Is it valid to do this?
I have $f(z)= z^i$,and $F(z)=\frac{z^{i+1}}{i+1}$ and assuming we're using principle values of $f$ and $F$ would it be correct to say that:
$\int_{-1}^{1} f(z) dz = \frac{z^{i+1}}{i+1}\mid_{z=1} - \lim_{z\to -1} \frac{z^{i+1}}{i+1} = F(1)-\lim_{z\to -1}F(z)$
where we know that $F(z)$ is the antiderivative of $f$ on $\mathbb{C}\backslash(-\infty,0]$.
I am unclear about whether it is ok to use the fundamental theorem here since the domain of the principal $log$ being analytic is $\mathbb{C}\backslash(-\infty,0]$ and for the fundamental theorem to be applied, we need to have an analytic function on the domain. Clearly $f$ is not analytic on $(-1, 0]$.