4038Supplement_Legendre_Bessel_Su2010

4038Supplement_Legendre_Bessel_Su2010 - Norms and Zeros 1...

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Norms and Zeros Math 4038 Summer 2010 1 k P n k :TheNo rmo fthe n th Legendre Polynomial We will prove that k P n k = r 2 2 n +1 for each nonnegative integer n , leaving numerous details for the reader to check. We will use Rodriguez’s Formula 1 , established in class: P n ( x )= 1 2 n n ! d n u dx n ,whe re u = ( x 2 1 ) n . We will apply integration by parts repeatedly. (2 n n !) 2 Z 1 1 P 2 n dx = Z 1 1 u ( n ) u ( n ) dx = u ( n ) u ( n 1) ± ± ± 1 1 Z 1 1 u ( n +1) u ( n 1) dx = Z 1 1 u ( n +1) u ( n 1) dx = ... =( 1) n (2 n )! Z 1 1 ( x 2 1 ) n dx ( A ) =( 1) n (2 n )! Z π 2 π 2 cos 2 n +1 θdθ ( B ) =( 1) n (2 n )! 2(2 n n !) 2 (2 n +1)! where step (A) is a trigonometric substitution, and step (B) is explained as follows. We prove by induction that Z π 2 π 2 cos 2 n +1 θdθ =2 Z π 2 0 cos 2 n +1 θdθ = (2 n n !) 2 (2
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This note was uploaded on 01/06/2012 for the course MATH 4038 taught by Professor Staff during the Summer '08 term at LSU.

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4038Supplement_Legendre_Bessel_Su2010 - Norms and Zeros 1...

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