# Definition:Legendre Polynomial

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## Definition

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

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

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## Examples

The first five Legendre polynomials are:

- $\map {P_0} x = 1$

- $\map {P_1} x = x$

- $\map {P_2} x = \dfrac 1 2 \paren {3 x^2 - 1}$

- $\map {P_3} x = \dfrac 1 2 \paren {5 x^3 - 3 x}$

- $\map {P_4} x = \dfrac 1 8 \paren {35 x^4 - 30 x^2 + 3}$

## 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

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$\ds \norm {\map {P_n} x} = \sqrt {\int_{-1}^1 \paren {\map {P_n} X}^2 \rd x} = \sqrt {\frac 2 {2 n + 1} }$

## 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 $\map {P_n} x$:

\(\ds \norm {\map {P_n} x}^2\) | \(=\) | \(\ds \int_{-1}^1 \map {P_n} x \map {P_n} x \rd x\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \int_{-1}^1 \map {P_n} x \paren {\frac {2 n - 1} n x \map {P_{n - 1} } x - \frac {n - 1} n \map {P_{n - 2} } x} \rd x\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {2 n - 1} n \int_{-1}^1 x \map {P_n} x \map {P_{n - 1} } x \rd x - \frac {n - 1} n \int_{-1}^1 \map {P_n} x \map {P_{n - 2} } x \rd x\) | Linear Combination of Integrals |

From Orthogonality of Legendre Polynomials:

- $\ds \int_{-1}^1 \map {P_n} x \map {P_m} x \rd x = 0 \iff n \ne m$

so:

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

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 $(1)$:

\(\ds \norm {\map {P_n} x}^2\) | \(=\) | \(\ds \frac {2 n - 1} n \frac {n + 1} {2 n + 1} \int_{-1}^1 \map {P_{n + 1} } x \map {P_{n - 1} } x \rd x + \frac {2 n - 1} {2 n + 1} \int_{-1}^1 \map {P_{n - 1} } x \map {P_{n - 1} } x \rd x\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac{2 n - 1} {2 n + 1} \int_{-1}^1 \map {P_{n - 1} } x \map {P_{n - 1} } x \rd x\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {2 n - 1} {2 n + 1} \norm {\map {P_{n - 1} } x}^2\) |

Thus:

\(\ds \norm {\map {P_n} x}^2\) | \(=\) | \(\ds \frac {2 n - 1} {2 n + 1} \norm {\map {P_{n - 1} } x}^2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {2 n - 1} {2 n + 1} \frac {2 n - 3} {2 n - 1} \norm {\map {P_{n - 2} } x}^2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \cdots\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {2 n - 1} {2 n + 1} \frac {2 n - 3} {2 n - 1} \frac {2 n - 5} {2 n - 3} \dotsm \frac 3 5 \frac 1 3 \norm {\map {P_0} x}^2\) |

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:

\(\ds \norm {\map {P_0} x}^2\) | \(=\) | \(\ds \int_{-1}^1 1 \rd x\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \bigintlimits x {-1} 1\) | Primitive of Constant | |||||||||||

\(\ds \) | \(=\) | \(\ds 2\) |

Thus:

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

and so taking the square root:

- $\norm {\map {P_n} x} = \sqrt {\dfrac 2 {2 n + 1} }$

$\blacksquare$

## Source of Name

This entry was named for Adrien-Marie Legendre.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Legendre polynomials** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Legendre's differential equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Legendre's differential equation** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Legendre polynomials**