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Math Help - Uniqueness of Gamma Function

  1. #1
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    Uniqueness of Gamma Function

    Is the \Gamma function uniquely determined (as a meromorphic function on \mathbb{C}) by the functional equation \Gamma(s + 1) = s\Gamma(s) and a single value, say \Gamma(1) = 1?

    And/or is there any other characterization of the Gamma function that uniquely determines it on \mathbb{C}?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by EinStone View Post
    Is the \Gamma function uniquely determined (as a meromorphic function on \mathbb{C}) by the functional equation \Gamma(s + 1) = s\Gamma(s) and a single value, say \Gamma(1) = 1?

    And/or is there any other characterization of the Gamma function that uniquely determines it on \mathbb{C}?
    Here are the criterion that uniquely determine  \Gamma(z) in  \mathbb{C} :

    1.)  \Gamma(1)=1

    2.)  \Gamma(s+1)=s\Gamma(s)

    3.)  \lim_{n\to\infty}\frac{\Gamma(s+n)}{n^s\Gamma(n)}=  1

    Ask if you'd like to see why this is true.

    This is a nice theorem, because if you want to show something is equal to the Gamma function, all you need to do is show it satisfies these three properties!
    Last edited by chiph588@; March 27th 2010 at 02:57 PM.
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    Can you give a counter example why my first definition does not work?
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by EinStone View Post
    Can you give a counter example why my first definition does not work?
    Let  F(s) = \Gamma(s)\cos(2\pi s)

    1.)  F(1) = \Gamma(1)\cos(2\pi) = 1

    2.)  F(s+1) = \Gamma(s+1)\cos(2\pi(s+1)) = s\Gamma(s)\cos(2\pi s) = sF(s)

    But  F(s)\neq\Gamma(s)
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by chiph588@ View Post
    Let  F(s) = \Gamma(s)\cos(2\pi s)

    1.)  F(1) = \Gamma(1)\cos(2\pi) = 1

    2.)  F(s+1) = \Gamma(s+1)\cos(2\pi(s+1)) = s\Gamma(s)\cos(2\pi s) = sF(s)

    But  F(s)\neq\Gamma(s)
    To support my uniqueness claim of  \Gamma(s) , let's compute  \lim_{n\to\infty}\frac{F(s+n)}{n^sF(n)} .


     \lim_{n\to\infty}\frac{F(s+n)}{n^sF(n)} = \lim_{n\to\infty}\frac{\Gamma(s+n)\cos(2\pi(s+n))}  {n^s\Gamma(n)\cos(2\pi n)} = 1\cdot\lim_{n\to\infty}\frac{\cos(2\pi(s+n))}{\cos  (2\pi n)} \not\equiv 1
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  6. #6
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    Another characterization (on \mathbb{R}) : \Gamma is the only function on (0,+\infty) that is log-convex (i.e. \log\Gamma is convex) and satisfies \Gamma(1)=1 and \Gamma(s+1)=s\Gamma(s) for all s>0. This is called Artin's theorem I think. (and the proof is really nice; probably you can find it somewhere online)
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  7. #7
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Laurent View Post
    Another characterization (on \mathbb{R}) : \Gamma is the only function on (0,+\infty) that is log-convex (i.e. \log\Gamma is convex) and satisfies \Gamma(1)=1 and \Gamma(s+1)=s\Gamma(s) for all s>0. This is called Artin's theorem I think. (and the proof is really nice; probably you can find it somewhere online)
    You are referring to the Bohr–Mollerup theorem.

    Here's the proof.
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  8. #8
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    Thank you guys. Was looking for this...
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