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 $(1-x^{2})^{m/2}$, we find the associated Legendre differential equation:

$(1-x^{2}){\frac {d^{2}P_{\ell }^{(m)}(x)}{dx^{2}}}-2x{\frac {dP_{\ell }^{(m)}(x)}{dx}}+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]P_{\ell }^{(m)}(x)=0.$

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

${\frac {1}{\sin \theta }}{\frac {d}{d\theta }}\sin \theta {\frac {d}{d\theta }}P_{\ell }^{(m)}+\left[\ell (\ell +1)-{\frac {m^{2}}{\sin ^{2}\theta }}\right]P_{\ell }^{(m)}=0.$

Extension to negative m

By the Rodrigues formula, one obtains

$P_{\ell }^{(m)}(x)={\frac {1}{2^{\ell }\ell !}}(1-x^{2})^{m/2}\ {\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell }.$

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

$(1-x^{2})^{-m/2}{\frac {d^{\ell -m}}{dx^{\ell -m}}}(x^{2}-1)^{\ell }=c_{lm}(1-x^{2})^{m/2}{\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell },\qquad 0\leq m\leq \ell ,$

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

${\frac {d^{\ell -m}}{dx^{\ell -m}}}(x^{2}-1)^{\ell }=c_{lm}(1-x^{2})^{m}{\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell },\qquad 0\leq m\leq \ell ,$

and it follows that the proportionality constant is

$c_{lm}=(-1)^{m}{\frac {(\ell -m)!}{(\ell +m)!}},\qquad 0\leq m\leq \ell ,$

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

$P_{\ell }^{(-|m|)}(x)=(-1)^{m}{\frac {(\ell -|m|)!}{(\ell +|m|)!}}P_{\ell }^{(|m|)}(x).$

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

$\int _{-1}^{1}P_{\ell }^{(m)}(x)P_{\ell '}^{(m)}(x)dx={\frac {2\delta _{\ell \ell '}(\ell +m)!}{(2\ell +1)(\ell -m)!}}$

$\int _{-1}^{1}P_{\ell }^{(m)}(x)P_{\ell }^{(n)}(x){\frac {dx}{1-x^{2}}}={\frac {\delta _{mn}(\ell +m)!}{m(\ell -m)!}}$

Recurrence relations

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

$(\ell -m+1)P_{\ell +1}^{(m)}(x)-(2\ell +1)xP_{\ell }^{(m)}(x)+(\ell +m)P_{\ell -1}^{(m)}(x)=0$

$xP_{\ell }^{(m)}(x)-(\ell -m+1)(1-x^{2})^{1/2}P_{\ell }^{(m-1)}(x)-P_{\ell -1}^{(m)}(x)=0$

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]