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

  • Thread starter julypraise
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Homework Statement



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?



Homework Equations





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.
 

Answers and Replies

  • #2
Ray Vickson
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Homework Statement



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?



Homework Equations





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.
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
 
  • #3
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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
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]?
 
  • #4
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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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