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Math Help - Bessel Function of order 1

  1. #1
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    Bessel Function of order 1

    The function


    $J_{1} (x) = \sum_{n=0}^{\infty} (-1)^{n}\frac{x^{2n+1}}{2^{2n+1} (n!)(n+1)!}$


    is called a Bessel function of order 1. Show that it satisfies the differential equation

     x^{2} J''_{1}(x) +xJ'_{1}(x)  + (x^{2} - 1)J_{1}(x) = 0

    I have tried differentiating the series and plugging in but I cannot get the right answer.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    We can check your computations.


    Fernando Revilla
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  3. #3
    MHF Contributor chisigma's Avatar
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    May be that a more confortable way is to valuate what are the solution of the DE...

    \displaystyle y^{''} + \frac{y^{'}}{x} + (1-\frac{\nu^{2}}{x^{2}}) =0 (1)

    ... that can be expressed in the form...

    \displaystyle y(x)= x^{\nu}\ \sum_{k=0}^{\infty} a_{k}\ x^{2k} (2)

    With a little of patience You find that is...

    \displaystyle y(x)= a_{0}\ x^{\nu}\ \{1+ \sum_{k=1}^{\infty} (-1)^{k}\ \frac {(\frac{x}{2})^{2 k}}{k!\ (1+\nu)\ (2+\nu)...(k+\nu)}\} (3)

    Now You set in (3) \nu=1 and a_{0}=1 and Youn obtain J_{1}(x)...

    Kind regards

    \chi \sigma
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