Confused on Bessel Proof

  • Thread starter Void123
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  • #1
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Homework Statement



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

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



Homework Equations



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

J(m) should be obvious.



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.
 

Answers and Replies

  • #2
lanedance
Homework Helper
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is m an integer?

maybe try showing your working & what special properties of the gamma function can you use to help?
 
  • #3
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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.
 
  • #4
141
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I think I figured it out.
 
  • #5
lanedance
Homework Helper
3,304
2
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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