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| The Legendre polynomials are a set of polynomials $\{P_i\}_{i=0}^{\infty}$ each of order $i$ that satisfy Legendre's ODE: $$\frac{d}{dx}[(1-x)^2 P_n'(x)] + n(n+1)P_n(x) = 0$$. |
The Legendre polynomials are a set of polynomials $\{P_i\}_{i=0}^{\infty}$ each of order $i$ that satisfy Legendre's ODE: $$\frac{d}{dx}[(1-x)^2 P_n'(x)] + n(n+1)P_n(x) = 0$$. |
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Alternatively $P_n$ is an eigenfunction of the self-adjoint differential operator $\frac{d}{dx}(1-x)^2 \frac{d}{dx}$ with eigenvalue $-n(n+1)$.
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Alternatively $P_n$ is an eigenfunction of the self-adjoint differential operator $\frac{d}{dx}(1-x)^2 \frac{d}{dx}$ with eigenvalue $n(n+1)$.
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| The Legendre polynomials are also known as Legendre Functions of the first kind. |
The Legendre polynomials are also known as Legendre Functions of the first kind. |
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| By Sturm-Liouville theory, this means they're orthogonal over some interval with |
By Sturm-Liouville theory, this means they're orthogonal over some interval with |
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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
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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 gram-schmidt orthogonalization of the monomials $\{x^i\}$. The normalization used
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| is $\langle P_n \| P_n \rangle = 2 / (2n + 1)$, which makes $P_n(\pm 1) = (\pm 1)^n$ |
is $\langle P_n \| P_n \rangle = 2 / (2n + 1)$, which makes $P_n(\pm 1) = (\pm 1)^n$ |
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| Rodrigues's Formula (which can be generalized to some other polynomial sets) is a sometimes convenient form of $P_n$ in terms of derivatives: |
Rodrigues's Formula is a sometimes convenient form of $P_n$ in terms of |
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derivatives: |
| $$P_n(x) = \frac{1}{2^n n!} \left( \frac{d}{dx} \right)^n (x^2 - 1)^n$$ |
$$P_n(x) = \frac{1}{2^n n!} \left( \frac{d}{dx} \right)^n (x^2 - 1)^n$$ |
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| The first few explicitly are: |
The first few explicitly are: |
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| \begin{eqnarray*} |
\begin{eqnarray*} |
| P_0(x) &=& 1 \\ |
P_0(x) &=& 1 \\ |
| P_1(x) &=& x \\ |
P_1(x) &=& x \\ |
| P_2(x) &=& \frac{1}{2} (3x^2 - 1) \\ |
P_2(x) &=& \frac{1}{2} (3x^2 - 1) \\ |
| P_3(x) &=& \frac{1}{2} (5x^3 - 3x) \\ |
P_3(x) &=& \frac{1}{2} (5x^3 - 3x) \\ |
| P_4(x) &=& \frac{1}{8} (35x^4 - 30x^2 + 3) \\ |
P_4(x) &=& \frac{1}{8} (35x^4 - 30x^2 + 3) \\ |
| ... |
... |
| \end{eqnarray*} |
\end{eqnarray*} |
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| As all orthogonal polynomials do, these satisfy a three-term recurence relation: |
As all orthogonal polynomials do, these satisfy a three-term recurence relation: |
| $$(n+1)P_{n+1}(x) = (2n+1)xP_{n}(x) - (n-1)P_{n}(x)$$ |
$$(n+1)P_{n+1}(x) = (2n+1)xP_{n}(x) - (n-1)P_{n}(x)$$ |
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| The Legendre Functions of the second kind also satisfy the Legendre ODE but are not regular at the origin. |
The Legendre Functions of the second kind also satisfy the Legendre ODE but are not regular at the origin. |
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| Related are the Associated Legendre Functions, and Spherical Harmonics. |
Related are the Associated Legendre Functions, and Spherical Harmonics. |
