Search results
Results From The WOW.Com Content Network
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 ...
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 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.
(Note that it makes sense for such an equation to have a polynomial solution. Each term in the equation is a polynomial, and the degrees are consistent.) This is a Sturm–Liouville type of equation. Such equations generally have singularities in their solution functions f except for particular values of λ.
Then the Frobenius method based on the indicial equation may be applied to find possible solutions that are power series times complex powers (z − a) r near any given a in the complex plane where r need not be an integer; this function may exist, therefore, only thanks to a branch cut extending out from a, or on a Riemann surface of some ...
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 .
Similar formulae hold for many other sequences of orthogonal functions arising from Sturm–Liouville equations, and these are also called the Rodrigues formula (or Rodrigues' type formula) for that case, especially when the resulting sequence is polynomial.