Graduate Can Gamma Functions Be Evaluated Analytically for Non-Integer Values?

[LagrangeEuler]

I have two questions related Gamma functions

1. Finding $\Gamma$ analytically. Is that possible only for integers and halfintegers? Or is it possible mayble for some other numbers? For example is it possible to find analytically $\Gamma(\frac{3}{4})$?

2. Integral $\Gamma(x)=\int^{\infty}_0 \xi^{x-1}e^{-\xi}d \xi$ converge only for $x>0$ in real analysis. How can we then write $\Gamma(\frac{1}{2})=\Gamma(-\frac{1}{2}+1)$ when relation $\Gamma(x+1)=x\Gamma(x)$ is derived from partial integration?

 

[fresh_42]

How about to use Gauß' formula for $x \in ℂ \backslash \{0, -1, -2, \dots \}$ instead:

$$Γ(x) =\lim_{n→\infty} \frac{n!n^x}{x(x+1) \cdots (x+n)}$$

Edit: It's sufficient to require $Re(x) > 0$ for the integral formula.

 

[mathman]

The gamma function has singularities at 0 and negative integers. Using analytic continuation the function can be defined elsewhere.