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Please help proving this Bessel identity.

  1. Nov 23, 2009 #1
    I have been working on this for a few days and cannot prove this:

    J-3/2 (x)=[tex]\sqrt{\frac{2}{\pi x}}[/tex][[tex]\frac{-cos(x)}{x}[/tex] - sin(x) ]

    Main reason is [tex]\Gamma[/tex](n-3/2+1) give a negative value for n=0 and possitive value for n=1,2,3.... I cannot find a series representation of this gamma function.


    Please advice me how to solve this problem. This is not a school homework.

    thanks a million

    Alan
     
    Last edited: Nov 23, 2009
  2. jcsd
  3. Nov 23, 2009 #2
    Maybe instead of the series representation for the Bessel function: show that your right-hand side satisfies Bessel's differential equation, and has the proper initial values, so that it therefore equals the Bessel function required.
     
  4. Nov 24, 2009 #3
    For n > 0 we apply this formula
    [tex]\Gamma (n+1) = n \Gamma (n)[/tex]

    but if n < 0 we apply this formula
    [tex]\Gamma (n)=\frac{\Gamma (n+1)}{n}[/tex]
     
  5. Nov 24, 2009 #4
    I know this formula!!!!!!! This is embarassing!!!! How can I over looked this and spent 3 days on this.....Even joined two more math forums!!!! I even plug in the numbers and hope this is not that simple!!!! I use

    [tex]\Gamma (-3/2)=\frac{\Gamma (-1/2)}{-3/2}[/tex] all the time!!! Just never try with n in it!!!!

    Thanks a million.......Even though you make me look really really bad!!!!!

    Cheers.
    Alan
     
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