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

In summary, the absolute value of the gamma function tends to zero as x tends to -\infty. However, there is another way to prove this, by looking at the graph of the gamma function.
  • #1
julypraise
110
0

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.
 
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  • #2


julypraise said:

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


Ray Vickson said:
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


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.
 

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

1. What is the definition of the Gamma Function?

The Gamma Function, denoted as \Gamma(x), is a mathematical function that generalizes the factorial function to complex and real numbers. It is defined as \Gamma(x) = \int_{0}^{\infty} t^{x-1}e^{-t} dt, where x is a complex or real number.

2. How do you prove that \Gamma(x) \to 0 as x \to -\infty?

To prove that \Gamma(x) \to 0 as x \to -\infty, we can use the fact that the integral defining the Gamma Function is convergent for all complex and real numbers. This means that as x approaches -\infty, the value of \Gamma(x) will approach 0 since the value of the integral decreases as the upper limit decreases.

3. What is the significance of proving \Gamma(x) \to 0 as x \to -\infty?

Proving that \Gamma(x) \to 0 as x \to -\infty is important because it shows that the Gamma Function is well-behaved and has a finite value at negative infinity. This is useful in many areas of mathematics, including complex analysis, number theory, and statistics.

4. Can you provide an example of a function that has a similar behavior as the Gamma Function at negative infinity?

Yes, the factorial function, n! can be used as an example. As n approaches negative infinity, the value of n! also approaches 0. This is because the factorial function is closely related to the Gamma Function, with n! = \Gamma(n+1).

5. Are there other ways to prove \Gamma(x) \to 0 as x \to -\infty besides using the convergence of the integral?

Yes, there are other methods to prove \Gamma(x) \to 0 as x \to -\infty, such as using the reflection formula for the Gamma Function and using properties of the exponential and logarithmic functions. However, the convergence of the integral is the most commonly used method for this proof.

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