emsol06 - Electromagnetic Theory I Solution Set 6 Due 3...

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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 ) ≡ ∞ X k =0 (- ) k k !Γ( 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 ) ≡ ∞ X k =0 (- ) k k !Γ( k + m ) x 2 2 k + m- 1 J m +1 ( x ) ≡ ∞ X k =0 (- ) k k !Γ( k + m + 2) x 2 2 k + m +1 = ∞ X k =1- (- ) k ( k- 1)!Γ( k + m + 1) x 2 2 k + m- 1 Then J m- 1 ± J m +1 = 1 Γ( m ) x 2 m- 1 + ∞ X k =1 (- ) k ( k + m ∓ k ) k !Γ( k + m + 1) x 2 2 k + m- 1 = ∞ X k =0 (- ) k k !Γ( k + m + 1) m 2 k + m x 2 2 k + m- 1 = 2 m/x 2 d/dx ∞ X k =0 (- ) k k !Γ( 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 ± 1 = J m ± 1 cos( m ± 1) π- J- m ∓ 1 sin( m ± 1) π = J m ± 1 cos mπ + J- m ∓ 1 sin mπ N m- 1 ± N m +1 = ( J m- 1 ± J m +1 ) cot mπ + ( J- m +1 ± J- m- 1 ) csc mπ = 2 m/x 2 d/dx J m cot mπ ±- 2 m/x 2 d/dx J- m csc mπ = 2 m/x 2 d/dx ( J m cot mπ- J- m csc mπ ) = 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 ± 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 ) = πi 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 ) ± I m +1 ( x ) = i- m +1 J m- 1 ( ix ) ± i- m- 1 J m +1 ( ix ) = i- m +1 ( J m- 1 ( ix ) ∓ 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 ) ± K m +1 ( x ) = π 2 ( i m H (1) m- 1 ( ix ) ± i m +2 H (1) m +1 ( ix )) = π 2 i m ( H (1) m- 1 ( ix ) ∓ H (1) m +1 ( ix )) = π 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 Γ( z ) → ∞ when z → 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 ) ≡ ∞ X k =0 (- ) k k !Γ( k + 1- m ) x 2 2 k- m (5) When m → a nonnegative integer, 1 / Γ( k + 1- m ) = 0 when k < m . Thus we can drop those terms from the sum: J- m ( x ) ≡ ∞ X k = m (- ) k k !Γ( k + 1- m ) x 2 2 k- m = ∞ X k =0 (- ) k + m ( k + m )!Γ( k + 1) x 2 2 k + m = (- ) m ∞ X k =0 (- ) k Γ( k + m + 1) k !...
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emsol06 - Electromagnetic Theory I Solution Set 6 Due 3...

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