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).