how to find the branch points and cut

Solution 1:

Your solution is correct, but since you are guessing, I will explain it.

The values of $z$ that make the expression under the square root zero will be branch points; that is, $z = \pm i$ are branch points. Let $z - i = r_1e^{i\theta_1}$ and $z +i = r_2e^{i\theta_2}$. Then $f(z) = \sqrt{z^2 + 1} = \sqrt{r_1r_2}e^{i(\theta_1+\theta_2)/2}$.

1. If we don't encircle any branch point, after one revolution, $f(z)\mapsto f(z)$.
2. If we encircle $z=i$ but not $z = -i$, then $$ \sqrt{r_1}e^{i(\theta_1+2\pi)/2} = \sqrt{r_1}e^{i\theta_1/2}e^{\pi i} = -\sqrt{r_1}e^{i\theta_1/2} $$ Therefore, $f(z)\mapsto -f(z)$ which is multiple valued
3. Same thing happens when we encircle $z=-i$ but not $z=i$
4. Lets encircle both branch points. $$ \sqrt{r_1r_2}e^{i(\theta_1+\theta_2+2\pi+2\pi)/2} = \sqrt{r_1r_2}e^{i(\theta_1+\theta_2)/2}e^{2\pi i} = \sqrt{r_1r_2}e^{i(\theta_1+\theta_2)/2}\cdot 1 $$ So $f(z)\mapsto f(z)$ still single valued.

We could choose $[i, \infty)$ and $[-i, -\infty)$, but from item 4, we have seen traversing around both points returns the function to its original value. Therefore, we can choose a finite branch cut, namely, $[-i, i]$.