Associated Legendre function: Difference between revisions

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imported>Paul Wormer
imported>Paul Wormer
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</math>
</math>
and we bring the factor (1-''x''&sup2;)<sup>-''m''/2</sup> to the other side.  
and we bring the factor (1-''x''&sup2;)<sup>-''m''/2</sup> to the other side.  
Equate the coefficients of the same powers 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>
\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 \le m \le \ell,  
\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 \le m \le \ell,  
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and it follows that the proportionality constant is
and it follows that the proportionality constant is
:<math>
:<math>
c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,
c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,\qquad  0 \le m \le \ell,
</math>
</math>
so that the associated Legendre functions of same |''m''| are related to each other by  
so that the associated Legendre functions of same |''m''| are related to each other by  

Revision as of 09:16, 22 August 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

we find, after multiplying through with , that the associated Legendre differential equation holds for the associated Legendre functions

In physical applications usually x = cosθ, then then 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.