I want to evaluate the integral $\int_Cf(z)dz$, where C is the unit circle centered at the origin, and $f(z)$ = log(z+2).
I know that the integral of a function over a simple, closed contour is $0$ when the function is analytic on and inside of the contour. $log(z+2)$ is analytic in the region C, so would this integral evaluate to $0$? Or does that theorem not apply when the contour/region contains a branch cut?
Thank you!