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The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre [3] as the coefficients in the expansion of the Newtonian potential where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′.
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 the Diophantine equation. The equation is named for Adrien-Marie Legendre who proved 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 ...
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation. or equivalently. where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on [−1, 1] only ...
In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, [1] is an involutive transformation on real -valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its ...
In numerical analysis, Gauss–Legendre quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating over the interval [−1, 1], the rule takes the form: where. n is the number of sample points used, wi are quadrature weights, and. xi are the roots of the n th Legendre polynomial.
Legendre form. In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because [ 1] the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity (the ellipse being defined parametrically ...
Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates. [2] The radial part of the solution varies from one potential to another, but the harmonics are always the same and are a consequence of spherical symmetry.