Associated Legendre function: Difference between revisions

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m (polynomial --> function)
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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: &minus;''l'' &le; ''m'' &le; ''l''.  
This equation allows extension of the range of ''m'' to: <font style="vertical-align: text-top;"> <math>-m \le \ell \le m</math></font>.


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.  
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.  

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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 functions 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: .

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]