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Math Help - Bessel function ??

  1. #1
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    Post Bessel function ??



    n : integer .
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  2. #2
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    Quote Originally Posted by amro05 View Post


    n : integer .
    What have you tried?
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by amro05 View Post


    n : integer .
    You need to mess around with the series for the Bessel function of the first kind:

    J_n(x) = \displaystyle\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m+n}}{2^{2m+n}m!\Gamma(m+n+1)}

    Clearly,

    J_{-n}(x) = \displaystyle\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m-n}}{2^{2m-n}m!\Gamma(m-n+1)}

    Now, make a substitution. Let m=m^{\prime}+n. Then,

    \begin{aligned}\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m-n}}{2^{2m-n}m!\Gamma(m-n+1)} &=\sum\limits_{m^{\prime}+n=0}^{\infty}\dfrac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p  rime}+n+1)}\\ &= \sum\limits_{m^{\prime}=-n}^{-1}\dfrac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p  rime}+n+1)}+\sum\limits_{m^{\prime}=0}^{\infty}\df  rac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p  rime}+n+1)}\end{aligned}

    A couple observations need to be made to finish it up.

    Can you proceed?
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  4. #4
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    Quote Originally Posted by Chris L T521 View Post
    You need to mess around with the series for the Bessel function of the first kind:

    J_n(x) = \displaystyle\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m+n}}{2^{2m+n}m!\Gamma(m+n+1)}

    Clearly,

    J_{-n}(x) = \displaystyle\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m-n}}{2^{2m-n}m!\Gamma(m-n+1)}

    Now, make a substitution. Let m=m^{\prime}+n. Then,

    \begin{aligned}\sum\limits_{m=0}^{\infty}\dfrac{(-1)^m x^{2m-n}}{2^{2m-n}m!\Gamma(m-n+1)} &=\sum\limits_{m^{\prime}+n=0}^{\infty}\dfrac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p  rime}+n+1)}\\ &= \sum\limits_{m^{\prime}=-n}^{-1}\dfrac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p  rime}+n+1)}+\sum\limits_{m^{\prime}=0}^{\infty}\df  rac{(-1)^{m^{\prime}+n} x^{2m^{\prime}+n}}{2^{2m^{\prime}+n}m!\Gamma(m^{\p  rime}+n+1)}\end{aligned}

    A couple observations need to be made to finish it up.

    Can you proceed?
    ----------------------
    yes I think I got it ( thank you very very mach )



    because the Gamma function of the negative integer = infinity
    so the first part = zero
    and the 2.nd part ( will be ) = (-1)^n Jn(x) ( but need some work )
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  5. #5
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    is it correct ??
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by amro05 View Post


    is it correct ??
    Looks good to me! (but note your sum should start at zero..you forgot to put that in.)
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