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'Legendre polynomial'
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| Title of object: |
Legendre polynomial |
| Canonical Name: |
LegendrePolynomials |
| Type: |
Definition |
| Created on: |
2005-04-22 01:25:54 |
| Modified on: |
2006-02-18 20:06:41 |
| Classification: |
msc:33C45 |
| Defines: |
Rodrigues's Formula |
Revision comment (for changes between this and next version):
| Changes for correction #8681 ('hmmm?'). |
Preamble:
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Content:
The Legendre polynomials are a set of polynomials $\{P_i\}_{i=0}^{\infty}$ each of order $i$ that satisfy Legendre's ODE: $$\frac{d}{dx}[(1-x^2) P_n'(x)] + n(n+1)P_n(x) = 0.$$
Alternatively $P_n$ is an eigenfunction of the self-adjoint differential operator $\frac{d}{dx}(1-x^2) \frac{d}{dx}$ with eigenvalue $-n(n+1)$.
The Legendre polynomials are also known as Legendre functions of the first kind.
By Sturm-Liouville theory, this means they're orthogonal over some interval with
some weight function. In fact it can be shown that, they're orthogonal on $[-1, 1]$ with weight function $W(x) = 1$. As with any set of orthogonal polynomials, this can be used to generate them (up to normalization) by Gramm-Schmidt orthogonalization of the monomials $\{x^i\}$. The normalization used
is $\langle P_n \| P_n \rangle = 2 / (2n + 1)$, which makes $P_n(\pm 1) = (\pm 1)^n$
Rodrigues's Formula (which can be generalized to some other polynomial sets) is a sometimes convenient form of $P_n$ in terms of derivatives:
$$P_n(x) = \frac{1}{2^n n!} \left( \frac{d}{dx} \right)^n (x^2 - 1)^n$$
The first few explicitly are:
\begin{eqnarray*}
P_0(x) &=& 1 \\
P_1(x) &=& x \\
P_2(x) &=& \frac{1}{2} (3x^2 - 1) \\
P_3(x) &=& \frac{1}{2} (5x^3 - 3x) \\
P_4(x) &=& \frac{1}{8} (35x^4 - 30x^2 + 3) \\
...
\end{eqnarray*}
As all orthogonal polynomials do, these satisfy a three-term recurrence relation:
$$(n+1)P_{n+1}(x) = (2n+1)xP_{n}(x) - (n)P_{n-1}(x)$$
The Legendre functions of the second kind also satisfy the Legendre ODE but are not regular at the origin.
Related are the associated Legendre functions, and spherical harmonics. |
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