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Confused on Bessel Proof

  1. Oct 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove J(-m) = [(-1)^m][J(m)]

    (Note: by "J(-m)" I mean "subscript (-m)")



    2. Relevant equations

    J(-m) = sum [((-1)^n) * (x/2)^(2n-m)]/[n! [tex]\Gamma[/tex](n - m + 1)]

    J(m) should be obvious.



    3. The attempt at a solution

    I tried just plugging in the above formulas hoping to get a simplified answer, but I know I'm missing something in that denominator.
     
  2. jcsd
  3. Oct 13, 2009 #2

    lanedance

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    Homework Helper

    is m an integer?

    maybe try showing your working & what special properties of the gamma function can you use to help?
     
  4. Oct 13, 2009 #3
    Yes, m is an integer. I am also told that [tex]\Gamma (-k) = \infty[/tex] and that I should, therefore, eliminate the terms from the sum that equal zero. Also, 1/[(2^m)([tex]\Gamma[/tex] (m + 1))] is reduced to some 'a' constant.

    That's all I got.

    Not entirely sure what to do.
     
  5. Oct 13, 2009 #4
    I think I figured it out.
     
  6. Oct 13, 2009 #5

    lanedance

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    Homework Helper

    cool, yeah its true that for negative integers
    [tex]|\Gamma (-k)| \rightarrow \infty[/tex]
    the other handy ones were for integers:
    [tex]\Gamma (n) = (n-1)![/tex]
    and so
    [tex]\Gamma (n+1) = n \Gamma(n)[/tex]
     
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