Let $C$ be the circle of radius $2$ centered at origin. Let $f(z)$ be the branch cut of the function $z^{2−i}$ on the domain $−π < θ < π$. Compute the integral $$ \int_C f(z) dz$$
My attempt:
$f(z) = z^{2−i} = e^{(2-i)logz} $ $C$ is defined as $2e^{i\theta}$ on the interval $−π < θ < π$
so we get: $\int_C f(z) dz = \int_{-\pi}^\pi e^{(2-i)(\ln2 + i\theta)}*2ie^{i\theta} d\theta $
Does this look right? Any help is appreciated.