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Unformatted text preview: Math Methods Solutions for Assignment 9 Nov. 01, 2010 Fall 2010 Special Functions Due Nov. 08, 2010 1 (6). By differentiating the Legendre polynomial generating function g ( x,t ) = (1- 2 xt + t 2 )- 1 / 2 with respect to t and with respect to x , obtain the recurrence relations for the Legendre polynomials, and prove that P n ( x ) satisfy the Legendre differential equation. 2 (6). By differentiating the Bessel generating function g ( x,t ) = e ( x/ 2)( t- 1 /t ) with respect to t and with respect to x , obtain the recurrence relations for the Bessel functions, and prove that J n ( x ) satisfy the Bessel differential equation. 3 (6). Prove the normalization condition for the Bessel functions Z ∞ dr r · J n ( kr ) · J n ( k r ) = 1 k δ ( k- k ) This is easiest to show using Z ∞ J n ( λ ) dλ = 1. 1 = r Z ∞ J n ( kr ) dk and 1 k = Z ∞ J n ( k r ) dr Substituting the first identity into the second equation and changing the order of integration we get 1 k = Z ∞ J n ( k r ) dr...
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This note was uploaded on 09/28/2011 for the course PHYSICS 801 taught by Professor Ivanov during the Fall '10 term at Kansas State University.
- Fall '10