I'm trying to compute, for the principal branch, the following integral where $C$ is the circle of radius $R$ centered at $0$ and $\alpha$ is a complex number different from zero
$$\int_{C} z^{\alpha-1} \, \mathrm{d}z $$
My attempt:
I define the parameterization for C as follows: $C:[0,2 \pi] \to \mathbb{C}$ and $C(\theta)=Re^{i \theta}$.
therefore $z^{\alpha-1}= e^{(\alpha-1)\text{Log(z)}}$
So,
$$\int_{C} z^{\alpha-1}dz = \int_{0}^{2\pi} e^{(\alpha-1)\text{Log}(Re^i\theta)} \cdot iRe^{i\theta} d\theta= i R\int_{0}^{2\pi} e^{(\alpha-1)\text{Log}(Re^i\theta)} \cdot e^{i\theta} d\theta$$
From here, I'm not sure how to proceed. Any advice?
$z=Re^{i\theta}\\ dz = iRe^{i\theta}\ d\theta$
$\int_0^{2\pi} R^{a-1}e^{(a-1)i\theta}(iRe^{i\theta})\ d\theta\\ \int_0^{2\pi} iR^{a}e^{ai\theta}\ d\theta$
And I will let you take it home from here.