I would like to have some explanation (or pehaps some intuition) on the Hankel's contour for the following contour integral: $$ F(n)= \oint_{C} \frac{e^{ -z} }{z^{n+1}} \,dz$$
,where n is non-integer.
Arfken's textbook shows how this end up relating to the gamma function, which it is not what I am asking here. Also I understand that the cut is chosen to be on the positive real axis for convergence of the integrand.
My question/confusion is that we usually try to avoid enclosing a branch cut while evaluating complex integrals (Hankel's contour being an exception), but on the picture below, we end up including the branch cuts and branch points and never closing the contour (I also understand this has a residue at infinity for n non-integer):
If I could get explanation on why I am allowed to include branch cuts (counterclockwise direction instead of excluding them) and why not closing the contour (I am assuming it is because the branch cut goes to infinity and there is a residue at infinity, but I am not sure), it would be nice.