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Unformatted text preview: Electromagnetic Theory I Solution Set 6 Due: 3 October 2011 21. a) Use the power series definition of the Bessel function J m ( x ) â‰¡ âˆž X k =0 ( ) k k !Î“( k + 1 + m ) x 2 2 k + m (1) to derive the recursion formulas for Bessel functions: J m 1 ( x ) + J m +1 ( x ) = 2 m x J m ( x ) , J m 1 ( x ) J m +1 ( x ) = 2 dJ m ( x ) dx (2) Solution : The power series for J m 1 and J m +1 can be written J m 1 ( x ) â‰¡ âˆž X k =0 ( ) k k !Î“( k + m ) x 2 2 k + m 1 J m +1 ( x ) â‰¡ âˆž X k =0 ( ) k k !Î“( k + m + 2) x 2 2 k + m +1 = âˆž X k =1 ( ) k ( k 1)!Î“( k + m + 1) x 2 2 k + m 1 Then J m 1 Â± J m +1 = 1 Î“( m ) x 2 m 1 + âˆž X k =1 ( ) k ( k + m âˆ“ k ) k !Î“( k + m + 1) x 2 2 k + m 1 = âˆž X k =0 ( ) k k !Î“( k + m + 1) m 2 k + m x 2 2 k + m 1 = 2 m/x 2 d/dx âˆž X k =0 ( ) k k !Î“( k + m + 1) x 2 2 k + m = 2 m/x 2 d/dx J m ( x ) b) Explain why these same recursion formulas are valid for N m , H (1) m , H (2) m . Solution : From the definition of N m , N m Â± 1 = J m Â± 1 cos( m Â± 1) Ï€ J m âˆ“ 1 sin( m Â± 1) Ï€ = J m Â± 1 cos mÏ€ + J m âˆ“ 1 sin mÏ€ N m 1 Â± N m +1 = ( J m 1 Â± J m +1 ) cot mÏ€ + ( J m +1 Â± J m 1 ) csc mÏ€ = 2 m/x 2 d/dx J m cot mÏ€ Â± 2 m/x 2 d/dx J m csc mÏ€ = 2 m/x 2 d/dx ( J m cot mÏ€ J m csc mÏ€ ) = 2 m/x 2 d/dx N m (3) 1 Establishing the identities for N m . But then since H (1) , (2) m = J m Â± iN m are linear combinations of J m , N m the identities immediately apply to them as well. c) Using the definitions I m ( x ) = i m J m ( ix ) , K m ( x ) = Ï€i m +1 2 H (1) m ( ix ) (4) obtain the analogous recursion formulas for I m , K m . Solution : We simply write out I m 1 ( x ) Â± I m +1 ( x ) = i m +1 J m 1 ( ix ) Â± i m 1 J m +1 ( ix ) = i m +1 ( J m 1 ( ix ) âˆ“ J m +1 ( ix )) = i m +1 2 d/d ( ix ) 2 m/ix J m ( ix ) = 2 d/dx 2 m/x I m ( x ) K m 1 ( x ) Â± K m +1 ( x ) = Ï€ 2 ( i m H (1) m 1 ( ix ) Â± i m +2 H (1) m +1 ( ix )) = Ï€ 2 i m ( H (1) m 1 ( ix ) âˆ“ H (1) m +1 ( ix )) = Ï€ 2 i m 2 d/d ( ix ) 2 m/ix H (1) m ( ix ) = 2 d/dx 2 m/x K m ( x ) Notice the sign differences for I, K compared to the others. d)Use the power series definition of J m , and the fact that Î“( z ) â†’ âˆž when z â†’ a nonpositive integer, to prove that J m ( x ) = ( ) m J m ( x ) when m is an integer. Solution : Assuming first that m is not an integer, the power series definition reads J m ( x ) â‰¡ âˆž X k =0 ( ) k k !Î“( k + 1 m ) x 2 2 k m (5) When m â†’ a nonnegative integer, 1 / Î“( k + 1 m ) = 0 when k < m . Thus we can drop those terms from the sum: J m ( x ) â‰¡ âˆž X k = m ( ) k k !Î“( k + 1 m ) x 2 2 k m = âˆž X k =0 ( ) k + m ( k + m )!Î“( k + 1) x 2 2 k + m = ( ) m âˆž X k =0 ( ) k Î“( k + m + 1) k !...
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This note was uploaded on 01/08/2012 for the course PHY 6346 taught by Professor Staff during the Spring '08 term at University of Florida.
 Spring '08
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