Gamma function convergence of an integral

[LagrangeEuler]

$\Gamma(x)=\int^{\infty}_0 t^{x-1}e^{-t}dt$ converge for $x>0$. But it also converge for negative noninteger values. However many authors do not discuss that. Could you explain how do examine convergence for negative values of $x$.

 

[Gigaz]

I'm pretty sure it doesn't converge for negative noninteger values and that the formula is not applicable there. Use the recursive formula for the Gamma function instead to get those values.

 

[mathman]

There is a general concept called analytic continuation, where a function has a particular representation in some domain can be extended outside this domain.