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Associated Legendre Functions

The associated Legendre functions, $ P_l^{\,m}(x)$ , are the well-behaved solutions of the differential equation

$\displaystyle \frac{d}{dx}\!\left[(1-x^{\,2})\,\frac{dP_l^{\,m}}{dx}\right]+\left[l\,(l+1)-\frac{m^{\,2}}{1-x^{\,2}}\right]P_l^{\,m} = 0,$ (292)

for $ x$ in the range $ -1\leq x\leq +1$ . Here, $ l$ is a non-negative integer (known as the degree), and $ m$ is an integer (known as the order) lying in the range $ -l\leq m\leq l$ . The functions themselves take the form[*]

$\displaystyle P_l^{\,m}(x)= \frac{(-1)^{\,l+m}}{2^{\,l}\,l!}\,(1-x^{\,2})^{\,m/2}\,\frac{d^{\,l+m}}{dx^{\,l+m}}\,(1-x^{\,2})^{\,l},$ (293)

which implies that

$\displaystyle P_l^{\,-m}(x)= (-1)^{\,m}\,\frac{(l-m)!}{(l+m)!}\,P_l^{\,m}(x).$ (294)

Assuming that $ 0\leq m\leq l$ , the $ P_l^{\,m}(x)$ satisfy the orthogonality condition

$\displaystyle \int_{-1}^1P_l^{\,m}(x)\,P_k^{\,m}(x)\,dx = \frac{2\,(l+m)!}{(2\,l+1)\,(l-m)!}\,\delta_{lk},$ (295)

where $ \delta_{lk}$ is a Kronecker delta symbol.

The associated Legendre functions of order 0 (i.e., $ m=0$ ) are called Legendre polynomials, and are denoted the $ P_l(x)$ : that is, $ P_l^0(x)=P_l(x)$ . It follows that[*]

$\displaystyle \int_{-1}^1P_l(x)\,P_k(x)\,dx = \frac{2}{(2\,l+1)}\,\delta_{lk}.$ (296)

It can also be shown that

$\displaystyle \frac{1}{(1-2\,x\,t+t^{\,2})^{1/2}} = \sum_{l=0,\infty} P_l(x)\,t^{\,l},$ (297)

provided $ \vert t\vert< 1$ and $ \vert x\vert\leq 1$ .

All of the associated Legendre functions of degree less than 3 are listed below:

$\displaystyle P_0^{\,0}(x)$ $\displaystyle = 1,$ (298)
$\displaystyle P_{1}^{\,-1}(x)$ $\displaystyle = (1/2)\,(1-x^{\,2})^{1/2},$ (299)
$\displaystyle \ P_1^{\,0}(x)$ $\displaystyle = x,$ (300)
$\displaystyle P_1^{\,+1}(x)$ $\displaystyle = -(1-x^{\,2})^{1/2},$ (301)
$\displaystyle P_2^{\,-2}(x)$ $\displaystyle =(1/8)\,(1-x^{\,2}),$ (302)
$\displaystyle P_2^{\,-1}(x)$ $\displaystyle =(1/2)\,x\,(1-x^{\,2})^{1/2},$ (303)
$\displaystyle P_2^{\,0}(x)$ $\displaystyle =(1/2)\,(3\,x^{\,2}-1),$ (304)
$\displaystyle P_2^{\,+1}(x)$ $\displaystyle =-3\,x\,(1-x^{\,2})^{1/2},$ (305)
$\displaystyle P_2^{\,+2}(x)$ $\displaystyle =3\,(1-x^{\,2}).$ (306)

next up previous
Next: Spherical Harmonics Up: Potential Theory Previous: Introduction
Richard Fitzpatrick 2014-06-27