Legendre Polynomial Orthogonality
Compare orthogonal polynomials under unit weight and inspect their endpoint slopes.
Free theory articleCompare orthogonal polynomials under unit weight and inspect their endpoint slopes.
Concept & interpretation
The Legendre polynomials, denoted Pn(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.
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].