If we have the gamma function in integral form $\Gamma(x)=\int_\limits0^\infty e^{-t}t^{x-1}dt$ and have proven that it converges for real $x>0$ (to a point), can we then immediately conclude that $\Gamma(z)$ converges to a point if Re($z$)$>0$?
I'm asking this because we note that $t^{z-1}$ is complex and thus can be written in polar form $re^{i\varphi}$. The distance between the origin and $t^{z-1}$ is equal to $r>0$. We can now easily associate this distance with the distance between $x=r$ and the origin. We already know that $\Gamma(x)$ converges, thus $\Gamma(z)$ must converge to a given distance from the origin. However, we have completely ignored the angle $\varphi$. Do we know that the angle converges? If so, how?
If $\Gamma(z)$ converges, can we somehow generalise the idea?