Gamma Function, Gamma 1/2=root pi

In summary, the author is having trouble finding a proof for gamma of 1/2 = root pi. They suggest using the Beta-function to help. When they substitute u for t^{\frac{1}{2}} and du for 2du, they are able to find gamma of 1/2 = \sqrt pi.
  • #1
leila
19
0
Hiya,

I'm having trouble finding a simple proof for gamma of 1/2 = root pi?

Any suggestions
 
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  • #2
One approach would be to show

[itex]\Gamma(x)\Gamma(1-x)[/itex] = [itex]\frac{\pi}{sin(\pi x)}[/itex]

then let x = 1/2.
 
  • #3
If you know the Beta-function too, and the relation with the Gamma-function:

[tex]B\left( {u,v} \right) = \frac{{\Gamma \left( u \right)\Gamma \left( v \right)}}{{\Gamma \left( {u + v} \right)}}[/tex]

Then it's easy to use the definition of the Beta-function to compute B(1/2,1/2) which gives [itex]\pi[/itex], so:

[tex]B\left( {\frac{1}{2},\frac{1}{2}} \right) = \pi = \frac{{\Gamma \left( {\frac{1}{2}} \right)\Gamma \left( {\frac{1}{2}} \right)}}{{\Gamma \left( 1 \right)}} = \Gamma \left( {\frac{1}{2}} \right)^2 \Leftrightarrow \Gamma \left( {\frac{1}{2}} \right) = \sqrt \pi [/tex]
 
  • #4
Oh I use to tell people that (-1/2)!^2=pi and then they'd ask me to show them why so I kept this one in memory. (of course this doesn't prove that (-1/2)!^2 = pi since factorial isn't really defined on non-negative non-natural numbers).

Start with the Gamma Function:
[tex]\Gamma (x)= \int \limits_0^\infty \exp (-t) t^{x-1} dt [/tex]
[tex]\Gamma (\frac{1}{2})= \int \limits_0^\infty \exp (-t) t^{-\frac{1}{2}} dt[/tex]

Then make the following substitution:
[tex]u=t^{\frac{1}{2}} [/tex]
[tex]du=\frac{1}{2}t^{-\frac{1}{2}}dt \Rightarrow t^{-\frac{1}{2}}dt=2du[/tex]

And the original equation becomes:
[tex]\Gamma (\frac{1}{2})= 2 \int \limits_0^\infty \exp (-u^2) du[/tex]

The next part is the 'trick.' The trick is to then square gamma so you have two integrals with two different variables of integration (say, x and y). Because of their form, you should be able to combine them into one integral and then change it into another two-variable coordinate system where pi's are used. Becareful when changing limits (hint: when you integrate over two variables from 0 to infinity, you are effectively integrating over the first quadrant. Therefore, you should change your limits to your new coordinate system to make sure you are also integrating over the first quadrant)(EDIT from before: sorry I forgot this was the homework help section; As you can see, I've curtailed my answer :wink:)
 
Last edited:
  • #5
Yeah, you'll need to switch to polar coordinates and the evaluating the double integral is pretty straightfoward. You can Google "Gaussian Integral" to see the technique used.
 

1. What is the Gamma Function?

The Gamma Function, denoted by Γ(x), is a mathematical function that is an extension of the factorial function to complex and real numbers. It has many applications in mathematics and statistics, particularly in the areas of probability and number theory.

2. How is the Gamma Function related to the factorial function?

The Gamma Function is an extension of the factorial function, meaning that for any positive integer n, Γ(n) = (n-1)!. However, the Gamma Function is defined for all complex and real numbers, not just positive integers.

3. What does the notation "Gamma 1/2 = root pi" mean?

The notation "Gamma 1/2 = root pi" refers to a specific value of the Gamma Function, where the input is 1/2. This value is equal to the square root of pi, or approximately 1.77245385091.

4. How is the Gamma Function used in statistics?

The Gamma Function is commonly used in statistics to calculate probabilities and to model the distribution of continuous data. It is also used in calculating confidence intervals and in hypothesis testing.

5. Can the Gamma Function be calculated by hand?

The Gamma Function can be calculated by hand for some values, such as positive integers. However, for most values, it requires specialized techniques and computer algorithms to calculate accurately. Many scientific calculators also have the Gamma Function built-in for easy calculation.

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