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Coefficients of a Fourier-Bessel series

  1. Aug 2, 2013 #1
    When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for [itex]k_1[/itex]and [itex]k_2[/itex] both zeroes of [itex]J_n(t)[/itex], the orthogonality relation given by:
    $$\int_0^1 J_n(k_1r)J_n(k_2r)rdr = 0, (k_1≠k_2)$$
    and for [itex]k_1 = k_2 = k[/itex]:

    $$\int_0^1 J_n^2(kr)rdr = \frac12J_n^{'2}(k)$$

    I understand how to get the first result since the Bessel's equation can be interpreted as a Sturm-Liouville problem, but how can I show the second one?
     
  2. jcsd
  3. Aug 3, 2013 #2

    SteamKing

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    This paper might offer a clue:
     

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  4. Aug 3, 2013 #3
    But how exactly i can evaluate the integral and get the result given by: [tex] I = \frac{R^2}{\alpha_m^2 - \alpha_n^2}[\alpha_mJ_0(\alpha_n)J_1(\alpha_m) - \alpha_nJ_0(\alpha_m)J_1(\alpha_n)][/tex] or another more general formula for Bessel functions of different order?
     
  5. Aug 3, 2013 #4

    SteamKing

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    I believe an integration by parts is called for, using certain relations of Bessel functions to get over the tricky bits:

    http://home.comcast.net/~rmorelli146/U3150/Bessel.pdf [Broken]
     
    Last edited by a moderator: May 6, 2017
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