Legendrepolynomer

Legendrepolynomer, the Danish term for Legendre polynomials, are a sequence of polynomials P_n(x) defined on the interval [-1, 1] and orthogonal with respect to the weight function 1. They arise in problems with spherical symmetry and are named after Adrien Legendre.

Each Legendre polynomial P_n satisfies the Legendre differential equation (1 - x^2) P_n''(x) - 2 x P_n'(x) + n(n+1) P_n(x) = 0. They can be generated by Rodrigues' formula: P_n(x) = (1 / (2^n n!)) d^n/dx^n [(x^2 - 1)^n]. They also possess a generating function given by 1 / sqrt(1 - 2 x t + t^2) = sum_{n=0}^\infty P_n(x) t^n, valid for |t| < 1.

Initial values and a standard recurrence define the sequence: P_0(x) = 1, P_1(x) = x, and (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x) for n ≥ 1. They satisfy the parity relation P_n(-x) = (-1)^n P_n(x). Orthogonality on [-1, 1] is expressed by ∫_{-1}^1 P_m(x) P_n(x) dx = 2/(2n+1) δ_{mn}, where δ_{mn} is the Kronecker delta.

Legendre polynomials form a complete orthogonal basis for functions on [-1, 1] and are widely used in numerical integration via Gauss-Legendre quadrature, in solving Laplace’s equation in spherical coordinates, and in expansions of functions with axial symmetry. They are a special case of Jacobi polynomials P_n^{(0,0)} and are related to the angular part of spherical harmonics.

Common first polynomials include P_0(x) = 1, P_1(x) = x, P_2(x) = (3x^2 - 1)/2, and P_3(x) = (5x^3 - 3x)/2.