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Math Help - The Gamma Function

  1. #1
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    The Gamma Function

    Hello,

    I'm trying to teach myself the Gamma Function from here:

    The Gamma Function

    However, I'm a tad confused at some parts.

    I don't understand at all how they figured out (derived) the inequality where it says

    "A careful analysis of the Gamma function (especially if we notice that is a convex function) yields the inequality"

    ...and I don't know what a convex function is. I don't really see the significance in deriving the function in all these different ways.

    Also, when they give the intial formula, I don't understand how it was derived/where it came from. Does anyone know how it was formulated?

    \int^{\infty}_0 e^{-t}t^n\,dt=n!

    Thanks for all your help.
    Last edited by RobLikesBrunch; September 9th 2009 at 07:02 PM.
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  2. #2
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    Note: this is somewhat urgent, so if anyone knows anything about it, please help me out.
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  3. #3
    MHF Contributor chisigma's Avatar
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    If you represent the \Gamma(*) as ‘infinite product’…

    \Gamma (x) = \frac{e^{-\gamma x}}{x} \cdot \prod_{n=1}^{\infty} \frac{e^{\frac{x}{n}}}{1+\frac{x}{n}} (1)

    … then the \ln \Gamma(*) is simply the sum of the logarithms of all factors…

    \ln \Gamma (x) = -\gamma x - \ln x + \sum_{n=1}^{\infty} \{\frac{x}{n} - \ln (1 + \frac{x}{n})\} (2)

    Kind regards

    \chi \sigma
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