Associated Legendre function: Difference between revisions

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imported>Paul Wormer
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imported>Paul Wormer
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==Recurrence relations==
==Recurrence relations==


The functions satisfy the following difference equations, which are taken from Edmonds.
The functions satisfy the following difference equations, which are taken from Edmonds.<ref>A. R. Edmonds, ''Angular Momentum in Quantum Mechanics'', Princeton University Press, 2nd edition (1960)</ref>


:<math>
:<math>
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=(\ell+m)P_{\ell-1}^{m}(x)-\ell x P_{\ell}^{m}(x)
=(\ell+m)P_{\ell-1}^{m}(x)-\ell x P_{\ell}^{m}(x)
</math><!-- Edmonds 2.5.25 -->
</math><!-- Edmonds 2.5.25 -->
==Reference==
<references />

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In mathematics and physics, an associated Legendre function Plm 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 Πml

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

    [Proof]

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)