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The Absolute Value of Gamma Function Tends To Zero As x Tends To Negative Infinity

  1. Sep 26, 2012 #1
    1. The problem statement, all variables and given/known data

    The absolute value of the gamma function [itex] \Gamma (x) [/itex] that is defined on the negative real axis tends to zero as [itex] x \to - \infty [/itex]. Right? But how do I prove it?



    2. Relevant equations



    3. The attempt at a solution

    I've tried to use Gauss's Formula:

    [tex] \Gamma(x)=\lim_{n\to\infty}\frac{n!n^{z}}{z(z+1) \cdots (z+n)}. [/tex]

    Should I keep going in this direction?

    But frankly, the calculation gets too technical so it'd be better if there is a bit easier way.
     
  2. jcsd
  3. Sep 26, 2012 #2

    Ray Vickson

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    Re: The Absolute Value of Gamma Function Tends To Zero As x Tends To Negative Infinit

    Have you ever looked at the graph of the Gamma function on the real line? Look in here:
    http://en.wikipedia.org/wiki/Gamma_function . Does it look to you that ##\Gamma(x) \rightarrow 0 ## as ##x \rightarrow -\infty?##

    RGV
     
  4. Sep 26, 2012 #3
    Re: The Absolute Value of Gamma Function Tends To Zero As x Tends To Negative Infinit

    Ah.. I know what you mean. Maybe I need to modify my problem first. I know it has poles on non-positive integers. But excluding poles, it seems the absolute value of the gamma function tends to zero as [itex] x \to - \infty [/itex].

    (http://en.wikipedia.org/wiki/File:Complex_gamma_function_abs.png)

    May I define

    [itex] f(x) = \Gamma (x) [/itex] only for [itex] x<0 \quad \mbox{and} \quad x \neq -1, -2, -3, -4, \dots [/itex]

    and then prove [itex] |f(x)| \to 0 [/itex] as [itex] x \to - \infty [/itex]?
     
  5. Sep 27, 2012 #4
    Re: The Absolute Value of Gamma Function Tends To Zero As x Tends To Negative Infinit

    Ah.... MY BAD!! sorry.. what was I thinking.... Let me clarify once more:

    Take [itex] x_{n} \in (-n,1-n) [/itex]. Then [itex] \Gamma (x_{n}) \to 0 [/itex] as [itex] n \to \infty [/itex].

    I think I have an idea to solve it without using Gauss's Formula. After I try, I will put on the thread.

    Anyway thanks for reminding me.
     
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