Legendre Polynomial/Examples/P2
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Example of Legendre Polynomial
The $2$nd Legendre polynomial is:
- $\map {P_2} x = \dfrac 1 2 \paren {3 x^2 - 1}$
Proof
\(\ds \forall n \in \N: \, \) | \(\ds \paren {n + 1} \map {P_{n + 1} } x\) | \(=\) | \(\ds \paren {2 n + 1} x \map {P_n} x - n \map {P_{n - 1} } x\) | Bonnet's Recursion Formula | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \map {P_2} x\) | \(=\) | \(\ds \paren {2 \times 1 + 1} x \map {P_1} x - 1 \map {P_0} x\) | setting $n = 1$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 3 x \map {P_1} x - \map {P_0} x\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 x \paren x - 1\) | Legendre Polynomial $P_0$ and Legendre Polynomial $P_1$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {P_2} x\) | \(=\) | \(\ds \dfrac 1 2 \paren {3 x^2 - 1}\) | simplifying |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generating function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generating function