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Using Bessel generating function to derive a integral representation of Bessel functi

  1. Dec 7, 2008 #1
    1. The problem statement, all variables and given/known data

    The Bessel function generating function is
    [tex]
    e^{\frac{t}{2}(z-\frac{1}{z})} = \sum_{n=-\infty}^\infty J_n(t)z^n
    [/tex]

    Show
    [tex]
    J_n(t) = \frac{1}{\pi} \int_0^\pi cos(tsin(\vartheta)-n\vartheta)d\vartheta
    [/tex]

    2. Relevant equations



    3. The attempt at a solution

    So far I have been able to use an analytic function theorem to write

    [tex]
    J_n(t)=\frac{1}{2\pi i} \oint e^{\frac{t}{2}(z-\frac{1}{z})}z^{-n-1}dz
    [/tex]
    (we are required to use this)
    But now I have no idea where to go from here.
     
    Last edited: Dec 7, 2008
  2. jcsd
  3. Dec 7, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Using Bessel generating function to derive a integral representation of Bessel fu

    It looks to me like you want to insert a specific contour. Like z=exp(i*theta).
     
  4. Dec 7, 2008 #3
    Re: Using Bessel generating function to derive a integral representation of Bessel fu

    Thanks can't believe I missed it
     
  5. Aug 26, 2011 #4
    Re: Using Bessel generating function to derive a integral representation of Bessel fu

    Using Bessel generating function to derive a integral representation of Bessel function
     
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