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Appendix A: List of associated Legendre functions

Let x = cos($x=\cos(\theta)$), and img104.gif. The associated Legendre functions Pnm = Pnm(x) are listed below, for n = 0,8 and m = 0,n, in Table 3 (n = 0,5) and Table 4 (n = 6,8). The associated Legendre functions with m = -M < 0 are simply given in terms of the equivalent functions with M > 0 according to

\begin{displaymath}P_n^m=P_n^{-M} = (-1)^M \frac{(n-M)!}{(n+M)!} P_n^M.
\end{displaymath} (30)


Table 3: List of associated Legendre polynomials from n = 0 to n = 5.
n m Function
0 0 1
1 0 x
1 1 s
2 0 img106.gif
2 1 3 x s
2 2 3 (1-x2)
3 0 $\frac{1}{2} x (5x^2-3)$
3 1
3 2 15 x (1-x2)
3 3 15 s3
4 0
4 1
4 2
4 3 105xs3
4 4 105 s4
5 0
5 1
5 2
5 3
5 4 945xs4
5 5 945s5



Table 4: List of associated Legendre polynomials from n = 6 to n = 8.
n m Function
6 0
6 1
6 2 $\frac{105}{8}s^2(33x^4-18x^2+1)$
6 3 $\frac{315}{2}(11x^2-3)xs^3$
6 4 $\frac{945}{2}s^4(11x^2-1)$
6 5 10395xs5
6 6 10395s6
7 0 $\frac{1}{16}x(429x^6-693x^4+315x^2-35)$
7 1 $\frac{7}{16}s(429x^6-495x^4+135x^2-5)$
7 2 $\frac{63}{8} x s^2(143x^4-110x^2+15)$
7 3 $\frac{315}{8}s^3(143x^4-66x^2+3)$
7 4 $\frac{3465}{2}xs^4(13x^2-3)$
7 5 $\frac{10395}{2}s^5(13x^2-1)$
7 6 135,135xs6
7 7 135,135s7
8 0 $\frac{1}{128}(6435x^8-12,012x^6+6930x^4-1260x^2+35)$
8 1 $\frac{9}{16}xs(715x^6-1001x^4+385x^2-35)$
8 2 $\frac{315}{16}s^2(143x^6-143x^4+33x^2-1)$
8 3 $\frac{3465}{8}xs^3(39x^4-26x^2+3)$
8 4 $\frac{10395}{8}s^4(65x^4-26x^2+1)$
8 5 $\frac{135,135}{2}xs^5(5x^2-1)$
8 6 $\frac{135,135}{2}s^6(15x^2-1)$
8 7 2,027,025xs7
8 8 2,027,025s8


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