Definition:Legendre Polynomial

Definition

Consider the Legendre's differential equation:

$(1): \quad \paren {1 - x^2} \dfrac {\d^2 y} {\d x^2} - 2 x \dfrac {\d y} {\d x} + n \paren {n + 1} y = 0$

for $n \in \N$.

The solutions to $(1)$ are called the Legendre polynomials of order $n$ and denoted $\map {P_n} x$.

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

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Length of Legendre Polynomial

Let $\map {P_n} x$ denote the Legendre polynomial of order $n$.

The length of $\map {P_n} x$ is defined and denoted as:

$\ds \norm {\map {P_n} x} := \sqrt {\int_{-1}^1 \paren {\map {P_n} x}^2 \rd x}$

Examples

The first few Legendre polynomials are:

Legendre Polynomial $\map {P_0} x$

$\map {P_0} x = 1$

Legendre Polynomial $\map {P_1} x$

$\map {P_1} x = x$

Legendre Polynomial $\map {P_2} x$

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

Legendre Polynomial $\map {P_3} x$

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

Legendre Polynomial: $\map {P_4} x$

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

Also see

• Generating Function for Legendre Polynomials
• Bonnet's Recursion Formula

• Results about the Legendre polynomials can be found here.

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): Legendre polynomials
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Legendre's differential equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Legendre's differential equation
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Legendre polynomials