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Associated Legendre Functions
The associated Legendre functions,
, are the well-behaved solutions of the
differential equation
![$\displaystyle \frac{d}{dx}\!\left[(1-x^{\,2})\,\frac{dP_l^{\,m}}{dx}\right]+\left[l\,(l+1)-\frac{m^{\,2}}{1-x^{\,2}}\right]P_l^{\,m} = 0,$](img673.png) |
(292) |
for
in the range
.
Here,
is a non-negative integer (known as the degree), and
is an integer (known as the order) lying in the range
.
The functions themselves take the form
![$\displaystyle P_l^{\,m}(x)= \frac{(-1)^{\,l+m}}{2^{\,l}\,l!}\,(1-x^{\,2})^{\,m/2}\,\frac{d^{\,l+m}}{dx^{\,l+m}}\,(1-x^{\,2})^{\,l},$](img677.png) |
(293) |
which implies that
![$\displaystyle P_l^{\,-m}(x)= (-1)^{\,m}\,\frac{(l-m)!}{(l+m)!}\,P_l^{\,m}(x).$](img678.png) |
(294) |
Assuming that
, the
satisfy the orthogonality condition
![$\displaystyle \int_{-1}^1P_l^{\,m}(x)\,P_k^{\,m}(x)\,dx = \frac{2\,(l+m)!}{(2\,l+1)\,(l-m)!}\,\delta_{lk},$](img680.png) |
(295) |
where
is a Kronecker delta symbol.
The associated Legendre functions of order 0 (i.e.,
) are called Legendre polynomials, and
are denoted the
: that is,
. It follows that![[*]](footnote.png)
![$\displaystyle \int_{-1}^1P_l(x)\,P_k(x)\,dx = \frac{2}{(2\,l+1)}\,\delta_{lk}.$](img685.png) |
(296) |
It can also be shown that
![$\displaystyle \frac{1}{(1-2\,x\,t+t^{\,2})^{1/2}} = \sum_{l=0,\infty} P_l(x)\,t^{\,l},$](img686.png) |
(297) |
provided
and
.
All of the associated Legendre functions of degree less than 3 are listed below:
![$\displaystyle P_0^{\,0}(x)$](img689.png) |
![$\displaystyle = 1,$](img690.png) |
(298) |
![$\displaystyle P_{1}^{\,-1}(x)$](img691.png) |
![$\displaystyle = (1/2)\,(1-x^{\,2})^{1/2},$](img692.png) |
(299) |
![$\displaystyle \ P_1^{\,0}(x)$](img693.png) |
![$\displaystyle = x,$](img694.png) |
(300) |
![$\displaystyle P_1^{\,+1}(x)$](img695.png) |
![$\displaystyle = -(1-x^{\,2})^{1/2},$](img696.png) |
(301) |
![$\displaystyle P_2^{\,-2}(x)$](img697.png) |
![$\displaystyle =(1/8)\,(1-x^{\,2}),$](img698.png) |
(302) |
![$\displaystyle P_2^{\,-1}(x)$](img699.png) |
![$\displaystyle =(1/2)\,x\,(1-x^{\,2})^{1/2},$](img700.png) |
(303) |
![$\displaystyle P_2^{\,0}(x)$](img701.png) |
![$\displaystyle =(1/2)\,(3\,x^{\,2}-1),$](img702.png) |
(304) |
![$\displaystyle P_2^{\,+1}(x)$](img703.png) |
![$\displaystyle =-3\,x\,(1-x^{\,2})^{1/2},$](img704.png) |
(305) |
![$\displaystyle P_2^{\,+2}(x)$](img705.png) |
![$\displaystyle =3\,(1-x^{\,2}).$](img706.png) |
(306) |
Next: Spherical Harmonics
Up: Potential Theory
Previous: Introduction
Richard Fitzpatrick
2014-06-27