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Latest revision as of 17:00, 13 July 2024
In mathematics and physics, an associated Legendre function Pℓm is related to a Legendre polynomial Pℓ by the following equation
![{\displaystyle P_{\ell }^{m}(x)=(1-x^{2})^{m/2}{\frac {d^{m}P_{\ell }(x)}{dx^{m}}},\qquad 0\leq m\leq \ell .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2772bc830d8e6117915b3bba08d70000c09d7996)
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
![{\displaystyle \Pi _{\ell }^{m}(x)\equiv {\frac {d^{m}P_{\ell }(x)}{dx^{m}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4bc5e76fc6572cb9cf78ce7ae255b66b8b430c)
where Pℓ(x) is a Legendre polynomial.
Differentiating the Legendre differential equation:
![{\displaystyle (1-x^{2}){\frac {d^{2}\Pi _{\ell }^{0}(x)}{dx^{2}}}-2x{\frac {d\Pi _{\ell }^{0}(x)}{dx}}+\ell (\ell +1)\Pi _{\ell }^{0}(x)=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73f7055579e067bf2b3e61a3870fbc637589c380)
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
![{\displaystyle \Pi _{\ell }^{m}(x)=(1-x^{2})^{-m/2}P_{\ell }^{m}(x),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50adcee89e068c2bb954044e662925ccfd9eadd0)
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
![{\displaystyle \left((1-x^{2})\;y\,'\right)'+\left(\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right)y=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1437f6cbbd903bbaba983882d00b3c23d94e8c6f)
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)
Extension to negative m
By the Rodrigues formula, one obtains
![{\displaystyle P_{\ell }^{m}(x)={\frac {1}{2^{\ell }\ell !}}(1-x^{2})^{m/2}\ {\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/facf858e1025b1c31e08a949c848596ddd5f7317)
This equation allows extension of the range of m to: −m ≤ ℓ ≤ m.
Since the associated Legendre equation is invariant under the substitution m → −m, the equations for Pℓ ±m, resulting from this expression, are proportional.[4]
To obtain the proportionality constant we consider
![{\displaystyle (1-x^{2})^{-m/2}{\frac {d^{\ell -m}}{dx^{\ell -m}}}(x^{2}-1)^{\ell }=c_{lm}(1-x^{2})^{m/2}{\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell },\qquad 0\leq m\leq \ell ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84e0a48c47ddd350c70a906e09c2cdaf1c086c32)
and we bring the factor (1−x²)−m/2 to the other side.
Equate the coefficient of the highest power of x on the left and right hand side of
![{\displaystyle {\frac {d^{\ell -m}}{dx^{\ell -m}}}(x^{2}-1)^{\ell }=c_{lm}(1-x^{2})^{m}{\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell },\qquad 0\leq m\leq \ell ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90f4fd149954bfc866dad3a02f11609ffe51312c)
and it follows that the proportionality constant is
![{\displaystyle c_{lm}=(-1)^{m}{\frac {(\ell -m)!}{(\ell +m)!}},\qquad 0\leq m\leq \ell ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbcf796af7999bf581bf7c5c7421892562455e9)
so that the associated Legendre functions of same |m| are related to each other by
![{\displaystyle P_{\ell }^{-|m|}(x)=(-1)^{m}{\frac {(\ell -|m|)!}{(\ell +|m|)!}}P_{\ell }^{|m|}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcc9542e38b04e9e58cf877afa7fbe9e00d9f32)
Note that the phase factor (−1)m arising in this expression is not due to some arbitrary phase convention, but arises from expansion of (1−x²)m.
Orthogonality relations
Important integral relations are:
![{\displaystyle \int \limits _{-1}^{1}P_{l}^{m}\left(x\right)P_{k}^{m}\left(x\right)dx={\frac {2}{2l+1}}{\frac {\left(l+m\right)!}{\left(l-m\right)!}}\delta _{lk},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5bded4f4f600081d4326b01a30d3ee197e0c54)
and:
![{\displaystyle \int _{-1}^{1}P_{\ell }^{m}(x)P_{\ell }^{n}(x){\frac {dx}{1-x^{2}}}={\frac {\delta _{mn}(\ell +m)!}{m(\ell -m)!}},\qquad m\neq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a6251b11b9eddb8c6ab48470cadcf4bd4ad36a)
The latter integral for n = m = 0
![{\displaystyle \int _{-1}^{1}P_{\ell }^{0}(x)P_{\ell }^{0}(x){\frac {dx}{1-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12f8c9d8f5862e6cbe41ecaa28ebd3be38a0837f)
is undetermined (infinite). (see the subpage Proofs for detailed proofs of these relations.)
Recurrence relations
The functions satisfy the following difference equations, which are taken from Edmonds.[5]
![{\displaystyle (\ell -m+1)P_{\ell +1}^{m}(x)-(2\ell +1)xP_{\ell }^{m}(x)+(\ell +m)P_{\ell -1}^{m}(x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913c1af16ef6dc0dbdc7b3a6f5c504cda9a2b846)
![{\displaystyle xP_{\ell }^{m}(x)-(\ell -m+1)(1-x^{2})^{1/2}P_{\ell }^{m-1}(x)-P_{\ell -1}^{m}(x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6a4b80ed1e7e34cce4ff17d505f9fa6ea3370d)
![{\displaystyle P_{\ell +1}^{m}(x)-xP_{\ell }^{m}(x)-(\ell +m)(1-x^{2})^{1/2}P_{\ell }^{m-1}(x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7896802f07516d0140f1867e45fc6f3a11668f16)
![{\displaystyle (\ell -m+1)P_{\ell +1}^{m}(x)+(1-x^{2})^{1/2}P_{\ell }^{m+1}(x)-(\ell +m+1)xP_{\ell }^{m}(x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b32215a5f0951c20b6d2a7ca69050fe6e208088a)
![{\displaystyle (1-x^{2})^{1/2}P_{\ell }^{m+1}(x)-2mxP_{\ell }^{m}(x)+(\ell +m)(\ell -m+1)(1-x^{2})^{1/2}P_{\ell }^{m-1}(x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80e7eb83d3e625791ceaf40276eb7dd7cd009908)
![{\displaystyle =(\ell +m)P_{\ell -1}^{m}(x)-\ell xP_{\ell }^{m}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80d837d4c2501afb7af08e8e76c0eb42897a58ee)
Reference
- ↑ 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
- ↑ The associated Legendre differential equation being of second order, the general solution is of the form
where
is a Legendre polynomial of the second kind, which has a singularity at x = 0. Hence solutions that are regular at x = 0 have B = 0 and are proportional to
. The Rodrigues formula shows that
is a regular (at x=0) solution and the proportionality follows.
- ↑ A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)