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Homework Help: Why is the gamma function equal to (n-1)! ?

  1. Sep 27, 2010 #1
    1. The problem statement, all variables and given/known data

    Why is the equality below true?

    [tex]\Gamma(n) = (n-1)![/tex]

    Where [tex]\Gamma(n) = \int^{\infty}_{0} x^{n-1} e^{-x}dx[/tex]


    2. Relevant equations



    3. The attempt at a solution

    I've read the article on wikipedia but I cannot understand it. Is there any special properties in calculus that I must know in order to comprehend this?

    Thank you
    M
     
  2. jcsd
  3. Sep 27, 2010 #2
    It is true only if n is integer

    Solve:

    [tex]\int_0^\infty t^{4-1}e^{-t} dt[/tex]

    Compare if it's equal to
    [tex](4-1)![/tex]

    Gamma function is really beautiful beacuse it extends the concept of factorial out of integer numbers.
    Yep, being able to integrate. :()
     
  4. Sep 28, 2010 #3

    fzero

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    Science Advisor
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    Gold Member

    You can use integration by parts to show that

    [tex]\Gamma(n) = (n-1)\Gamma(n-1)[/tex]

    If n is an integer, you can use this to prove by induction that

    [tex]\Gamma(n) = (n-1)![/tex]
     
  5. Sep 28, 2010 #4

    hunt_mat

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    Homework Helper

    I went to a talk by John Chapman on this and he said said that the Gamma function is related to "Runge phenomenon".
     
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