Legendre polynomials: P_n(x) orthogonal on [-1,1] with weight 1. P_0=1, P_1=x, P_2=(3x²−1)/2, P_3=(5x³−3x)/2. Bonnet: (n+1)P_{n+1}=(2n+1)xP_n−nP_{n-1}. Rodrigues: P_n=(1/2^n·n!)d^n/dx^n(x²−1)^n.
How to Use
Enter n and x:
- P_n(x): Evaluation
- Sequence: P_0..P_n
- Coefficients: Polynomial form
Gauss-Legendre
Roots of P_n are the Gauss-Legendre quadrature nodes. n-point Gauss quadrature is exact for polynomials up to degree 2n-1 — the maximum possible! This makes it the most efficient quadrature method.
Physics
Spherical harmonics Y_l^m(θ,φ) = P_l^m(cos θ)e^{imφ}. Multipole expansion: potential at distance r from charge distribution uses Legendre polynomials. Fundamental in electrostatics, gravity, quantum mechanics.
Step-by-Step Instructions
- 1Enter degree n.
- 2Enter x value.
- 3Compute P_n(x).
- 4View sequence.
- 5See Gauss nodes.