In my studies I've come across the concept of branch cuts, and I'm having a little bit of trouble digesting the topic. As I understand it, in this example:
$$f(z) = \sqrt{z^2 + 1} = \sqrt{z + i}\sqrt{z - i}$$
With the substitution $ z + i = r_1e^{i\theta_1} $ and $ z - i = r_2e^{i\theta_2} $
Then:
$$ f = \sqrt{r_1r_2}e^{\frac{i(\theta_1 + \theta_2)}{2}} $$
The branch points are located at $ z =i $ and $z = -i $. If we define the branch cut to be from $i$ to $-i$ then any contour enclosing one point but not the other will "pick up" a factor of $ e^{i\pi} = -1$, and any contour enclosing both will "pick up" a factor of $ (-1)^2 = 1$.
My question is, how do we define a sensible value to attribute to $\theta$ on each side of the branch cut? Would $$\theta = \begin{cases} \frac{\pi}{2}, & \text{for $x \lt 0$} \\ \frac{5\pi}{2} & \text{for $x \ge 0$ is odd} \end{cases}$$ be a sensible choice?
