Legendrepolynomien

Legendrepolynomien are a sequence of orthogonal polynomials that play a significant role in the solution of boundary value problems for Laplace's equation in spherical coordinates. They are named after the French mathematician Adrien-Marie Legendre.

The Legendre polynomials, denoted by P_n(x), can be defined by a recurrence relation or by Rodrigues' formula. The recurrence relation is (n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x), with P_0(x) = 1 and P_1(x) = x. Rodrigues' formula is given by P_n(x) = (1/(2^n n!)) * d^n/dx^n (x^2 - 1)^n.

These polynomials satisfy the Legendre differential equation: (1-x^2)y'' - 2xy' + n(n+1)y = 0. The orthogonality property states that the integral of P_m(x)P_n(x) from -1 to 1 is zero if m is not equal to n, and it is 2/(2n+1) if m equals n.

Legendre polynomials are fundamental in areas such as quantum mechanics, where they appear in the solution of the hydrogen atom, and in numerical analysis, particularly in Gaussian quadrature for approximating integrals. Their applications extend to various fields of physics and engineering involving spherical symmetry.