Need reference or derivation of Gamma function for half-integer orders

[jrenfree]

Hi all,

I'm looking at the http://en.wikipedia.org/wiki/Gamma_function#General" for the gamma function, and it lists equations for the gamma function of half-integer orders (i.e. gamma(0.5+n) and gamma(0.5-n)).

But, it doesn't list a reference as to where this equation comes from. Does anyone know where I can find a reference for this equation, or perhaps how to derive it?

Thanks!

 

[Mute]

By the property of the Gamma function

$$\Gamma(z + 1) = z\Gamma(z)$$

all you really need to calculate is $\Gamma(1/2)$. If you make the change of variables $t = y^2$ in the definition of the Gamma function, it will give you a (hopefully) familiar looking integral.

This will let you calculate $\Gamma(n+1/2)$; for $\Gamma(1/2-n)$, you'll need to use one of the reflection formulas that allow you to calculate $\Gamma(z)$ for $\mbox{Re}(z) < 0$ (but z not an integer).