Knowra Legendre polynomials Legendre polynomials Legendre polynomials are an orthogonal sequence of polynomials on the interval from −1 to 1 with constant weight. Each degree gives a solution of Legendre’s differential equation.
Orthogonal polynomials : Polynomials orthogonal under an inner product defined by integration against a weight function. Legendre polynomials are a central example, orthogonal on [−1, 1] with weight one.
Orthogonality relation : An integral identity stating that distinct members of a function family have zero inner product. For Legendre polynomials, ∫₋₁¹Pℓ(x)Pm(x)dx vanishes when ℓ and m differ.
Spherical harmonics : An orthogonal basis of functions on the sphere built from associated Legendre functions and azimuthal factors. Legendre polynomials provide the zero-order angular dependence in this basis.
Chebyshev polynomials : Orthogonal polynomials associated with the weight (1−x²)^−1/2 on [−1, 1]. Their endpoint-weighted orthogonality differs from Legendre polynomials’ constant weight.
Legendre differential equation : The differential equation (1−x²)y″−2xy′+ℓ(ℓ+1)y=0, whose polynomial solutions are Legendre polynomials. Its degree-indexed polynomial solutions define the sequence.
Sturm–Liouville theory : A framework for self-adjoint differential equations whose eigenfunctions are orthogonal under a weight. It explains why solutions of Legendre’s equation with distinct degrees are orthogonal.
Multipole expansion : A representation of a distant field as a series of contributions ordered by spatial angular complexity. Legendre polynomials express the angular dependence of axisymmetric potentials.
Hermite polynomials : Orthogonal polynomials on the real line with a Gaussian weight. They solve a different classical differential equation and suit unbounded, Gaussian-weighted problems.
Rodrigues' formula : A derivative formula that generates classical orthogonal polynomials from powers of a variable-weight expression. For Legendre polynomials it gives Pℓ(x) = (1/(2^ℓℓ!)) d^ℓ/dx^ℓ (x²−1)^ℓ.
Legendre polynomial zeros : The roots of a Legendre polynomial, which are real, simple, and lie in (−1, 1). Their locations underpin both interpolation nodes and Gaussian quadrature.
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