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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 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 ...
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.
Legendre polynomials occur in the solution of Laplace's equation of the static potential, ∇ 2 Φ(x) = 0, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle).
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre . It is also sometimes known as de Polignac's formula , after Alphonse de Polignac .
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 ...
Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001): ″ (+) ′ + (+) = When α = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials.
He is also the namesake of the Legendre polynomials, solutions to Legendre's differential equation, which occur frequently in physics and engineering applications, such as electrostatics. Legendre is best known as the author of Éléments de géométrie , which was published in 1794 and was the leading elementary text on the topic for around ...