sociology - Legendre polynomials

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:

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

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:

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

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.

These are the first few Legendre polynomials:

n	$P n (x)$
0	1
$1$	$x$
2	$(1 / 2)(3 x 2 - 1)$
3	$(1 / 2)(5 x 3 - 3 x)$
4	$(1 / 8)(35 x 4 - 30 x 2 + 3)$
5	$(1 / 8)(63 x 5 - 70 x 3 + 15 x)$
6	$(1 / 16)(231 x 6 - 315 x 4 + 105 x 2 - 5)$

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

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

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

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
$1$	$2 x - 1$
2	$6 x 2 - 6 x + 1$
3	$20 x 3 - 30 x 2 + 12 x - 1$

References

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapters 22 and 8.)

Categories: Polynomials | Special functions | Numerical analysis