imported>Subpagination Bot |
imported>Paul Wormer |
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| :<math>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.</math> | | :<math>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.</math> |
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| This equation allows extension of the range of ''m'' to: -''l'' ≤ ''m'' ≤ ''l''. | | This equation allows extension of the range of ''m'' to: −''l'' ≤ ''m'' ≤ ''l''. |
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| Since the associated Legendre equation is invariant under the substitution ''m'' → -''m'', the equations for ''P''<sub>''l''</sub><sup>( ±''m'')</sup>, resulting from this expression, are proportional. | | Since the associated Legendre equation is invariant under the substitution ''m'' → −''m'', the equations for ''P''<sub>''l''</sub><sup>( ±''m'')</sup>, resulting from this expression, are proportional. |
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| To obtain the proportionality constant we consider | | To obtain the proportionality constant we consider |
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| (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 \le m \le \ell, | | (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 \le m \le \ell, |
| </math> | | </math> |
| and we bring the factor (1-''x''²)<sup>-''m''/2</sup> to the other side. | | and we bring the factor (1−''x''²)<sup>−''m''/2</sup> to the other side. |
| Equate the coefficient of the highest power of ''x'' on the left and right hand side of | | Equate the coefficient of the highest power of ''x'' on the left and right hand side of |
| :<math> | | :<math> |
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| P^{(-|m|)}_\ell(x) = (-1)^m \frac{(\ell-|m|)!}{(\ell+|m|)!} P^{(|m|)}_\ell(x). | | P^{(-|m|)}_\ell(x) = (-1)^m \frac{(\ell-|m|)!}{(\ell+|m|)!} P^{(|m|)}_\ell(x). |
| </math> | | </math> |
| Note that the phase factor (-1)<sup>''m''</sup> arising in this expression is ''not'' due to some arbitrary phase convention, but arises from expansion of (1-''x''²)<sup>m</sup>. | | Note that the phase factor (−1)<sup>''m''</sup> arising in this expression is ''not'' due to some arbitrary phase convention, but arises from expansion of (1−''x''²)<sup>m</sup>. |
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| ==Orthogonality relations== | | ==Orthogonality relations== |
| Important integral relations are | | Important integral relations are |
Revision as of 09:01, 3 October 2007
In mathematics and physics, an associated Legendre function Pl(m) is related to a Legendre polynomial Pl by the following equation
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 polynomials are important in quantum mechanics and potential theory.
Differential equation
Define
where Pl(x) is a Legendre polynomial.
Differentiating the Legendre differential equation:
m times gives an equation for Π(m)l
After substitution of
and after multiplying through with , we find the associated Legendre differential equation:
In physical applications it is usually the case that x = cosθ, then the associated Legendre differential equation takes the form
Extension to negative m
By the Rodrigues formula, one obtains
This equation allows extension of the range of m to: −l ≤ m ≤ l.
Since the associated Legendre equation is invariant under the substitution m → −m, the equations for Pl( ±m), resulting from this expression, are proportional.
To obtain the proportionality constant we consider
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
and it follows that the proportionality constant is
so that the associated Legendre functions of same |m| are related to each other by
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
Recurrence relations
The functions satisfy the following difference equations, which are taken from Edmonds[1]
-
Reference
- ↑ A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)
External link
Weisstein, Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web Resource. [1]