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Math Help - Gamma function, zeros

  1. #1
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    Gamma function, zeros

    Show that \Gamma(z) has no zeros.

    I am not sure how to prove this. I would appreciate a few hints or suggestions. \Gamma(z) is the gamma function. The gamma function is meromorphic. It is defined in the right half-plane by \Gamma(z)= \int_0^{\infty} e^{-t}t^{z-1}dt for \text{Re}(z)>0. There is also another representation of \Gamma(z)=\frac{\Gamma(z+m)}{(z+m-1) \cdots (z+1)z} where the right-hand side is defined and meromorphic for \text{Re}(z)>-m with simple poles at z=0, -1, \ldots, -m+1. Thank you.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    What about \Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin{\pi z}}?
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  3. #3
    MHF Contributor chisigma's Avatar
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    The most simple way to demonstrate that \Gamma (z) has no zeroes is to expand its inverse as 'infinite product'...

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

    The (1) shows that \frac{1}{\Gamma (z)} is an entire function, so that it has no poles... and therefore \Gamma (z) has no zeroes...

    Kind regards

    \chi \sigma
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