19a - Lecture (n) (n) Interlude: Calculate Pn (t) = Pn (0),...

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Lecture Interlude : Calculate P ( n ) n ( t ) = P ( n ) n (0), a constant. Fact : P n ( t ) = 1 2 n n ! d n dt n ( t 2 - 1) n , the Rodrigues Formula for Legendre Polynomials. This implies that P ( n ) n = (2 n )(2 n - 1) ··· ( n + 1) n ( n - 1) ··· (2)(1) 2 n n ! = (2 n )! 2 n n ! . Alternative Approach to P ( n ) n : Recall the generating function for the Legendre Polynomials, P 0 ( t ) + P 1 ( t ) u + P 2 ( t ) u 2 + ··· + P n ( t ) u n + ··· = (1 - 2 tu + u 2 ) - 1 / 2 Apply d n dt n d n dt n (LHS) = P ( n ) n ( t ) u n + P ( n ) n +1 ( t ) u n +1 + ··· So, 1 u n d n dt n (LHS) = P ( n ) n ( t ) + X k =1 P ( n ) n + k ( t ) u k Therefore, P ( n ) n = 1 u n d n dt n (LHS) ± ± ± ± u =0 = 1 u n d n dt n (RHS) ± ± ± ± u =0 . 1
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Differentiating the generating function with respect to t gives ∂t (1 - 2 tu + u 2 ) - 1 / 2 = - 1 2 ( - 2 u )(1 - 2 tu + u 2 ) - 3 / 2 = u (1 - 2 tu + u 2 ) - 3 / 2 , n ∂t n (1 - 2 tu + u 2 ) - 1 / 2 = u n (1)(3) ··· (2 n - 1)(1 - 2 tu + u 2 ) - (2 n +1) 2 . Thus, 1 u n n ∂t n (1 - 2 tu + u 2 ) - 1 / 2 ± ± ± ± u =0 = (2 n )! 2
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.

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19a - Lecture (n) (n) Interlude: Calculate Pn (t) = Pn (0),...

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