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The general Legendre equation reads ″ ′ + [(+)] =, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. . The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P n; and when λ is an integer (denoted n), and μ = m is also an integer with | m | < n are the associated Legendre ...
In mathematics, Legendre's equation is a Diophantine equation of the form: + + = The equation is named for Adrien-Marie Legendre who proved it in 1785 that it is solvable in integers x, y, z, not all zero, if and only if −bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwise relatively prime integers and also not all ...
The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics.
The differential equation admits another, non-polynomial solution, the Legendre functions of the second kind. A two-parameter generalization of (Eq. 1) is called Legendre's general differential equation, solved by the Associated Legendre polynomials.
Download as PDF; Printable version; ... (called Pearson's differential equation), ... Rodrigues stated his formula for Legendre polynomials ...
Since z = 1 − x, the solution of the hypergeometric equation at x = 1 is the same as the solution for this equation at z = 0. But the solution at z = 0 is identical to the solution we obtained for the point x = 0, if we replace each γ by α + β − γ + 1. Hence, to get the solutions, we just make this substitution in the previous results.
In quantum mechanics, they are the solutions of Schrödinger's equation for the harmonic oscillator. They are also eigenfunctions (with eigenvalue (− i n ) of the continuous Fourier transform . Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of e − x 2 / 2 ...
2.2 Legendre differential equation. ... Download as PDF; ... It is found in the solution of Laplace's equation in spherical coordinates: ...