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Math Help - Bessel integration by parts

  1. #1
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    Bessel integration by parts

    Hi guys. I'm totally stuck on this problem. Any help would be appreciated:

    Using the definitions for the derivatives of Jp(x) (Bessel Function), along with integration by parts, demonstrate the following...



    Thanks in advance!

    Edit: Embedding of image isn't working for some reason. Here's a link...

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  2. #2
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    Quote Originally Posted by canopy View Post
    Hi guys. I'm totally stuck on this problem. Any help would be appreciated:

    Using the definitions for the derivatives of Jp(x) (Bessel Function), along with integration by parts, demonstrate the following...



    Thanks in advance!

    Edit: Embedding of image isn't working for some reason. Here's a link...

    http://imgfreehost.com/out.php?i26646_Picture4.png
    I'll do the first, the second follows similarly. We have the following property

    xJ_n' = - n J_n + x J_{n-1}

    so

    xJ_{n+1}' = - (n+1) J_{n+1} + x J_{n}

    then

    x^mJ_{n+1}' = - (n+1) x^{m-1}J_{n+1} + x ^mJ_{n}

    so

    \int x^mJ_{n+1}' dx = - (n+1)\int x^{m-1}J_{n+1} dx+ \int x ^mJ_{n}dx

    integration by parts on the frist integral gives

    x^m J_{n+1} - m \int x^{m-1} J_{n+1}' dx = - (n+1)\int x^{m-1}J_{n+1} dx+ \int x ^mJ_{n}dx

    and re-arranging terms gives

    \int x ^mJ_{n}dx = x^m J_{n+1} - ( m - n-1)\int x^{m-1}J_{n+1} dx

    For the second, try using

    xJ_n' = n J_{n} - xJ_{n+1}
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  3. #3
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    Thanks!

    In the interest of good taste, I'll refrain from using multiple exclamation points and all capital letters, but it's very hard to resist.

    Thanks!
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