Gamma Function Limits: Proving \Gamma(x) \to 0 as x \to -\infty

[julypraise]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I've tried to use Gauss's Formula:

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

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.

 

[Ray Vickson]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I've tried to use Gauss's Formula:

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

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

 

[julypraise]

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 x \to - \infty.

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

May I define

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

and then prove |f(x)| \to 0 as x \to - \infty?

 

[julypraise]

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

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

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.