LegendrePolynomials - Legendre Polynomials Rodrigues...

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1 Legendre Polynomials – Rodrigues’ Formula. Ian R. Gatland, Georgia Institute of Technology. Physics 3122 Notes, June 17, 2010. The differential equation for the Legendre polynomials, P ( x ) , is (1 x 2 ) ʹஒ ʹஒ P ( x ) 2 x ʹஒ P ( x ) + ( + 1) P ( x ) = 0 (1) and a solution is given by Rodrigues’ formula P ( x ) = 1 2 ! d dx x 2 1 ( ) (2) To prove that Rodrigues’ formula, equation (2), is a solution of Legendre’s equation, (1), we start with v = ( x 2 1) (3) The differential of equation (3) is dv / dx = 2 x ( x 2 1) 1 (4) Then multiplying equation (4) by ( x 2 1) and substituting from equation (3) we find (1 x 2 ) dv / dx + 2 xv = 0 (5) Differentiating equation (5) and using superscript values in parentheses to specify derivatives we get (1 x 2 ) v (2) + 2( 1) xv (1) + (2 ) v = 0 (6) After k differentiations we arrive at (1 x 2 ) v ( k + 2) + 2( k 1) xv ( k + 1) + ( k + 1)(2 k ) v ( k ) = 0 (7) This may be verified by a further differentiation (1 x 2 ) v ( k + 3) 2 xv ( k + 2) + 2( k 1) xv ( k + 2) + 2( k 1) v ( k + 1) + ( k + 1)(2
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