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

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

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

$\Pi _{\ell }^{(m)}(x)\equiv {\frac {d^{m}P_{\ell }(x)}{dx^{m}}},$

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

$(1-x^{2}){\frac {d^{2}\Pi _{\ell }^{(0)}(x)}{dx^{2}}}-2x{\frac {d\Pi _{\ell }^{(0)}(x)}{dx}}+\ell (\ell +1)\Pi _{\ell }^{(0)}(x)=0,$

m times gives an equation for Π^(m)_l

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

After substitution of

$\Pi _{\ell }^{(m)}(x)=(1-x^{2})^{-m/2}P_{\ell }^{(m)}(x),$

we find, after multiplying through with $(1-x^{2})^{m/2}$, that the associated Legendre differential equation holds for the associated Legendre functions

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

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

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