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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.
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections ...
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 ...
Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.
Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial P m ℓ (cos θ ) . Finally, the equation for R has solutions of the form R ( r ) = A r ℓ + B r − ℓ − 1 ; requiring the solution to be regular throughout R 3 forces B = 0 .
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.
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues ( 1816 ), Sir James Ivory ( 1824 ) and Carl Gustav Jacobi ( 1827 ).
also P 0 n are the Legendre polynomials and P m n for 1 ≤ m ≤ n ... one in the page about the associated Legendre polynomials, ... sci_lemoine_1m.pdf Archived ...