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mathematical physics > special function > associated Legendre polynomial

mathematical analysis > functional analysis > special function > associated Legendre polynomial

... > function > elementary function > polynomial > orthogonal polynomials > associated Legendre polynomial

algebra > polynomial > orthogonal polynomials > associated Legendre polynomial

Terme préférentiel

associated Legendre polynomial

Définition(s)

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
${\\displaystyle \\left(1-x^{2}\ ight){\ rac {d^{2}}{dx^{2}}}P_{\\ell }^{m}(x)-2x{\ rac {d}{dx}}P_{\\ell }^{m}(x)+\\left[\\ell (\\ell +1)-{\ rac {m^{2}}{1-x^{2}}}\ ight]P_{\\ell }^{m}(x)=0,}$
or equivalently
${\\displaystyle {\ rac {d}{dx}}\\left[\\left(1-x^{2}\ ight){\ rac {d}{dx}}P_{\\ell }^{m}(x)\ ight]+\\left[\\ell (\\ell +1)-{\ rac {m^{2}}{1-x^{2}}}\ ight]P_{\\ell }^{m}(x)=0,}$
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 if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. In general, when ℓ and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. The fully general class of functions with arbitrary real or complex values of ℓ and m are Legendre functions. In that case the parameters are usually labelled with Greek letters.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Associated_Legendre_polynomials)

Concept(s) générique(s)

Traductions

polynôme associé de Legendre

français

URI

http://data.loterre.fr/ark:/67375/PSR-MPTBN71B-H

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RDF/XML TURTLE JSON-LD Date de création 16/08/2023, dernière modif. 16/08/2023