} {d^n \over dx^n } \left[ (x^2 -1)^n \right]. </math>
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:
- <math>\int_{-1}^{1} P_m(x) P_n(x)\,dx = {2 \over {2n + 1}} \delta_{mn}</math>
(where δmn 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.
These are the first few Legendre polynomials:
| n | <math>P_n(x)\,</math> |
| 0 | <math>1\,</math> |
| 1 | <math>x\,</math> |
| 2 | <math>\begin{matrix}\frac12\end{matrix} (3x^2-1) \,</math> |
| 3 | <math>\begin{matrix}\frac12\end{matrix} (5x^3-3x) \,</math> |
| 4 | <math>\begin{matrix}\frac18\end{matrix} (35x^4-30x^2+3)\,</math> |
| 5 | <math>\begin{matrix}\frac18\end{matrix} (63x^5-70x^3+15x)\,</math> |
| 6 | <math>\begin{matrix}\frac1{16}\end{matrix} (231x^6-315x^4+105x^2-5)\,</math> |
The graphs of these polynomials (up to n = 5) are shown below:
Contents |
Legendre polynomials in multipole expansions
Image:Point axial multipole.png
Legendre polynomials are useful in expanding functions of the form
- <math>
\frac{1}{1 + \eta^{2} - 2\eta x} = \sum_{k=0}^{\infty} \eta^{k} P_{k}(x) </math>
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 <math>\Phi(r, \theta)</math> (in spherical coordinates) due to a point charge located on the z-axis at <math>z=a</math> (Fig. 2) varies like
- <math>
\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^{2} + a^{2} - 2ar \cos\theta}} </math>
If the radius r of the observation point P is much greater than a, the potential may be expanded in the Legendre polynomials
- <math>
\Phi(r, \theta) \propto \frac{1}{r} \sum_{k=0}^{\infty} \left( \frac{a}{r} \right)^{k} P_{k}(\cos \theta) </math>
where we have taken <math>\eta \equiv a/r < 1</math> and <math>x \equiv \cos \theta</math>. 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
- <math>P_k(-x) = (-1)^k P_k(x). \,</math>
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
- <math>P_k(1) = 1. \,</math>
The derivative at the end point is given by
- <math>P_k'(1) = \frac{k(k+1)}{2}. \, </math>
Shifted Legendre polynomials
The shifted Legendre polynomials <math>\tilde{P_n}(x)</math> are defined as being orthogonal on the unit interval [0,1]
- <math>\int_{0}^{1} \tilde{P_m}(x) \tilde{P_n}(x)\,dx = {1 \over {2n + 1}} \delta_{mn}.</math>
An explicit expression for these polynomials is given by
- <math>\tilde{P_n}(x)=(-1)^n \sum_{k=0}^n {n \choose k} {n+k \choose k} (-x)^k.</math>
The analogue of Rodrigues' formula for the shifted Legendre polynomials is:
- <math>\tilde{P_n}(x) = ( n!)^{-1} {d^n \over dx^n } \left[ (x^2 -x)^n \right].\, </math>
The first few shifted Legendre polynomials are:
| n | <math>\tilde{P_n}(x)</math> |
| 0 | 1 |
| <math>1</math> | <math>2x-1</math> |
| 2 | <math>6x^2-6x+1</math> |
| 3 | <math>20x^3-30x^2+12x-1</math> |
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.
External links
References
- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 8 Legendre Functions and Chapter 22 Orthogonal Polynomials.)de:Legendre-Polynom
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