Associated Legendre function

In mathematics and physics, an associated Legendre function P_l^(m) is related to a Legendre polynomial P_l by the following equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^{(m)}_\ell(x) = (1-x^2)^{m/2} \frac{d^m P_\ell(x)}{dx^m}, \qquad 0 \le m \le \ell. }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi^{(m)}_\ell(x) \equiv \frac{d^m P_\ell(x)}{dx^m}, }

where P_l(x) is a Legendre polynomial. Differentiating the Legendre differential equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2) \frac{d^2 \Pi^{(0)}_\ell(x)}{dx^2} - 2 x \frac{d\Pi^{(0)}_\ell(x)}{dx} + \ell(\ell+1) \Pi^{(0)}_\ell(x) = 0, }

m times gives an equation for Π^(m)_l

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2) \frac{d^2 \Pi^{(m)}_\ell(x)}{dx^2} - 2(m+1) x \frac{d\Pi^{(m)}_\ell(x)}{dx} + \left[\ell(\ell+1) -m(m+1) \right] \Pi^{(m)}_\ell(x) = 0 . }

After substitution of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi^{(m)}_\ell(x) = (1-x^2)^{-m/2} P^{(m)}_\ell(x), }

and after multiplying through with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2)^{m/2}} , we find the associated Legendre differential equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2) \frac{d^2 P^{(m)}_\ell(x)}{dx^2} -2x\frac{d P^{(m)}_\ell(x)}{dx} + \left[ \ell(\ell+1) - \frac{m^2}{1-x^2}\right] P^{(m)}_\ell(x)= 0 . }

In physical applications it is usually the case that x = cosθ, then the associated Legendre differential equation takes the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sin \theta}\frac{d}{d\theta} \sin\theta \frac{d}{d\theta}P^{(m)}_\ell +\left[ \ell(\ell+1) - \frac{m^2}{\sin^2\theta}\right] P^{(m)}_\ell = 0. }

Extension to negative m

By the Rodrigues formula, one obtains

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.}

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 P_l^{( ±m)}, resulting from this expression, are proportional.

To obtain the proportionality constant we consider

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2)^{-m/2} \frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^{m/2} \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell},\qquad 0 \le m \le \ell, }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, }

and it follows that the proportionality constant is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,\qquad 0 \le m \le \ell, }

so that the associated Legendre functions of same |m| are related to each other by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^{(-|m|)}_\ell(x) = (-1)^m \frac{(\ell-|m|)!}{(\ell+|m|)!} P^{(|m|)}_\ell(x). }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-1}^{1} P^{(m)}_{\ell}(x) P^{(m)}_{\ell'}(x) d x = \frac{2\delta_{\ell\ell'}(\ell+m)!}{(2\ell+1)(\ell-m)!} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-1}^{1} P^{(m)}_{\ell}(x) P^{(n)}_{\ell}(x) \frac{d x}{1-x^2} = \frac{\delta_{mn}(\ell+m)!}{m(\ell-m)!} }

Recurrence relations

The functions satisfy the following difference equations, which are taken from Edmonds

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\ell-m+1)P_{\ell+1}^{(m)}(x) - (2\ell+1)xP_{\ell}^{(m)}(x) + (\ell+m)P_{\ell-1}^{(m)}(x)=0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xP_{\ell}^{(m)}(x) -(\ell-m+1)(1-x^2)^{1/2} P_{\ell}^{(m-1)}(x) - P_{\ell-1}^{(m)}(x)=0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\ell+1}^{(m)}(x) - x P_{\ell}^{(m)}(x)-(\ell+m)(1-x^2)^{1/2}P_{\ell}^{(m-1)}(x)=0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\ell-m+1)P_{\ell+1}^{(m)}(x)+(1-x^2)^{1/2}P_{\ell}^{(m+1)}(x)- (\ell+m+1) xP_{\ell}^{(m)}(x)=0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2)^{1/2}P_{\ell}^{(m+1)}(x)-2mxP_{\ell}^{(m)}(x)+ (\ell+m)(\ell-m+1)(1-x^2)^{1/2}P_{\ell}^{(m-1)}(x)=0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2)\frac{dP_{\ell}^{(m)}}{dx}(x) =(\ell+1)xP_{\ell}^{(m)}(x) -(\ell-m+1)P_{\ell+1}^{(m)}(x) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =(\ell+m)P_{\ell-1}^{(m)}(x)-\ell x P_{\ell}^{(m)}(x) }

External link

Weisstein, Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web Resource. [1]