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Math Help - second derivative of Bessel Function in terms of higher and lower orders of Bessel fn

  1. #1
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    Question second derivative of Bessel Function in terms of higher and lower orders of Bessel fn

    I have been trying to replicate a result given in a textbook that says

    J_{n}^{''}(x)=\frac{1}{4}\{J_{n-2}(x)-2J_{n}(x)+J_{n+2}(x)\}

    where J_{n}(x) is the Bessel Function of the First Kind.

    Can someone show me how to get this from the recurrence formulae for Bessel derivatives found in the literature as

    (\frac{1}{x}\frac{d}{dx})^{m}(x^{n}J_{n}(x))=x^{n-m}J_{n-m}(x)

    and

    (\frac{1}{x}\frac{d}{dx})^{m}(x^{-n}J_{n}(x))=(-1)^{m}x^{-n-m}J_{n+m}(x)
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  2. #2
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    Re: second derivative of Bessel Function in terms of higher and lower orders of Besse

    Hi !
    have a look at attachment :
    Attached Thumbnails Attached Thumbnails second derivative of Bessel Function in terms of higher and lower orders of Bessel fn-besselj.jpg  
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  3. #3
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    Re: second derivative of Bessel Function in terms of higher and lower orders of Besse

    Thanks JJacquelin!

    My problem was in using the recurrence formulae for the case m=2. Not sure how to interpret (d/dx)^2, but your way avoids this.
    Last edited by billm; October 31st 2012 at 01:47 PM.
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