From this answer I know the choice of continuous extention $\ \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt\ $ is not unqiue.
But is that particular extension the unique best choice in some sense? E.g. is it the only one that makes $\ z\mapsto1/\Gamma(z)\ $ entire, or does it minimize some uglyness-measuring functional?
