Say i have the following integral:
$$\int_\gamma \frac{1}{z}dz$$
where $\gamma$ is half of the unit circle lying in the region defined by $Re(z)<0$. It's pretty clear that the integral crosses the branch cut of the the antiderivative $log(z)$. My solution to avoid this problem is to define $\gamma_1$ & $\gamma_2$, where $\gamma_1$ is the path on the same unit circle as $\gamma$, but starting at $i$ and ending at $e^{i(\pi-\epsilon)}$ and $\gamma_2$ would be similar to $\gamma_1$, but starts at $e^{i(-\pi+\epsilon)}$ and ends at $-i$ .Now to compute the value of the integral I would like to say that: $$\int_\gamma \frac{1}{z}dz=\lim_{\epsilon\rightarrow 0}\left(\int_{\gamma_1} \frac{1}{z}dz+\int_{\gamma_2} \frac{1}{z}dz\right)$$ and then proceed to evaluate the integrals using the fundamental theorem of calculus and take the limit. Is this approach valid?