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Homework Help: Gamma Function Convergence

  1. Jun 5, 2005 #1
    I just learned induction in another thread and I'm curious if it can be used to prove that the gamma function converges for [itex]p\geq0[/itex]. I'm not sure if it can be used in this way. Is this wrong?

    Gamma Function is defined as:
    [tex]\Gamma(p+1)=\int_0^\infty e^{-x}x^p \,dx[/tex] We're trying to show that this converges for [itex]p\geq0[/itex]

    Smallest case, p=0:
    [tex]\Gamma(1)=1[/tex] converges

    Assume the following converges:
    [tex]\Gamma(p)=\int_0^\infty e^{-x}x^{p-1} \,dx[/tex]

    Using integration by parts we find:
    [tex]\Gamma(p+1)=p\Gamma(p)[/tex]

    So since
    [tex]\Gamma(p)[/tex] converges
    then
    [tex]\Gamma(p+1)=p\Gamma(p)[/tex] must also converge
     
  2. jcsd
  3. Jun 5, 2005 #2

    Galileo

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    For which values of p are you assuming it holds? Let's assume it holds for all [itex]1\leq p \leq p'[/itex] for some real number [itex]p'\geq 1[/itex].
    for [itex]p\in [1,p'][/itex]
    for [itex]p\in[1,p'][/itex].

    Together with the first step of the induction process, you've shown that [itex]\Gamma(p+1)[/itex] converges for p=0,1,2,3,4,..., but not for the real numbers in between.
    If you can show that [itex]\Gamma(p+1)[/itex] converges for [itex]0\leq p <1[/itex] your induction shows it to be true for all real numbers [itex]p\geq 1[/itex]. Can you see why?
     
    Last edited: Jun 5, 2005
  4. Jun 5, 2005 #3
    Ah right, that was my assumption that it was true for some real number greater than 1. I should have been more explicit.

    To prove that [itex]\Gamma(p+1)[/itex] converges for [itex]0\leq p <1[/itex] couldn't I just say that [itex]\Gamma(1+1) \geq \Gamma(0+1)[/itex] and since [itex]\Gamma(1+1)[/itex] converges then the gamma function must converge for [itex]0 \leq p \leq 1[/itex]?
     
  5. Oct 11, 2010 #4
    looks like u only showed that it converges for all natural numbers
     
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