Legendre Polynomial Orthogonality

The Legendre polynomials, denoted P_n(x), form a sequence of orthogonal polynomials on the interval [-1, 1] with respect to the unit weight function w(x) = 1. They arise naturally in the solution of Laplace's equation in spherical coordinates and are a fundamental example of orthogonal functions used in physics and numerical analysis, such as Gaussian quadrature. Unlike the Chebyshev polynomials which use the weight 1/√(1-x²), the Legendre polynomials use a flat unit weight.

Worked Example

Let's calculate the Legendre polynomial of degree 2, P₂(x), using Bonnet's recursion formula: (n+1)P_n+1(x) = (2n+1)xP_n(x) - nP_n-1(x). We start with the base cases P₀(x) = 1 and P₁(x) = x.

For n=1, the recurrence gives: (1+1)P₂(x) = (2(1)+1)xP₁(x) - 1P₀(x)
2P₂(x) = 3x(x) - 1(1) = 3x² - 1
Therefore, P₂(x) = (3x² - 1)/2 = 1.5x² - 0.5

At x = 0.5, P₂(0.5) = 1.5(0.25) - 0.5 = 0.375 - 0.5 = -0.125.

Interactive Experiment

Use the interactive module to visualize the polynomials and their orthogonality.

Set the Degree n control to 2.
Set the Compare m control to 3.
Set the Probe x control to 0.5.

Notice that the integral of their product, ∫ P₂P₃ dx, evaluates to exactly 0, confirming their orthogonality. The signed area plot (shaded) shows equal positive and negative regions over [-1, 1].

Limitations

This visualization only plots up to degree 5. Higher degrees have rapid oscillations that require more dense sampling than the standard steps used here for real-time interaction. The standard normalization P_n(1) = 1 is used, which means the squared norm decreases as 2/(2n+1).

References

For further reading on orthogonal polynomials and the Legendre recurrence:

Legendre Polynomial Orthogonality