Definition:Legendre Polynomial

Definition
The Legendre polynomials are the solutions to Legendre's differential equation.

These solutions form a polynomial sequence of orthogonal polynomials on the interval $\closedint {-1} 1$.

Bonnet's Recursion Formula
Legendre polynomials can be found using Bonnet's Recursion Formula.


 * $\paren {n + 1} \map {P_{n + 1} } x = \paren {2 n + 1} x \map {P_n} x - n \map {P_{n - 1} } x$

Length of Legendre Polynomial
$\displaystyle \norm {\map {P_n} x} = \sqrt {\int_{-1}^1 \paren {\map {P_n} X}^2 \rd x} = \sqrt {\frac 2 {2 n + 1} }$

Proof of Length
Applying Bonnet's Recursion Formula for $n - 1$:


 * $n \map {P_n} x = \paren {2 n - 1} x \map {P_{n - 1} } x - \paren {n - 1} \map {P_{n - 2} } x$

so:
 * $\map {P_n} x = \dfrac {2 n - 1} n x \map {P_{n - 1} } x - \dfrac {n - 1} n \map {P_{n - 2} } x$

Substituting for $\map {P_n} x$:

From Orthogonality of Legendre Polynomials:
 * $\displaystyle \int_{-1}^1 \map {P_n} x \, \map {P_m} x \rd x = 0 \iff n \ne m$

so:
 * $\displaystyle (1): \quad \norm {\map {P_n} x}^2 = \frac {2 n - 1} n \int_{-1}^1 x \map {P_n} x \, \map {P_{n - 1} } x \rd x$

From Bonnet's Recursion Formula:


 * $\displaystyle x \map {P_n} x = \frac {n + 1} {2 n + 1} \map {P_{n + 1} } x + \frac n {2 n + 1} \map {P_{n - 1} } x$

Substituting for $x \map {P_n} x$ in $(1)$:

Thus:

Most of this cancels out, leaving:


 * $\norm {\map {P_n} x}^2 = \dfrac {\norm {\map {P_0} x}^2} {2 n + 1}$

It remains to compute the length of the first Legendre polynomial:

Thus:
 * $\norm {\map {P_n} x}^2 = \dfrac 2 {2 n + 1}$

and so taking the square root:
 * $\norm {\map {P_n} x} = \sqrt {\dfrac 2 {2 n + 1} }$