Deriving properties of the Gamma Function

[millwallcrazy]

I was just curious as to how I can show the following properties of the Gamma Function, they came up in some lecture notes but were just stated?

Notation: G(z) = Gamma function
2^(z) = 2 to the power of z
I = Integral from 0 to infinity

(1) G(z)*G(1-z) = pi*cosech(pi*z)
(2) (2^(2z-1))*G(z)*G(z+(1/2)) = G(2z)*G(z/2)


Taking into consideration that the definition of G(z) = I(exp(-u)*u^(z-1)du)

Thanks

 

[Gib Z]

The first is called Euler's reflection formula, the second is called the Duplication formula. You could try Google-ing those terms for a proof, or flick through a complex analysis book in your library.