< CZ:Featured articleRevision as of 09:28, 23 March 2012 by imported>Chunbum Park
by Paul Wormer
In mathematics and physics, an associated Legendre function Pℓm is related to a Legendre polynomial Pℓ by the following equation

Although extensions are possible, in this article ℓ and m are restricted to integer numbers. For even m the associated Legendre function is a polynomial, for odd m the function contains the factor (1−x ² )½ and hence is not a polynomial.
The associated Legendre functions are important in quantum mechanics and potential theory.
According to Ferrers[1] the polynomials were named "Associated Legendre functions" by the British mathematician Isaac Todhunter in 1875,[2] where "associated function" is Todhunter's translation of the German term zugeordnete Function, coined in 1861 by Heine,[3] and "Legendre" is in honor of the French mathematician Adrien-Marie Legendre (1752–1833), who was the first to introduce and study the functions.
Differential equation
Define

where Pℓ(x) is a Legendre polynomial.
Differentiating the Legendre differential equation:

m times gives an equation for Πml
![{\displaystyle (1-x^{2}){\frac {d^{2}\Pi _{\ell }^{m}(x)}{dx^{2}}}-2(m+1)x{\frac {d\Pi _{\ell }^{m}(x)}{dx}}+\left[\ell (\ell +1)-m(m+1)\right]\Pi _{\ell }^{m}(x)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0646d66b5288c6fcb0a223d4bd1a138025408a4)
After substitution of

and after multiplying through with
, we find the associated Legendre differential equation:
![{\displaystyle (1-x^{2}){\frac {d^{2}P_{\ell }^{m}(x)}{dx^{2}}}-2x{\frac {dP_{\ell }^{m}(x)}{dx}}+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]P_{\ell }^{m}(x)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65e0b6b6c103256a203530be670df71713f3ecd1)
One often finds the equation written in the following equivalent way

where the primes indicate differentiation with respect to x.
In physical applications it is usually the case that x = cosθ, then the associated Legendre differential equation takes the form
![{\displaystyle {\frac {1}{\sin \theta }}{\frac {d}{d\theta }}\sin \theta {\frac {d}{d\theta }}P_{\ell }^{m}+\left[\ell (\ell +1)-{\frac {m^{2}}{\sin ^{2}\theta }}\right]P_{\ell }^{m}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3db05097c0b453bec6526c7657d198a6409784d)
.... (read more)
notes
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- ↑ N. M. Ferrers, An Elementary Treatise on Spherical Harmonics, MacMillan, 1877 (London), p. 77. Online.
- ↑ I. Todhunter, An Elementary Treatise on Laplace's, Lamé's, and Bessel's Functions, MacMillan, 1875 (London). In fact, Todhunter called the Legendre polynomials "Legendre coefficients".
- ↑ E. Heine, Handbuch der Kugelfunctionen, G. Reimer, 1861 (Berlin).Google book online
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