Concept information
Preferred term
associated Legendre polynomial
Definition
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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. 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)
![{\\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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/914d8bc4e6fd78db7db32609c345663927ad11ab)
![{\\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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8c86f72c4096479b00d25aaa318916c46e36c9)
Broader concept
In other languages
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French
URI
http://data.loterre.fr/ark:/67375/PSR-MPTBN71B-H
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