Gamma Function, Gamma 1/2=root pi

[leila]

Hiya,

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

Any suggestions

 

[Astronuc]

One approach would be to show

\Gamma(x)\Gamma(1-x) = \frac{\pi}{sin(\pi x)}

then let x = 1/2.

 

[TD]

If you know the Beta-function too, and the relation with the Gamma-function:

B\left( {u,v} \right) = \frac{{\Gamma \left( u \right)\Gamma \left( v \right)}}{{\Gamma \left( {u + v} \right)}}

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

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

 

[Jelfish]

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:
\Gamma (x)= \int \limits_0^\infty \exp (-t) t^{x-1} dt
\Gamma (\frac{1}{2})= \int \limits_0^\infty \exp (-t) t^{-\frac{1}{2}} dt

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

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

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 )

 

[Jameson]

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.