imported>Paul Wormer |
imported>Paul Wormer |
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| In [[mathematics]] and [[physics]], an '''associated Legendre function''' ''P''<sub>''l''</sub><sup>(''m'')</sup> is related to a [[Legendre polynomial]] ''P''<sub>''l''</sub> by the following equation | | In [[mathematics]] and [[physics]], an '''associated Legendre function''' ''P''<sub>''l''</sub><sup>(''m'')</sup> is related to a [[Legendre polynomial]] ''P''<sub>''l''</sub> by the following equation |
| :<math> | | :<math> |
| P^{(m)}_\ell(x) = (1-x^2)^{m/2} \frac{d P_\ell(x)}{dx^\ell}. | | P^{(m)}_\ell(x) = (1-x^2)^{m/2} \frac{d^m P_\ell(x)}{dx^m}, \qquad 0 \le m \le \ell. |
| </math> | | </math> |
| For even ''m'' the associated Legendre function is a polynomial, for odd ''m'' the function contains the factor (1-''x'' ² )<sup>½</sup> and hence is not a polynomial. | | For even ''m'' the associated Legendre function is a polynomial, for odd ''m'' the function contains the factor (1-''x'' ² )<sup>½</sup> and hence is not a polynomial. |
Revision as of 11:09, 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
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: -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.
Orthogonality relations
Important integral relations are
Recurrence relations
The functions satisfy the following difference equations, which are taken from Edmonds[1]
-
Reference
- ↑ 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]