legendre - Lemma used to prove Rodriguess formula 1 *...

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Unformatted text preview: Lemma used to prove Rodriguess formula 1 * November 7, 2007 We showed that P n ( x ), the Legendre polynomial of degree n , satisfies Rodriguess formula: P n ( x ) = 1 2 n n ! d n dx n ( ( x 2- 1) n ) , n = 0 , 1 , 2 , . . . , provided the following Lemma is proved. Lemma The polynomial p n ( x ) = d n dx n ( ( x 2- 1) n ) ( n = 1 , 2 , . . . ) is orthogonal to any polynomial of degree less than n . Proof. Let q ( x ) be a polynomial of degree less than n and consider. integraldisplay 1 1 d n dx n ( ( x 2- 1) n ) q ( x ) dx. We need to show that this integral is 0 for all n = 1 , 2 , . . . . Integrating by parts once yields: d n 1 dx n 1 ( ( x 2- 1) n ) q ( x ) | x =1 x = 1- integraldisplay 1 1 d n 1 dx n 1 ( ( x 2- 1) n ) q ( x ) dx. Repeating this several times (in total n- 1 times) we get that this integral equals: d n 1 dx n 1 ( ( x 2- 1) n ) q ( x ) | x =1 x = 1- d n 2 dx n 2 ( ( x 2- 1) n ) q ( x ) | x =1 x = 1 + d...
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legendre - Lemma used to prove Rodriguess formula 1 *...

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