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.