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Homework Help: Bessel Functions

  1. Dec 3, 2003 #1
    Hey guys I was wondering if you could help me out with a proof of the recursion relations of Bessel functions on my homework:

    Show by direct differentiation that

    [tex]


    J_{\nu}(x)=\sum_{s=0}^{\infty} \frac{(-1)^{s}}{s!(s + \nu)!} \left (\frac{x}{2}\right)^{\nu+2s}

    [/tex]

    obeys the important recursion relations

    [tex]
    J_{\nu-1}(x)+J_{\nu+1}(x) = \frac{2\nu}{x}J_{\nu}(x)
    [/tex]

    [tex]
    J_{\nu-1}(x)-J_{\nu+1}(x) = 2J_{\nu}(x)
    [/tex]

    I've tried differentiating with respect to x but I get a factor of 2s that's no good. And I've also tried replacing nu with nu plus one and nu minus one but that ends up with a lot of s terms as well. I am pretty much lost on what to do so if you could just point me in the right direction that'd be great. Thanks a lot.

    Csmines
     
  2. jcsd
  3. Dec 3, 2003 #2
    csmines,
    I'm not at all sure, but the first step in this could be to show that
    [tex]
    \frac{d}{dx}\left[x^{-\nu}J_{\nu}(x)\right]=-x^{-\nu}J_{\nu+1}(x)
    [/tex]
    The trick is probably to
    1) split off the s=0 term
    2) make an index shift s->s+1 in the rest sum
    3) differentiate (s=0 term vanishes).

    This is IMO not too difficult. But can we use it to show the desired relation? I wonder.
     
    Last edited: Dec 3, 2003
  4. Jun 25, 2010 #3
    hey let me know how to find out bessel transform of a sequence of numbers ,as in we calculate fourier transform of a sequence??

    this is urgent pls do reply..
    i need this for my project on "analog signal processing using bessel function using matlab"..
     
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