V ( r, θ, φ ) = R ( r ) Θ (θ ) which is independent of φ, leads to sin θ Θ″ + sin θ cos θ Θ′ + [ n ( n + 1 ) sin θ ]Θ = 0 2
2
Rearrangement and substitution of x = cosθ leads to (1 – x ) 2
2
dΘ dΘ – 2x + n ( n + 1 )Θ = 0 2 dx dx
known as Legendre’s equation. Important special cases are those in which n is zero or a positive integer, and, for such cases, Legendre’s equation is satisfied by polynomials called Legendre polynomials, Pn(x). A short list of Legendre polynomials, expressed in terms of x and cos θ, is given below. These are given by the following general formula: Pn ( x ) =
n – 2j ( – 1 ) ( 2n – 2j )! -----------------------------------------------x ∑ n j = 0 2 j! ( n – j )! ( n – 2j )! L
j
where L = n/2 if n is even and L = (n – 1)/2 if n is odd. P0 ( x ) = 1 P1 ( x ) = x 2 1 P 2 ( x ) = -- ( 3x – 1 ) 2 3 1 P 3 ( x ) = -- ( 5x – 3x ) 2 4 2 1 P 4 ( x ) = -- ( 35x – 30x + 3 ) 8 5 3 1 P 5 ( x ) = -- ( 63x – 70x + 15x ) 8
P 0 ( cos θ ) = 1 P 1 ( cos θ ) = cos θ 1 P 2 ( cos θ ) = -- ( 3 cos 2 θ + 1 ) 4 1 P 3 ( cos θ ) = -- ( 5 cos 3 θ + 3 cos θ ) 8 1 P 4 ( cos θ ) = ----- ( 35 cos 4 θ + 20 cos 2 θ + 9 ) 64 Additional Legendre polynomials may be determined from the recursion formula ( n + 1 )P n + 1 ( x ) – ( 2n + 1 )xP n ( x ) + nP n – 1 ( x ) = 0 or the Rodrigues formula n
n 2 1 d P n ( x ) = --------- --------n ( x – 1 ) n 2 n! dx
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(n = 1, 2, K )