Orthogonal Polynomials

Orthogonal polynomials

Explore Legendre Pₙ, first-kind Chebyshev Tₙ and physicists’ Hermite Hₙ through degree32. Roots are red; click a root or choose it from the list. Click or drag elsewhere to evaluate a free point.

Hermite defaults to ψₙ(x)=Hₙ(x)exp(−x²/2)/√hₙ. This separately defined normalized function reveals inner oscillations that large polynomial tails obscure. The polynomial view remains available; large values use a labeled scale factor.

The defining weights are1, (1−x²)^(−1/2) and exp(−x²), respectively. Squared norms are2/(2n+1), π (n=0) or π/2 (n>0), and √π·2ⁿ·n!. Hermite uses H₀=1,H₁=2x, not the probabilists’ convention.

Diagnostics numerically evaluate normalized weighted products using the matching Gaussian rule with max(n,m)+1 nodes. Unlike the finite plot interval, Hermite quadrature covers the whole real weighted domain. Exact-arithmetic Gaussian rules integrate these polynomial products exactly; floating-point residuals remain. Self-products approach1, mixed products approach0 unless n=m. The sampled plot and numerical roots are finite-precision illustrations.

Family, degree and view update directly; numeric edits follow after a short pause. Enable the comparison curve to see degree m on the same axes. Both curves share one scale; Hermite comparison is clipped to the interval chosen by n. Roots and point markers belong to n. Orthogonality is a weighted integral, not a requirement that curves never cross.

The session keeps unfinished fields, open sections and completed results in this browser. Restoring a completed result cheaply reconstructs the bounded polynomial plot; unfinished work remains unfinished.

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