First, your two cases are "n= 1" and " " which does not make sense. I assume you meant " ".
You also don't state what the curve is. Obviously, if the curve does not have in its interior, the the integrand is analytic there so the integral is 0. So I presume that C is a curve having in its interior. In that case, we can transform the curve into a circle with center at without changing the integral because the integrand is analytic at every point between the given curve and such a circle.
So we can assume that C is the circle with center at and radius R. In that case where goes from 0 to . Of course then , , and the integral becomes .
IF , then the integral is . But the exponential is 1 at both 0 and so the integral is 0.
IF ,then the integral is