Associated Legendre function: Difference between revisions
imported>Paul Wormer |
imported>Paul Wormer |
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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 | 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: -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.