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Associated Legendre function

by Paul Wormer

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

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

Although extensions are possible, in this article ℓ and m are restricted to integer numbers. 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 functions are important in quantum mechanics and potential theory.

According to Ferrers the polynomials were named "Associated Legendre functions" by the British mathematician Isaac Todhunter in 1875, where "associated function" is Todhunter's translation of the German term zugeordnete Function, coined in 1861 by Heine, and "Legendre" is in honor of the French mathematician Adrien-Marie Legendre (1752–1833), who was the first to introduce and study the functions.

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

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

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

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

One often finds the equation written in the following equivalent way

$\left((1-x^{2})\;y\,'\right)'+\left(\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right)y=0,$

where the primes indicate differentiation with respect to x.

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

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

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