Associated Legendre function: Difference between revisions

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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>


This equation allows extension of the range of ''m'' to: -''l'' &le; ''m'' &le; ''l''.  
This equation allows extension of the range of ''m'' to: &minus;''l'' &le; ''m'' &le; ''l''.  


Since the associated Legendre equation is invariant under the substitution ''m'' &rarr; -''m'', the equations for ''P''<sub>''l''</sub><sup>( &plusmn;''m'')</sup>, resulting from this expression, are proportional.  
Since the associated Legendre equation is invariant under the substitution ''m'' &rarr; &minus;''m'', the equations for ''P''<sub>''l''</sub><sup>( &plusmn;''m'')</sup>, resulting from this expression, are proportional.  


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''&sup2;)<sup>-''m''/2</sup> to the other side.  
and we bring the factor (1&minus;''x''&sup2;)<sup>&minus;''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''&sup2;)<sup>m</sup>.
Note that the phase factor (&minus;1)<sup>''m''</sup> arising in this expression is ''not'' due to some arbitrary phase convention, but arises from expansion of (1&minus;''x''&sup2;)<sup>m</sup>.
 
==Orthogonality relations==
==Orthogonality relations==
Important integral relations are
Important integral relations are

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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: −lml.

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

  1. 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]